FUNDAMENTALS OF VALUATION
FUNDAMENTALS OF VALUATION FI 4000
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Date Created: 09/21/15
NOTES FOR FI 4000VALUATION Stephen D Smith Fall 2003 Introduction The point of this course is to develop tools for analyzing most any basic valuation problem that you might encounter in finance This is an analytical course but the necessary tools are ones you should have already developed high school algebra 1eg how to determine the slope of a straight line and basic statistics calculating averages variances and covariances What do we mean by valuation This is a subjective notion of what someone thinks some object e g a security is worth in terms of money or some other common denominator at a point in time The market price is what the object is actually selling for in the market A m consists of a group of potential traders An eguilibriurn is said to occur when market pric subjective valuation for all potential traders Otherwise there is an incentive to trade What are we valuing in F1 4000 Mostly financial claims or securities of some type although the tools developed here can be used to value other assets as well e g artwork Almost all financial claims involve either the exchange of l something today eg money for something else today eg goods or 2 something later eg money for something else later e g goods or 3 something today e g money for something later e g hopefully more money The key point here is that regardless of when the transaction takes place the terms of trade are typically set today including contingency provisions for one or the other of the two parties What will we cover in this class The tools developed here will allow you to analyze determine the value of how many units of quotYquot that one unit of quotXquot is quotworthquot Examples of Unconditional Contracts a Xgood today39 Ygoods tomorrow Exchange rate real rate of interest b Xmoney today39 Ymoney tomorrow Exchange rate nominal rate of mist c Xgood today39 Ymoney today Exchange rate price level The in ation is the percentage change in this exchange rate over time d Xmoney 1 today39 Ymoney 2 today Exchange rate spot foreign exchange rate e Xmoney 1 tomorrow Ymoney 2 tomorrow Exchange rate forward foreigp exchange rate f Xmoney tomorrow Ygoodssecurities tomorrow Exchange rate orwardfutures price Examples of Conditional Contracts a money today for the u whatever money is left over tomorrow after delivering on other promises Eguig Contract b money today for the pigllto purchase security tomorrow at an exchange rate f1xed today Call option c money today for the rith to sell security tomorrow at an exchange rate fixed today Put option Since few groups can deliver unconditionally you are almost always faced with E or uncertainty when you attempt to value securities The class will mainly deal with risk and not uncertainty where risk is defined as the chance of an unfavorable to you event occurring An example of risk involves knowing that there are 50 red and 50 black balls in an urn betting on red but knowing that there is a 50 chance that you will draw black Uncertainty involves the situation where not knowing the percentage of red and black balls in the urn you bet on red but don t know the odds that you will draw black In the first case you know the distribution of possible outcomes but in the second case you do not know the distribution Since valuation is a subjective concept what is very valuable to me may not be very valuable to you the tools developed here for purposes of valuation start with an investor profile The ingredients to a profile include a Attitude toward certain money today for certain money tomorrow subjective rate of interesttime value of money b Attitude toward quotrandomnessquot risk over a fixed period of time Investors who ignore randomness are called risk neutral while those who demand higher compensation for increased randomness are called risk averse c Period of time that is relevant for purposes of decision making investor horizon could be in nite d The current endowment wealth of the investor Valuation involves combining this investor pro le with the properties of the financial contract These properties include a the of the payoffs including the last payoff matting b the anticipated m of the payoffs at a point in time c the degree of randomness of the payoffs the E of the contract d the contingencies if any in the contract When combined these eight ingredients can be used to establish the subjective value of a security The basic trading rule to be used in this class amounts to establishing whether subiective value gt market price or subiective value lt market price In the first case you should buy more of the security go long while in the second case you should sell the security go short Since the subjective value of something depends in part on what else you own many of the valuation exercises will be done in the context of the what something is worth in a portfolio of securities In particular since some of the randomness in payoffs is due to things specific to an individual security eg CEO leaves risk in the portfolio can be reduced by splitting funds up into many different securities Diversification eliminates these types of security specific risks leaving only unavoidable risks If all investors behave this way only nondiversif1able risks will be relevant for purposes of valuation This is the idea behind the asset pricing models that will be presented later in the course Before that however the class will cover an even more basic tool for purposes of valuing securitiesthe principle of quotno free lunchquot In the context of this class that concept will be called quotQ riskless arbitragequot and it simply means that if you put up none of your own wealth and take on no randomness then you should not be able to make a profit The next class involves portfolio theory and equilibrium but before that it will prove useful to look at the notion of riskless arbitrage more closely since it is probably one of the most basic tools used in valuation In fact the absence of riskless arbitrage is essential to establishing equilibrium in nancial markets The idea here is quite straightforward Intuitively if two securities or assets generate exactly the same cash ows over time then they must sell for the same price If they do not then there is the opportunity for a free lunch Consider the following simple example Suppose security X and security Y generate exactly the same cash ow say 1 next year but price X gt price Y then you can rig up a situation that has the following three characteristics 1 You put up none of your own money No investment 2 You face no randomness No risk and 3 You generate a positive profit either now or in the future Nonnegative cash flow These are the three components that make up a situation whereby there is Riskless Arbitrage Ruling out these kinds of free lunches the Q of riskless arbitrage will prove essential to establishing the value of a number of securities like options and other socalled derivative securities ie ones whose values depend on the value of some other security How in the example discussed above could you arrange a situation where 1 3 hold if price Xgt price Y Well rst you would try to borrow security X from someone and sell it now This is called shorting a security Then you would use the proceeds to purchase Y and have some money left over now Next year you would receive 1 from security Y but you must retum security X to its original owner But since X is paying 1 you have just enough to buy it its price at maturity is clearly 1 and return it to its owner So you have taken no risk invested none of your own money and generated a pro t Price X Price Y This can t last long39 eventually Price X Price Y since everyone will be trying to sell X drian it s price down and buy Y driving it s price up This concept of arbitrage or no arbitrage will be used repeatedly throughout the course Portfolio Theory It has been stressed that investors like higher eXpected returns and dislike more quotriskquot It would be nice to have a uniform measure of risk that could be applied to all securities and bundles of securities equally regardless of the source of risk What are the candidates in the quothorse racequot to be a quotgoodquot measure of risk First the most common definition of risk should re ect the fact that an investor faces risk if they face the possibility of realizing outcomes less than what was expected Second a reasonable measure of risk would recognize that not all bad outcomes are the same39 an outcome somewhat less than expected should be measured as quotless riskyquot than an outcome that is much less than expected Third it would be nice if the ranking of more to less risky were independent of the units of measurement39 that is the ranking doesn t change whether measured in say dollars or pounds Interestingly one common measure that you learned in basic statistics has almost all of these three characteristics This measure is the standard deviation which as you will recall is the sguare root of the variance First the rankings of standard deviation does not depend on how the units are measured Second with the standard deviation larger deviations from the mean get quotmore wei tquot in the risk measure The only catch is that the standard deviation also gives weight to outcomes that are higher than the mean and more weight to ever larger deviations above the expected outcome This problem is not as bad as it sounds If for example you are looking at a quotsymmetric distributionquot these shortcomings will not matter A risk ranking based only on deviations less than the mean will be the same as one where the ranking is based on all of the deviations om the mean While skewed distributions do arise and can be important the standard deviation will work as a good quotfirstquot approximation for measuring what is meant by risk The issue of skewness will come up again in the section covering derivatives Review of Means and Standard Deviations Recall from statistics that if there are say N different states of nature or scenarios that can occur with probabilities p1 p2pN and r is a random variable that takes on one of N different values rl r2 rN the expected value or average is given by 1 BO P1r1 P2r24T1Dr1l 1 2 Es PSIS where E means expected value or average and the term quot25quot means to sum over all possible scenarios quotsquot Furthermore the variance of r is given by 2 Var p1r1Erl2 p2r2Erl24T1Ur1l Erl2 2s pltsgtltrltsgtErgt2 Another way of writing equation 2 that some people nd easier to work with is to write Vain as 239 we 2s pltsgtltrltsgt2gt1 Eltrgt2 The standard deviation of i sri is given by 3 SW Varltrgt12 The p s of course need not be the same All that is needed is for something some state to occur so pl p2pN l probabilities must sum to one If you are however working with historical data the convention is to give each observation equal weight So if you have N years of data each year quotsquotshould get an equal quotprobabilityquot and since the p s must sum to one you should give each year a probability of ps lN for all scenarios quotsquot when using formulas 13 When using historical data the expected value is sometimes called the arithmetic average or average for short Also sometimes in this case pslN l is used in calculating Varr in equation 2 but for reasonably large amounts of data big N this difference doesn t really matter Examplel Suppose that r is the unknown return on a stock and there are three scenarios or states of the world a boom normal growth and a recession with the associated probabilities and payoffs for this random variable Table 1 my Probabiliggpg s21 Return on stock r1 sn boom 30 3 25 25 normal 50 5 10 1 recession 20 2 20 2 Then using equation 1 the eXpected value of r is Er plr1 p2r2 p3r3 325 51 22 Likewise using equation 2 you can calculate the variance as Varr P1r1Er2 p2r2Er2p3r3 Er2 325 0852 51 0852 2 2 0852 and the standard deviation is just SDr 024512 1566 Equations 1 to 3 are very general once you identity the states or scenarios the probabilities and the relevant random variable you can always use these formulas Example 2 Historical Data Suppose instead of the probabilities in Example 1 what you have observed is that over the past three years you have observed returns of 25 10 and 20 Using equations 1 3 you would use ps l3 for all of the outcomes So in this case you would get Erplr1 p2r2 p3r31325 131 132 Q Varm plt1gtltrlt1gtErgt2 plt2gtltrlt2gtErgt2plt3gtltrlt3gt Er1gt2 l3250852 13l0852 l3 20852 0362 and SDr 036212 1903 Notice that nothing in the calculations change except the probabilities In this class the random variables of interest are typically retums on securities and the notation here reflects to the extent possible the notation in Bodie Kane and Marcus The return on the safe asset with r the same for all scenarios s is denoted rf Since this is known it will often be referred to as a constant Another way to say this is to say that the expected return on the safe asset is Erp1rf p2rf p3rf p1 p2 p3rf If since p1 p2 4p3 1 Furthermore the variance is given by Var p1rfEr2 POIfEr243rfEr2 0 since Er rf in this case Also SDr Varr12 O The working assumption here is that investors like securities or combinations of securities with higher values of Er and lower values of SDr which is the same as lower values of Varr In figure 1 Er is plotted on the quotYquot aXis and SDr is plotted on the quotXquot aXis So investors are clearly getting quothappierquot as you move to the Northwest in this picture Indeed you could view the Mof portfolio theory as choosing combinations of securities that provide you with riskretum opportunities that are as far to the Northwest as possible In fact later on a number the quotSharpequot ratio will be developed for the purpose measuring how quotgoodquot a particular set of opportunities is in terms of making investors quothappierquot Since investors are not limited to buying or selling just one security some provision must be made for calculating means and standard deviations for groups or portfolios of securities Using the following simple rules from statistics for means and standard deviations will prove to be extremely helpful for this purpose Rule 1 If A and B are constants and rp is a random variable then 4 E A B rp A BErp and Rule2 If A and B are constants and rp is a random variable then 5 SDA Brp B snap where quot quot means the absolute value As mentioned earlier these two rules will prove extremely useful for purposes of calculating the mean and standard deviation of returns for portfolios of securities which are the needed inputs for guring out which combinations are getting you closer to the northwest in gure 1 The most obvious example given the discussion so far would be to calculate the mean and standard deviation of return for a portfolio that consists of some combination of the riskless asset think of this as a money market fund if you like with constant return rf and a risky asset with a random return rp where the subscript quotpquot means risg portfolio In this case the quotportfolioquot consists of only one risky security but the notation will stay the same later when more risky assets are added to the analysis Suppose you place quotyquot of your funds in the risky security and quot lyquot of your funds in the riskless asset These weights must sum to one and they are constants39 once chosen they are fixed known numbers as opposed to being random like rp So the return on this combination of securities rc where the subscript quotcquot stands for quotcombinationquot is just a weighted average of the returns on the two securities in this case 6 rc 1Yrf yrp Equation 6 is just a special case of the more general rule for calculating retums when you have a combination of many securities The following rule will always work for calculating the retums on combinations of securities Rule 3 If an investor has the opportunity to invest in quotKquot securities and places yiin the quotithquot security the return on this combination is given by 7 rc mm We yiri yKrK where of course y1y2 yjyK 1 Equation 7 is just a formal way to represent the idea that the return on a combination of securities is just the weighted average of the returns on the individual securities Notice that the r s in equation 7 can be random or constants Rule 3 will prove useful throughout the course but for the example of one risky and one riskless asset K2 and you get equation 6 where the subscripts are omitted Notice that y ly and rf are all constants while rp is a random variable So let 1 rf A B and r X Then Rule 1 sa s that the eX ected value of re Y Y p Y P can be calculated as 81360 EA BX E1Yrf Yrpl 1Yrf YErp and Rule 2 says that the standard deviation of re can be calculated as 9 SDrc SDA BX SD1Yrf Yrpl lyl SDrp Eguations 8 and 9 along with Rule 3 is all that will be needed to calculate the means and standard deviations of any combination of securities regardless of the number of risky assets How is this going to work Well the trick is to first use Rule 3 to calculate rp the return on the risky portfolio and then use equations 8 and 9 to calculate the mean and standard deviation of the return on the combination of the riskless security and the risky portfolio of securities This topic will be discussed more fully later but it may prove worthwhile to work through some numbers for the one risky and one riskless security case Example 3 Suppose we take the data from Table 1 as the return on the risky quotportfolioquot of one security so the last column represents rp Then it is possible to plot out for different values of y the eXpected retum and standard deviation of this combination of securities Suppose rf 5O5 Then using equations 8 and 9 you can plot out Erc and SDrc Table 2 Investment in risky security y Erc SDrc O 05 00 3 3 0616 0517 75 0763 1175 1 085 1566 125 0938 1958 As example 3 makes clear by changing y you can plot out all of the relevant values of expected return and risk for this combination of assets In fact what you will find is that the line as a function of y is mm fact focusing for the moment only on cases where y gt O Mrepresents short sales of the risky asset so y y equations 8 and 9 can be written as 839 Erc 1Yrf YErp If YErp rf and 939 SD60 YSDrp If on an quotXquot quotYquot graph you think of equation 8 as Y and 9 as X then you know from high school algebra that the rise over the run as you change y is given by 10 AErc Ay Erp rf quotA in Yquot and l l ASDrc Ay SDrp quotA in X quot So the rise over the run really is a constant independent of y and the slope of the line is given by 12 A in YA in X Erp rfSDrp equation 12 defines the quotSharpe ratioquot which will be denoted by quotSquot So in example 3 the slope of this line is 085051566 2234 Table 2 and Figure 2 represents the opportunity set for risk and expected return when the investor is long the risky asset In fact for any y so that l gt y gt0 the investor is said to be long both the risky and the riskless assetm Athe investor is said to be long the risky and short the riskless asset Finally for ylt0 the investor is said to be short the riskv asset and long the riskless asset Table 3 shows the mirror image of Table 2 except that y lt 0 The absolute value sign in equation 9 comes into play here the rise over the run now has a negative sign The Sharpe Ratio S M for y lt 0 This is plotted in Figure 3 Table 3 Investment in risky securityy Erc SDrc 0 05 00 33 0385 0517 75 0238 1175 1 015 1566 125 0063 1958 Clearly no one who liked expected return and disliked standard deviation would choose to invest in the quotopportunity setquot where y lt 0 There are two general lessons to be learned from this example The rst is It never makes sense when viewed in 39 quot to short a riskv securitv whose expected return is greater than the riskless rate39 ie v lt 0 if E69 gt rfmakes no sense This lesson has particular relevance later when you will leam that when viewed in the context of holding many securities this seemingly simple idea may not be true due to the logic of portfolio theory it is the standard deviation of your whole position and the not the standard deviation of any particular asset that matters to your happiness in riskexpected return space The second lesson to be learned is the notion of efficient portfolio choices In this example y lt O is inefficient from a riskexpected return perspective because there are other choices y gt O that provide a higher expected retum for the same level of standard deviation or risk This leads to a general definition of efficient choices An efficient choice for purposes of securig selection is one for which there is no other choice with lower risk and the same expected retum or higher expected return and the same risk The set of all efficient choices is called the ef cient set or asset allocation line So in this example any y gt O is an efficient choice and any y lt O is an inefficient choice Adding More Risa Assets In this section you will see that absent one additional step calculating riskretum opportunities with many risky assets is computationally identical to what you have just gone through with one risky and one riskless asset The intermediate step simply involves another application of Rule 3 The trick that will be used involves solving the efficient set of choices in two steps Step 1 Decide what percentage of your funds is to be invested in the riskless asset This determines ly in riskless and y total in risgg Step 2 Decide what percentage of y is to be invested in each of the individual risky assets Call these percentages or weights wi i 1 2 K1 remember that there are K assets and one of them is riskless where w1w2 wi wK1 1 After these two steps you can calculate the return on the risky portfolio using Rule 3 equation 7 except use the weights wj instead of yi notice that wj yjy This gives you 13 rp w1r1w2r2wjri wK1rK1 Finally just use y 1y rp and rf to calculate eXpected returns and standard deViations for different combinations of risky and riskless assets just as in the one asset case The returns on these combinations are calculated the same way and those equations are repeated below 8 Erc 1Yrf YErp 9 SD60 Y SDrp Example 4 Table 4 below is just Table 1 with the addition of a second risky asset along with a column for the riskless asset Table 4 State Probability Return Stockl Return Stock2 Retum Riskless s ps r1 r2 rf boom 3 25 20 05 normal 5 10 01 05 recession 2 20 05 05 Suppose that regardless of the choice of investment in risky assets y you place 40 of that money in asset 2 and 60 in asset 1 So w1 6 and W2 4 You will soon see the rule for optimally choosing the w s but for now just take these numbers as given So rp 6r14r2 so rp 625 42 Q with probability 3 a boom rp 610 401 m with probability 5 normal or rp 620 405 with probability 2 recession So now you can calculate the expected retum and standard deviation of rp Erp plrp1 p2rp2 p3rp3323 5064 2 14 and Vamp plt1gtltrplt1gtErpgt2 plt2gtltrplt2gtErpgt2 plt3gtltrplt3gt Elrpl2 323 0732 5064 0732 2 14 0732 m or SDrp Notice that you can rework the Sharpe Ratio S 07305 1286 mwhich is actually less than the Sharpe Ratio when you have only one asset w1 1 So a 6040 miX between risky asset one and two is clearly not the best miX The logic here is that since you can always choose to ignore asset 2 your best miX of assets 1 and 2 must be such that the Sharpe Ratio is no lower than that of holding asset l or asset 2 for that matter alone After all you are trying to get to the northwest and that is equivalent to increasing the slope of the riskreturn line So in general Increasing the Sharpe Ratio is Equivalent to improving the investment opportunity set For a given set of assets the opportunity set with the largest Sharpe ratio is the e icient set Naturally you might wonder how you can determine which way to move the percentages in order to increase the Sharpe Ratio Table 5 reproduces the returns on the risky assets and the return on the portfolio given the 6040 mix Table 5 6040 mix between asset 1 and 2 State Probability Retum Stockl ReturnStock2 Retum portfolio S ps r1 r2 rp boom 3 25 20 23 normal 5 10 01 064 recession 2 20 05 14 In order the answer the question you first need to know how to calculate the covariance between two random variables The covariance is a measure of association between two random variables If the covariance is positive then on average the two random variables move in the same direction while the opposite is true if the covariance is negative You need this concept of covariance because since you are worried about maximizing the expected excess returnrisk ratio the Sharpe Ratio for your whole portfolio the desirability of adding some or more of an asset to your 20 portfolio should naturally be determined based on how it changes the returnrisk ratio for your portfolio 14 C0Vrh rj p1ri1Erilrj1Erjl POri2Erilrj2Erjl pNriN ErilrjN Elrjl Zs PSriSErilrjSErjl Another way to write the covariance is 1439 C0Vrhrj 2s pSriSrjSlErleril Equation 14 is usually how I do the calculation but if you do it correctly the answers from 14 and 14 will be exactly the same Notice that the variance is just a special case of the covariance Using equation 14 you can calculate Cov inn This is given by 15 COV Teri Zs pSriSErilriSEril Zs pSriSEril2 Varri which is just equation 2 for the variance ExampleS Given the data in Table 5 it is straightforward to calculate the covariance between the returns on the two risky assets and the covariance between the returns on one of the risky assets and the return on the portfolio For example Covr1 r2 is given once you determine that Er2 m by C0Vr1 r2 ZS pSr1Sr2SlEr1lEr2l 3252 5101 2 2 05 085055 21 0128 F or a xed mix of risky assets you can also calculate the covariance of retums from an individual security and the portfolio retum This will be important in calculating the decision rule as to whether you should add or take away a certain asset in the portfolio For example C0Vr1 rpgt 2s pltsgtr1ltsgtrpltsgtEr1Erp 32523 51O64 2 2 14 085073 M and Com rpgt 2s pltsgtr2ltsgtrpltsgt1Er2Erp 3223 501O64 2 05 14 055073 M are the covariance of r1 and r2 with rp respectively for the 6040 weights Armed with the ability to calculate means variances and covariances you can always use the following portfolio decision rule to determine which way the weights should change to increase the expected returnrisk ratio Now that you know how to calculate the covariance we can look at an alternative way to calculate the variance of the return on a risky portfolio I will stick to the two asset case here but the result can be generalized to the N asset case as well 22 16 V r p WlVarr1 W2Varr2 2 W1 W2 Covr1r2 w1varr1 139 W1Var T2 2 w1139 W1 C0VT1T2 Esample We can use the information in Table 5 to calculate the variance using equation 16 It should turn out to be 0165 just the same as when we calculated it using our earlier method It is straightforward to calculate Varr1 0245 and Varr2 0095 We already know that Er1 085 Er2 055 and Covr1 r2 0128 We also know that the weights are W1 6 and W2 1 W1 4 Therefore we can calculate Vamp 620245 4 20095 2640128 0165 Which is exactly the same as the alternative method used earlier The standard deviation is given by 0165 2 1284 Of course the standard deviation is just the square root of this number Portfolio Decision Rule Let wi be the percentage of risky assets held in asset i Then you should follow the following rules in order to maximize the Sharpe ratio Increase wj from its current level i 16 BOD gt If C0Vri rpVarrplErp rfl Decrease wj from its current level i 23 17 130139 lt If C0Vri rpVarrplErp rfl Sometimes the term Covri rpVarrp is written in short hand notation as 18 Covrj rpVarrp Bip pronounced quotbetaquot where quotiquot and quotpquot refer to asset quotiquot and portfolio quotpquot So the decision rules can be written more compactly as Increase wj from its current level 19 Em gt If BipETp rfl Decrease wj from its current level 20 Em lt If BipETp rfl Notice that the only things in equations 19 and 20 that depend on asset quotiquot are Eri and Bip Notice that unlike the case of looking at an asset in isolation portfolio selection rules shows that you may wish to decrease wi from current levels which is the same thing for fixed y as reducing yj or even short asset i wi lt 0 Example 6 Using the data that you have you already know that Er1 085 Er1 055 Erp 073 rf 05 Varrp0165 Covr1rp0198 Covr2rp01 l5 Blp0198Ol65 12 and sz01 l5Ol6570 Applying the rule to both assets you can see that 24 Er1085 gt05 12O7305 O776 so you should add more to asset 1 Likewise doing the calculations for asset 2 you can see that Er2055lt05 70O7305 O66l so you should subtract from asset 2 Notice also that the weighted average B39s sum to 10 ie6l2 470 1 This will always be true no matter how many risky assets you look at Many Assets and Investors The Security Market Line If you think of rf ijErp rf as the amount that an investor requires to cover the time value of money rf and the premium for risk in the context of a given portfolio ijErp rf you might call the sum of these two terms the quotrequiredquot rate of return call it requiredri So in the context of your portfolio you should seek out assets whose expected return exceeds the required return and vice versa39 sell off or even short assets whose expected return is less than the required return39 again in the context of a given portfolio Now walk yourself through the following quotthought experimentquot Thought 1 Suppose that the whole universe of risky assets was available for investment so the relevant portfolio of risky assets is the quotmarketquot and the return on the portfolio rp is the same as the return on market call it rm Thought 2 Further suppose that investors all quotsee the same picturequot That is everyone has identical information so they calculate the same values for means variances and covariances 25 Then it will have to be the case that for all assets i when using portfolio m the following is true in equilibrium 2l Eri requiredri rf ijErp rf when portfolio p is the market or in other words 22 Eri required ri rf BjmErm rf Equation 22 is typically called the Capital Asset Pricing Model and the plot of requiredri on the quotYquot aXis against Bjm on the quotXquot aXis is sometimes called the Security Market Line Why given the thoughts 1 and 2 does it l to be that equation 22 holds for all assets The easiest way to see why it has to hold is to first suppose that it doesn t and show that this contradicts the assumption that you are in an equilibrium situation For example suppose that for some asset i it were the case that Eri gt required r1 Since everyone sees the same picture there will be a rush by everyone to purchase asset i which contradicts the assumption that you are in equilibrium Put another way there would be no potential sellers of asset i at given prices In fact potential sellers would not sell unless what they eXpect to lose Eri is less than or equal to what they eXpect to gain represented by requiredri The logic can also be worked through for the case where Eri ltrequiredrj In this case everyone would want to be sellers and there would be no potential buyers So the only equilibrium is one where Erj ltrequired r1 Notice that the intercept of the function is at Bjm 0 and required ri rf and the slope of the line is Erm rf So an asset that does not covary with the market has a required return equal to the riskfree rate How can this be 26 Well think about the more extreme case where Bjm lt 0 In this case requiredrj lt rf The asset actually has a required return of less than the riskless rate The logic is that since this asset is counter cyclical it has a negative covariance with the market has returns when the market return is low adding this asset to your portfolio reduces risk by more than that achieved by simply adding an asset whose returns are independent of the market39 in this case either the riskless asset g the asset for which Bjm 0 So the general point is that if 1 gt Bjm gt 0 then Erm gt requiredri gt rf and if Bjm gt 1 requiredrj gt Erm In this case the asset has more than average market risk and the required return will re ect this fact Of course in order to calculate the required return some estimate of Bjm the market portfolio and the riskless rate is required Usually time series data in used for this purpose Moreover in this class the return on quotlarge company stocksquot is typically going to be used as a proxy for the market The next example shows how these calculations can be done Example 7 Below are the returns on large company stocks small company stocks and TBills over the last ve years Table 6 Year large stock returns rm small stock retums ri riskless rate rf 1994 0131 0311 0390 1995 3743 3446 0560 27 1996 2307 1762 0521 1997 3336 2278 0526 1998 2858 0731 0486 So using this historical data you can calculate all the information you need for the security market line by doing the following Recall from statistics that you can always t a straight line between quotYquot and quotXquot that minimizes the sum of squared errors the line of best fit Using the data here each year gets a probability of 15 2 ps Furthermore let rj rf Y and rm rf X given in Table 7 and fit a line of theform 24YabXe where a and b are constants and quotequot is the error The best estimate of b call it b is given by the formula 25 b CovYXVarX and the best estimate of a call it a is given by 26 a EY b EX Table 7 Year pYear X rm rf Y rj rf 1994 2 0259 0079 28 1995 2 3183 2886 1996 2 1786 1241 1997 2 2810 1752 1998 2 2372 1217 So using the formulas for means variances and covariances you can see that EX 1978 EY 0917 VarX 0147 CovXY CovYX 0088 So b CovYXVarX 00880147 5973 and a EY b Boo 0917 59731978 0264 So the beta of the small stock portfolio is about 6 and the intercept of this regression is less than zero You can use the same logic as before in terms of portfolio selection rules Notice that if CAPM were true quotaquot would be zero since b CovYXVarX In this case a lt 0 so you should sell small stocks and reallocate your portfolio If a gt 0 you should add more of the security to your portfolio But this is just another way of looking at the portfolio selection rule that says add if the eXpected return exceeds the required retum and Vice versa Review of Time Value of Money 29 Suppose a security promises a cash flow of CFt in period t tl2 T where T is the w or last payment T infinity is possible of course For now assume that the discount rate that equates the present and future value of the cash flow is the same for all t39 call it r Then the present value of this security P is given by 1 P CFlll r CFZll r2 CFT11TltT391gt CFTllrT This is a general formulation Some common special cases are given below Examples i CFt A for all t tl2 Tl T annuity 2 P Al l r Alr2 All rT391 AlrT In order to get the present value interest factor for an annuity that you know multiply both sides of equation 2 by 1r This gives 3 Pan A Al l r AH r A1rT391 Now subtract 2 from 3 This gives 4 Pr A Al l rT Dividing both sides of 4 by r and grouping terms gives 5 P Arl llrT 6 You should all be familiar with equation 5 from Principles of Finance For example ifA 1 r 1 and T 25 P 111 11125 907 Also notice that if rgtO then as T goes to infinity the term lll rT goes to zero So a security paying A per year forever has a value of 7 P Ar 39 consol or perpetuity 30 So ifA and r are as above P11 10 ii CFt A1gt for t 1 2 T 39 growing annuity In this case the price is given by 7 P Ar1 11 rT Where r r g1 g You can get equation 7 the same way you got equation 5 but instead of 1 r use 1 r 1 r1 g as the discount factor and go through the same steps Notice that r gt 0 if and only ifr gt g since g can t be less than 1 So in this case as T goes to infinity the term 11 rT goes to zero and the price of the growing annuity becomes 8 PAJ r ltAlt1 ggtgtltr g Which when you define A1 g as the dividend at date 1 is the constant grth model for stock prices that you learned in Principles of Finance For example if T is infinity r 1 and A1 then a nongrowing perpetual security is worth P 11 10 But ifg 05 then P 11051 05 21 This provides an example of how valuation can be broken down into the present value of current income 10 and the present value of gowth iii CFt 0 fort 12 T1 and CFt F for t T zero coupon bond Where F is the face value or promised return of principal This is the simplest kind of bond and will be fundamental to valuation as you go through 31 this course The value of a zero coupon bond is of course just a simple present value problem That is 9 P Fl rT iv CFtCfort 12 T1 and CFt C F for t T ordinary coupon bond Where C is the coupon payment and F is as before the face value of the bond Usually bonds are sold in 1000 denominations and are issued at par value As you will see next this means that F is usually 1000 If the bond makes coupon payment once a year these are called Euro bonds39 bonds issued in the US typically pay half the annual coupon twice a year semiannual then 10 P C1 r C1r2 C1 rT391 c 1 0T F1 rT Using the fact that the coupon payments are an T period annuity equation 10 can be written use equation 5 pick A C and add equation 9 as 11 P Cr1 11rT F11 rT So the value of an ordinary coupon bond is the sum of a T period annuity offering C per year and zero coupon bond promising F at maturity date T For example ifC 90 F 1000 T 25 and r 1 then P 9011 11125 10001125 81693 9230 90923 Some of the terminology from the bond markets will prove useful later on These include the coupon rate or coupon yield defined as CF In this 32 example the coupon rate is 901000 09 Letting rc CF be the coupon rate the price in equation 11 can be written as 12 P Fror1 11rT F1rT So if the coupon rate is equal to the discount rate rc r Using equation 12 you can see that in this case P F39 the bond sells at par In this example if r09 then P 1000 check for yourself If r0 lt r P lt F and it is said that the bond sells at a discount to the par or face value Finally if rc gt r P gt F and the bond is said to sell at a premium relative to the par value For example if r 08 P 110675 By setting the coupon rate equal to current market rates at the time of issue firm s can make sure that initially their bonds sell at par generally 1000 as noted earlier Some Properties of Bond Prices The following fun facts conceming bond prices will be helpful throughout the course in the sense that they provide fundamean relationships between value P and the factors that determine P T r F and C Property 1 For fixed T F and C P moves inversely to r This is just a reminder from Principles of Finance that bond prices move inversely to discount or interest rates Property 2 For fixed T F and C a decrease in r results in a larger change in P than the change in P associated with an increase in r of the same absolute magpitude Property 2 simply points to the fact that the present value of a future payment is not linear in r Figure 1 shows why property 2 is true 33 Property 3 For fixed C and F the percentage changes in P associated with a change in r are larger in absolute value the larger T Property 3 simply points out the fact that for a given change in interest rates percentage price changes will be larger the longer the maturity of the bond You can make sense of this just by looking at equation 9 the equation for a zero coupon bond Fixing F a small change in r can cause a large change in P if T is very large but the change in P will not be very large if T is small Check this for yourself Property 4 For fixed C and F the percentage changes in P described in property 3 increase at a diminishing rate as T increases Property 4 is essentially a statement that while price changes are increasing in maturity the difference in the price changes is getting smaller the longer the maturity For example the difference in the percentage price changes for T 4 vs T 3 is smaller than the difference in the percentage price changes for T 3 and T 2 Property 5 For fixed T and F the percentage change in P for a given change in r is larger the smaller C The only exceptions to this rule involve cases where the bond has only one remaining payment T l or the bond is a perpetuity maturity is infinite Property 5 is a statement that reflects the fact that from a present value point of view a bond with a large coupon payment is a shorter term bond than one with a low or zero coupon You are simply getting more of your money back sooner The exceptions deal with the case where a there is only one payment so the difference between principal and interest is meaningless or b all of the payments come from coupons no repayment of principal 34 You can get a sense of these 5 facts by example Suppose you have three coupon bonds with maturities of one two and three years respectively All of these bonds have coupon rates of 1 and F 1000 You have another bond promising 1210 F in two years zero otherwise Initially let r1 for all of the bonds So all of the bonds initially sell for 1000 P Now increase r to 11 but leave everything else the same Then you can verify that now P lt 1000 for all of the bonds veri es property 1 Moreover you can check that for the three coupon bonds P1 year gt P2 year gt P3 year verifies property 3 Furthermore you can check that P1 year P2 year gt P2 year P3 year verifies property 4 To verify property 5 you can show that P2 year coupon bond gt P 2 year zero coupon bond Finally you can verify property 2 by reworking the problem by changing r from 1 to 09 and checking that the absolute value of the increase in prices is greater than the absolute value of the decrease in prices when r 11 Factors In uencing the Level and Structure of Interest Rates This lecture deals with the factors that in uence the level and structure of nominal or money based interest rates The structure of interest rates refers to the relationship between the time to maturity and the promised interest rate on the security The easiest way to discuss these issues is to first consider the case of certainty and then deal with the complications of uncertainty Certain Level of Rates In ation 35 Recall that quotrfquot is the money rate of interest What we want to know is what factors go into the determination of quotrquot The most basic relationship is sometimes called the quotFisher Hypothesis which simply asserts that investors do not suffer from money illusion Money is not valued for itself but rather what goods and services the money can purchase The Fisher Hypothesis simply states that the money rate of interest should re ect both the quotrealquot discount rate call it E as well as changes in the value of money relative to goods and services which is re ected in the in ation rate In particular if it costs I0 to buy some good today and I1 to buy the same good next year the in ation rate call it quotiquot is just the percentage change in the price level or i11 IOIO For example if a quart of orange juice is 1 today and tums out to cost 125 next year the in ation rate is i 125 11 25 or 25 The Fisher hypothesis simply asserts that under certainty the retum from investing in a bond that pays off in money with no in ation adjustment should equal the return on a bond that is in ation protected or else there will be arbitrage Consider the following two investments Option 1 Invest 1 in a default free bond that pays off a known nominal rate r At the end of the year you will have principal plus interest or 1rf as your payoff in money Option 2 Invest the same 1 in a default free bond that promises a rate of quotRquot plus an adjustment on principal and interest for in ation whatever it turns out to be These will be referred to as index bonds At the end of the year you will have 1R1i as your payoff in y This bond is similar to the in ation protection bonds TIP s issued by the US Treasury 36 The Fisher hypothesis simply says that the payoff on Option 1 and Option 2 must be the same39 ie 1 1rf 1R1i For example if R 025 and i 02 then l le 1025102 10455 or rf0455 Otherwise there will be arbitrage opportunities For example suppose that r04 In this case you could borrow 1 at 04 and invest the dollar in the index bond Next period the index bond will pay assuming in ation tums out to be 02 10455 in money Can pay back the dollar you borrowed plus interest or 104 leaving you with a profit of 0055 So you have made a pro t using none of your own money and taking no risk People will continue to borrow at quotrquot driving the nominal rate up and simultaneously investing in the index bonds driving quotRquot down until equation 1 holds Since equation 1 can be written as 1r 1RiRi the Fisher relationship is often approximated assuming Ri is quotsmallquot by 2 rt Ri So in order to discuss the factors that in uence the level of nominal interest rates one needs to look at the factors that are believed to in uence the real rate of interest R and the in ation rate i Looking first at in ation it is generally believed that at least in the long run in ation is caused by quottoo much money chasing too few goodsquot In other words if the grth rate in money exceeds the growth rate in real goods and services the economy will experience higher levels of in ation higher i and vice versa if money grows too slowly relative to output While other factors can in uence the in ation rate in the shortterm e g oil price shocks in the long run these simply cause relative price changes e g compared to labor if the supply of money 37 remains constant In this case firms will simply substitute labor for oil in their production processes until the relative price changes get corrected Tuming to the real rate R it is thought that at least three factors in uence the real rate39 a the degree of impatience on the part of investors A higher level of impatience means that other things the same investors who have a strong preference for quotI want it nowquot will demand higher retums higher R for putting off consumption of goods and services b the level of wealth in the economy Investors with higher levels of wealth will be willing to lend other things the same at lower rates lower R than those with lower levels of wealth c the quality of investment opportunities in the economy As investment opportunities improve there is greater demand for borrowing and this tends at least in the short run to increased levels of R the real rate Certain Term Structure of Interest Rates Spot Rates and Yields to Maturity In the time value of money review quotthequot discount rate r was taken as a constant regardless of when the payment occurred in the future However in reality it is seldom the case that discount rates are the same for bonds of all maturities What factors would cause longterm rates to be higher than short term rates for bonds issued by the same entity say the US government In order to look into this question first define rt as that rate which can be 38 earned from buying a security promising CFt at date t and zero otherwise at a present value of Pt In other words rt solves 3 Pt CFt1rtt where Pt is the present value of the cash flow and CH is the future value at date t As in any simple present value problem the issue can also be viewed as one where you invest Pt now for t periods at rt This will yield you CFt at date t or 4 Pt mot CFt The rate rt is sometimes called the yield or spot rate on a zero coupon or pure discount security The relationship between the rt39s and t maturity is called the term structure of interest rates Studying these securities is important for a number of reasons First the value of more compleX securities like coupon bonds can be established by knowing the values of these zero coupon securities This is the concept of value additivi Second rt represents the proper discount rate to be used to value cash flows to be received at date t Using the yield to maturity from a more compleX security like a coupon bond to discount the cash flow at date t generates biased answers for purposes of proper valuation This is true because in general the yield to maturity on a compleX many payment security with a maturity of quotnquot years will not equal the spot rate for a payment to be received in year quotnquot In fact as the following example shows the yield to maturity is a compleX average of the spot rates The following example may help to clarify this notion Suppose two zero coupon securities are trading in the market Each promises 100 at maturity 39 and zero otherwise The first security matures in one year while the second matures in two years Current prices are 95 and 90 respectively Solving for r1 one gets that 1001r1 95 gt r1 10095 1 0526 Likewise you can solve for r2 since 10014132 90 gt r2 1009012 1 0541 Next consider the price of an annuity that offers 100 per year for two years Well since the cash ows from this annuity are exactly the same as those you would receive from buying one of each of the zero coupon bonds the price of the annuity must equal 185 95 90 or there would be the possibility of riskless arbitrage Suppose for example that the annuity was selling for 180 Then you could borrow one each of the zero coupon bonds sell them now for 95 and 90 respectively remember that this constitutes two short sales and promise to return them at maturity In both cases the bonds will obviously sell for 100 at maturity Then use the proceeds to buy the annuity for 180 leaving you with 5 in your pocket now Next year you receive 100 from the annuity but must return the one year bond that is maturing whose price at maturity 100 Likewise in the second year you receive 100 from the annuity but must retum the two year bond now worth 100 So the future cash flows are a wash you have money in your pocket and it came from someone else This can t persist for very long as people buy the annuity and sell the zero coupon bonds Given this discussion it is therefore always possible to find a rate call it y such that the discounted cash ows from the annuity equals 185 Specifically you can solve the yield to maturity problem just like in Principles 40 of Finance ie find a y y2 yn is the notation that will be used to de ne the yield to maturity on a bond whose last payment is made at date n so that 185 100ly2 lOOly22 Solving for y2 using your calculator gives y2 m Notice that r1 0526 lt y2 0536 lt r2 0541 so y2 really is kind of an quotaveragequot of r1 and r2 But y2 should not be used to discount cash flows from date 2 unless the term structure is at in which case yr Otherwise using y2 to discount cash flows creates a relative to proper value of the discounted cash flow to be received at date 2 In this example 12 gt r1 and the term structure is ward Mg so using y2 to discount date 2 cash flows will overstate the value and vice versa when the term structure in downward sloping ie r2 lt r1 in this example ForwardBreakeven Rates In the above example the term structure is upward sloping39 the rt39s are getting larger the larger t The term structure could however be downward sloping or even humped first increasing and then decreasing or vice versa Two questions need to be addressed First why like today is the term structure often not at and second as always is there money to be made if the r s are different from each other Under certainty the answers are simple As for the first question the term structure will be upward sloping today if it is known that interest rates will increase in the future and downward sloping if it is known that rates will increase in the future As for the second question the answer in m as long as there are no possibilities for arbitrage 41 To see how this works work your way through the following mental exercise Suppose that initially the term structure is at but that everyone knows this is certainty that rates will be higher next year What are people going to do Well you know from property 1 of bond prices that the prices of bonds of all maturities will decrease as rates go up However form property 3 longer term bonds will fall by more in price than will shortterm bonds So rational people will try to dump their longterm bonds quicker than shortterm bonds But this selling pressure will drive the price of longterm bonds down and their interest rates up Today s term structure will now be upward sloping This answers the first question except that you might ask Wrates are going to rise next year Well from the discussion of the Fisher hypothesis it must be known that either i the in ation rate is going to be higher in the future say because the central bank prints too much money andor ii real rates are going to increase in the future say because it is known that investors are going to become more quotimpatientquot In order to answer the second question you need not look any further than to say that if there is to be no arbitrage the m from holding all bonds no matter the bonds maturity or how long it is held must be the same under certainty Even under uncertainty however you can calculate given today s rates what future rates must be in order for the retum to be the same for all bonds The future rates that equalize the retums are usually called forward rates or quotbreakevenquot rates of interest In order to see how this works suppose you start with zero coupon bonds paying 1 at maturig zero otherwise The bonds have maturities of tl 42 t2tTl andt T respectively T is the longest term bond Using the simple present value formula it is possible given the prices of these bonds to calculate the spot rates39 the rt s Suppose you want to compare the return from holding a 2 year bond for one year to that of holding a one year bond for one year What must the one year bond rate be neXt year in order for the m to be the same for the two investments Well the one year bond will provide a gm one plus the return of l r1 The two year bond is currently selling for ll 122 r2 is the two year rate but these rates are stated in annual terms However when it is sold in one year the two year bond will tum into a one year bond and the sales price will be ll X where quotXquot is the one year rate one year from now So the gross return from holding this bond for one year will be 100l X100l r22 l r22l X So if the gross returns are to be the same on the two bonds it must be the case that 5 1 r22 1 r11X If you know 12 and r1 it is easy enough to solve for X In the earlier eXample r2 0541 and r1 0526 so X 1054l2l0526 l m So given today s rates the one year rate one year from now must be 0556 in order for the two investments to earn the same return So X is the breakeven rate in this eXample In general quotth tquot will be used to denote the forward rate for a contract to start in periodt and having maturity Tt I gt t In this eXample X 1f 139 the one year forward rate to start one year from now Notice that under certainty we could have equivalently said that we wanted the payoff from investing 1 in two year bonds and holding it for two years to be the 43 same as the payoff from putting 1 into a one year bond and rolling over the proceeds into another one year bond Given the existence of a three year bond it is possible to calculate two more forward rates39 the one year forward rate to start two years from now 2f 1 as well as the two year forward rate to start one year from now 1f2 For example suppose that there is another bond promising 100 in three years zero otherwise and it is selling for 85 Solving for 13 gives 85 1001 r33 or r 3 M There are two possibilities that need to be dealt with The first is that an investment of 1 in the three year bond for three years should have the same payoff as that from investing 1 in a one year bond and reinvesting the money at maturity in a two year bond and holding it to maturity this two year rate is 1f2 So this means that 1f2 solves 6 11 133105573 11 r11 1f22 105261 1f22 or 1f2 M In other words given today s rates two year rates next year must be 0572 in order to prevent arbitrage In a similar way you can find that future one year rate two years form now that makes the payoff from investing 1 in the three year bond and holding it to maturity and the payoff investing 1 in a sequence of one year bonds Given that 1f 1 is known you can solve for 2f1 by solving 11133105573 11 r111f11 2n 10526105561 Zn or 2f1 0588 44 The general rule for calculating forward rates is that if you have a bond with maturity T and another with maturity t T gt t then you can always nd the forward or breakeven rate on a bond to start at t with maturity Tt by solving 7 1 ITT 1 rtt1 thtT39t for tht For this example for T3 and tl you have solved for 1f2 and for T3 and t2 you have solved for 2f 1 Finally for T2 and tl you solved for 1f1 which is where the example got started solving for quotxquot Uncertainty Level and Structure of Interest Rates Although the future is unknown in this case it is still possible to calculate forward or other quotbreakeven ratesquot First consider the level of rates Given that you know now the money interest rate r and the real interest rate R then you can always solve using equation 1 for the in ation rate that makes the two w the same So even though you don t know i which is random you can solve for some i call it i so that the returns on the two investments is the same The value that solves this is 8 i 1 rf1 R 1 So for example if r 05 and R 025 then you will have been better of buying the index bond ifi gt i 1051025 l and vice versa ifi lt 0244 So if investors are unconcerned about risk they will invest in the index bond if they expect in ation higher than 0244 and invest in the non indexed bond if they think that in ation will exceed 0244 But if investors are unconcemed about risk both securities should have the same expected retum so in this case i had better be the expected in ation rate and the Fisher Hypothesis can be written as 45 9 lrf 1Rli Using exactly the same reasoning given today s spot rates the r s one can always use equation 7 to calculate the breakeven or forward rates even under uncertainty Again assuming that investors are unconcerned about risk one version of the expectations hypothesis says that these forward rates should equal expected future interest rates This is sometimes referred to as the unbiased expectations hypothesis since forward rates are quotunbiasedquot forecasts of future interest rates The difference between uncertainty and certainty is that even if investors are unconcerned about risk the expected return cannot be the same for holding different bonds for any maturity Therefore the quotgeneralquot expectations hypothesis noted by Fabozzi simply cannot be true The expectations hypothesis that will be used here is what has come to be called the quotlocalquot expectations hypothesis This version says that the expected retum from holding a security for one period is the same for all securities This statement implies because of property 2 of bond prices remember that present value is not linear in the discount rate that forward rates will not equal what investor s expect as opposed to know in the case of certainty interest rates to be in the future The following example may help in clarifying this last point Suppose you consider two discount bonds each promising 1 at maturity and zero otherwise The maturities are Tl and T2 with current prices of 9259 and 8264 respectively Then the known return from holding the one year bond for one year is l92599259 Of course the two year bond will be a one year bond in one year Suppose there is a 50 chance that one 46 year bond prices one year from now will be 9259 and a 50 chance that they will be 8591 Then the return from holding the two year bond for one year since there is do coupon payment is either 925982648264 m or 859182648264 M The expected retum is just 5 1204 50396 E the same as that from holding the one year bond for one year Given these possible bond prices one year interest rates one year from now can also be determined using the simple present value formula Either it will be the case that 9259 11r or it will be the case that 8591 11r So r will be either Q or M and the expected interest rate is 508 5 1640 w However using equation 5 to calculate the one year forward rate to start one year from now you get that 1f 1 lt 1220 This is a general property of the local expectations hypothesis Forward rates will be less than expected future interest rates if the expected return from holding all securities is the same over one period What if instead of being indifferent to risk investors wanted to avoid uncertainty In the example above holding the two year bond for one year is a quot air gamblequot in the sense that on average the return from holding the two year bond for one year is the same as the known return from holding the one year bond for one year Recall that if an investor dislikes risk or uncertainty they are risk averse ie they are unwilling to take a fair gamble relative to a known amount of money If an individual investor is risk averse and has a horizon of one year then the two year bond is a fair gamble but risky This type of risk is often called 47 interest rate risk but the term used here will be price risk to signify that relative to this horizon of one year the sales price of all longer term bonds is unknown at this investor s horizon Put another way an investor with a one year horizon can eliminate or irnmunize him or her self against risk by purchasing the one year bond Likewise an investor with a two year horizon can irnmunize him or her self against risk by purchasing a two year zero coupon bond If default free the return from holding a two year zero coupon bond for two years is known today regardless of the path of future interest rates So for this investor the purchase of the one year bond would involve risk Specifically the risk that the proceeds will need to be reinvested at some unknown rate This type of risk is known as reinvestment rate risk The combination of price risk and reinvestment rate risk is what will be referred to as interest rate risk The general rule of thumb is that an investor with horizon H can eliminate interest rate risk by purchasing a zero coupon bond with maturity T H For T lt H the investor faces reinvestment rate risk and for T gt H the investor faces price risk Assuming that all investors want to avoid risk won t take fair gambles it is therefore sensible to believe that they will demand a higher expected return over their horizon than the known rate of return on a zero coupon bond with maturity T H But markets are just made up of a bunch of investors so which bonds require the higher expected return will depend on the distribution of investor horizons Suppose for example that all investors had horizons exactly equal to ten years Then all bonds with maturity less than ten years would need to eam more than the ten year rate the known return on a zero coupon bond 48 over ten years because of reinvestment rate risk Likewise bonds with maturities of greater than ten years would require a premium due to price risk In this case we would predict that the retum from holding ten year bonds is going to be the lowest among all bonds The li uidi reference theory of the term structure basically asserts that most investors have short horizons small H s so that on average bonds with higher maturities must earn higher returns because they carry more price risk relative to the short investor horizon Suppose that the market is dominated by investors with Hl Then all bonds with maturity T gt 1 must eam a premium The evidence at least in this century suggests that the average retum from holding bonds is indeed increasing in maturity When combined with property 3 of bond prices long term bonds have more price risk than shortterm bonds this is evidence in favor of the idea that investors have short horizons When viewed from this perspective the liquidity preference theory also implies that forward rates will be upward biased estimates of future spot rates if the time value of money in uence from property 2 is not quottoo largequot In order to see this keep in mind that the liquidity preference theory says that all bonds with maturities T gt 1 should eam a higher return than that for T 1 But the gross return from holding a zero coupon bond paying 1 at maturity for one year is 1 r1 and the return from holding the two year bond for one year is ll X11 122l l r22 1 X where X is the future one period interest rate So as long as X lt 1f1 l r22l r1 1 the retum from holding the two year bond will be greater than that from holding a one year bond for one year and vice 49 versa if X gt 1f 1 So roughly speaking the forward rate must usually exceed the future spot rate if the longer term bond is on average to have a higher return over one year than the short term bond Duration and ConveXity Value Additivity says that the value of any security is equal to the value of zero coupon bonds In general you can write the price of any default free asset promising CFt at date t t 1 2 T as 10 P CF1Pl crm CFTPT where Pt 11 rtt is the price of a zero coupon bond promising 1 at maturity and zero otherwise These will often be called simple securities Clearly if CFt is not zero for all t lt T an investor with a horizon of H who picks a bond with T H cannot immunize by buying a non zero coupon bond While there is no price risk the intermediate cash ows need to be reinvested at an unknown rate so that the investor faces reinvestment rate risk The problem is only amplified if T lt H since now the investor faces reinvestment rate risk for both principal and interest The only other possibility is T gt H In this case the investor faces both price and reinvestment rate risk but these work in opposite directions If rates rise above current discount rates and stay there the bond will have to be sold at a loss However the reinvested intermediate payments eam a higher return than at the old rates Conversely should rates fall below current rates the investor realizes a capital gain at sale but a relatively low return on reinvested cash flows In fact by picking the bond carefully an investor can rig up a situation where heshe will earn the same return whether rates go up or down since what is lost in capital interest if rates increase decrease is offset by what is gained in interest capital Thus by picking the quotri tquot bond an investor cam immunize themselves even with coupon bonds What is the quotri tquot bond Given some simplifying assumptions we can answer this question First let s assume for the moment that the term structure is flat so that all discount rates are equal and equal to the yield to maturity y Then given cash flows CFt and maturity T the investor should find a bond such that the investor s horizon H is equal to l l D w1l w22 wtt wTT Sumwtt where wt CFtlyP is the present value of date t cash ow divided by the sum of the present values the price P Notice that w1 w2 wt wT 1 so equation 1 l is just the time value weighted average maturity of the security and D is known as the Macaulay duration It is easy to check that the duration of a zero coupon bond with maturity T is D T For all other bonds with maturity T D lt T Let s see how one can immunize lock in their return by buying a coupon bond with a duration equal to their horizon Example Suppose we have a bond with a coupon of 40 per year and a yield to maturity of y 05 The maturity is 3 years and the face value F is equal as usual to 1000 This gives a price P of 97276 In order to calculate the duration we must first calculate the weights In this case w 40l0597276 0392 w 40l05297276 0373 and w 40 1000l05397276 9235 Notice that w1 w2 w3 039203739235 1 The duration is given by D 0392l 03732 92353 288 If the investor has a horizon H of 288 then this bond will immunize him or her against interest rate changes To see how this works we must de ne the total return from holding the bond for H periods The general formula is given by P total cash flow from bond1 returnH or return total cash flow from bondP 1 Obviously since y is the yield to maturity or promised retum if rates stay at 5 the return will equal y 05 You can check this for yourself However suppose that right after purchasing the bond interest rates jump to 6 What will be the total return Well we must first calculate the total cash flow from the bond In this case we have total cash ow 4010613988 40106 88 10401063912 111949 and the return is given by return 11194997226 123988 0501 which is approximately the promised yield to maturity You can check for yourself that the return will equal the promised return if rates fall to say 4 as well Why does this work Notice that reinvestment rate risk what you can reinvest cash ows at and price risk work in opposite directions If T gt H then if rates go up your get more than promised on your reinvestment However when you sell the bond after H periods you take a loss The opposite situation occurs if rates decline By setting D H these two effects exactly offset each other and the realized return is exactly equal to the promised retum It is in this sense that the investor is immunized against rate changes The general rule of thumb for comparing the promised to actual return is the following If D gt H and rates go up after you purchase the bond then return lt y If D gt H and rates go down after you purchase the bond then return gt y If D gt H and rates go down after you purchase the bond then return gt y If D lt H and rates go up after you purchase the bond then return gt y If D lt H and rates go down after you purchase the bond then return lt y Just like the fun facts for bonds we have some fun facts for duration A N DJ As noted earlier the duration of a zero coupon bond is equal to its maturity In other words w l or t T and zero otherwise Duration is a decreasing function of the coupon rate on a bond This makes sense because the higher the coupon rate the more weight that is given to the earlier payments Since the weights must add to one this implies less weight for more distant cash flows thus a lower duration measure Duration is a decreasing function of the coupon rate on a bond This makes sense because as the interest rate increases the present value of distant cash ows decrease more rapidly than those for earlier cash ows Thus the weights shift toward earlier cash flows and duration declines The opposite logic applies when interest rates decline 4 The duration of a portfolio is just the weighted average of the durations for the individual securities In addition to immunization duration has other useful purposes The most important is that when properly modified duration gives us approximately the price risk for a bond Specifically in order to calculate price changes associated with interest rate changes we typically use what is called modi ed duration which is just given by 15 D D1y What are we going to use D for Well it tums out that for very small changes in rates D can as noted earlier be used to get a good approximation to the interest sensitivity of a fixed income security It particular you can show that 17 APP z DAy or in words the percentage change in the bonds price is approximately equal to the negative of the modi ed duration multiplied by the change in interest rates So in our earlier three period bond example D 288 andy 05 then D 288105 274 Therefore if rates increase by say 100 basis points so Ay 01 then APP m 274 OlO274 or put in words a 100 basis point increase in rates will cause this ten year coupon bond to fall in price by approximately 274 A similar decrease in rates would cause the bond to increase in price by a similar percentage Notice that while the duration for our example coupon bond is positive there is no reason why duration cannot be negative Consider the following example Example Assume that the yield to maturity or internal rate of return on a two period project is 5 Cash flows in year one are 900 and cash ows in year two are 500 Thus the projects initial price is 900105 5001052 40363 What is the duration Well w 900 10540363 21236 and w2 500105240363 11236 Notice that w w2 1 and the duration is given by D 212361 112362 1236 The modified duration is 1177 What this means is that as interest rates go up the value of this project also goes up and vice versa While modi ed duration works OK as an approximation for very small changes in rates formally it works when Ay m 0 it can do a poor job of approximating the actual change in bond prices when Ay is big like it often is in today s environment To see what s going on let s go back to our ten year coupon bond example In this case new y old y Ay 05 01 06 and so the new P at y 06 can be found by plugging into the formula for the price of a coupon bond or new P 4006111063 10001063 94654 So the actual percentage change in price for this 100 basis point increase in rates is given by94654 97276 97276 30269 In other words the duration approximation gives an answer that is 5 basis points too large in absolute value in this example We can say something more general than this and that is that APP gt DAy if Ay lt O APP lt DAy if Ay gt0 In words the duration approximation over predicts price declines as rates increase and under predicts price increases as rates decline Lucky for us there is an adjustment an ugly ugly one at that that can be used to get closer to the actual price decrease In particular we can use 18APP z DAy ConvexityAy22 where the convexity measure is given by l9Convexity w1l2 w223 wtttl wTTT11y2 Sumwttt11y2 Example For the three period bond example used earlier we have that convexity 039212 037323 9235341052 1033 So we can calculate for our 100 basis point increase in rates that APP m DAy ConvexityAy22 27401 10332012 0274 0005 0269 which is equal up to one basis point to the actual price decline calculated earlier Notice that the addition of convexity improves the approximation of modi ed duration in both directions By adding a positive number when rates increase it makes the duration approximation less negative and by adding a positive number for rate increases it makes the duration approximation more positive Given our earlier comments in both cases the approximation is now closer to the actual price change FX Markets Earlier the idea that interest rates are just exchange rates for the same currency over time That is for example if you give up 1US you get 1US1 r a year from now where r is the one year interest rate in the US say on US Treasury bills In terms of purchasing a US dollar next period a current US dollar is worth llr P the price of a discount bond Now 1 P ll r the number of US dollars needed now to purchase one US dollar next period If for example rl then P90909 and it takes 90909 US dollars today to obtain 1 US next year The study of foreign exchange rates is nothing more than the study of the prices at which say US dollars can be exchanged for some other currency This exchange can take place now This is called the spot foreign exchange contract Or by agreement the exchange can take place in the future but at an exchange rate agreed upon now This is called a forward foreign exchange contract The quotation system used here is the one used by Shapiro the socalled American quote system whereby the exchange rate is quoted as the number of US dollars needed to purchase one unit of the foreign currency So use the notation 2 e the number of US dollars needed now to purchase one unit of the foreign currency now for the spot exchange rate and likewise let 3 f the number of US dollars needed to purchase one unit of the foreign currency T he exchange will take place at some tture time be the forward exchange rate The interval of time considered here will as usual be one year 360 days to be exact for this section but most real forward contracts for foreign exchange trade at shorter intervals than this forward contracts are available with maturities of 30 60 90 180 and 360 day s Using this quote convention a foreign currency is said to be depreciating against the US dollar in the spot or forward market if e or f is declining 39 that is it takes fewer dollars to buy one unit of the foreign currency Just the opposite would be true if using the European quote system of units of foreign currency needed to purchase 1 Cross Rates The concept here is that if you know how many units of currency quotAquot you need to purchase one unit of currency quotBquot and you know how many units of currency quotAquot are needed to purchase one unit of currency quotCquot it should be straightforward to calculate the cross rate as ABAC CB If the actual exchange rate for currency C to B is different there will be the opportunity for riskless arbitrage in this case called triangilar arbitrage Example Suppose it takes 150 Japanese yen to purchase 1 in the spot market Then eY 1150 M is the spot dollar to Yen Y exchange rate Furthermore suppose it takes 25 dollars to obtain one British pound L Then 6L 25 The cross rate of L to get one Y is given by 0066725 00267 This cross rate should equal the actual spot rate for pounds to yen If not there is an arbitrage opportunity Suppose for example that the actual L Y exchange rate is Q What to do Well it seems to take more pounds to buy a yen than it should using the cross rate So the pound is undervalued relative to the yen Here are the steps for the arbitrage Step 1 Borrow 1 and convert it into yen this whole process is only going to take a minute so you can mostly ignore the interest cost Convert it into yen You receive 15OY Step 2 Convert the yen to pounds at the actual exchange rate to get 150003 45L Step 3 Convert the pounds back into dollars to get 45251125 gt1 As long as the interest rate in dollars is less than 125 per minute you will make a riskless profit from this transaction The key here is to purchase the overvalued currency in this case yen with dollars convert the overvalued into the undervalued currency in this case pounds and then convert back into dollars This ability to engage in riskless arbitrage keeps cross rates very close to actual rates The only catch here is that the strategy is seguential so for example the yen to pound exchange rate in step 2 may change before you have the opportunity to finish step 1 Another Arbitrage Relationship Covered Interest Rate Parity If interest rates are just the exchange of one currency for itself over time and foreign exchange is just the exchange of one currency for another you might think that these quotexchangequot rates are linked You would be correct In fact if both forward and spot fx contracts trade the interest rates across countries will be linked via arbitrage arguments This notion is called covered interest rate parity Before looking into the link between f e r and r where r is the interest rate in the foreign country some terminology is in order If f lt e the foreign currency is said to trade a forward discountcontracting now it will take fewer dollars in one year to obtain a unit of the foreign currency than the number of dollars needed now to purchase the currency while if f gt e the currency is said to trade at a forward premium What determines whether the currency trades at a discount or premium in the forward market Covered interest rate parity makes sure that if there are no arbitrage opportunities f gt e if r gt r or rates in the US are higher than they 60 are in the foreign country f lt e if r lt r In fact covered interest rate parity allows for a more specific statement than even this If there is to be no arbitrage it must be the case that 4fe 1 r1 r Let s take a closer look at this relationship and see why it makes sense Consider the following two investment opportunities starting with 1 Option 1 Invest the 1 in US security and receive a payoff in dollars next year In particular you would get certain payoff in s next year 11 1111 Option 2 Take the 1 convert it into the foreign currency now at the spot rate invest the foreign currency in the foreign bond and write a forward contract now to convert back the proceeds of this investment into dollars next year at the forward rate The payoff in foreign currency will be certain payoff in foreign currency next year 11e g 1r1 If you convert this sum known today back at f you will have certain payoff in s next year 11e 1 1r1 Covered Interest Rate Parity says that the certain payoff from these two investments must be the same This is the same thing of course as saying that equation 4 must hold Working through an example may help to clarify why this is true Example Suppose that one year rates in the US are r while one year rates in the UK are r Then if e 25 covered interest rate parity says 61 that the forward rate f must be 25105l l 2386 in order to prevent arbitrage Suppose that this was not the case and the actual forward rate was say amp Then checking back to options 1 and 2 you can safely conclude that at these actual exchange rates the certain payoff in s next year from option 1 gt certain payoff in s next year from option 2 In particular amp gt 22525l l 2 Under these circumstances it is quotfree lunch timequot So you should go long option 1 and short option Remember to always go long the undervalued asset and short the overvalued asset But an undervalued asset under certainty is simply one that yields a higher return than the overvalued asset for the same amount of investment How can this be accomplished Here are the steps Actions Now Step 1 Borrow 1L today and agree to repay it in one year at a rate r lO Step 2 Convert the L to dollars at the spot exchange rate 6 25 So you now have 25 Invest this in the US bond market at r 5 Step 3 Write a forward contract to mature in one year that allows you to convert 2625 back into pounds at f 225 Payoffs Next Year Step 4 Collect 2625225 1167L from your actions in steps 2 and 3 Step 5 Pay back your loan in pounds from step 1 You owe 11L This leaves you with a profit of O67L regardless of what happens to exchange rates or interest rates over the coming year 62 Clearly investors are going to want to borrow pounds buy dollars spot invest in British bonds and sell dollars forward As there is more selling pressure on s forward the dollar will naturally depreciate in the forward market until the forward rate remember that this is L so that the dollar depreciating means an increase inf increases to f 25 and equation 4 holds Keep in mind the importance of the fact that these bonds are assumed to be M in the local currency Think of these bonds as one year US Treasury bills and say bonds issued by the British government This is an extremely important point because if this was not true you would not be able to construct options 1 and 2 so that the payoffs next year were certain For example if the securities were instead equity investments in the US and the UK the returns would be random and you would not know how many s to convert back into pounds in Step 3 above Given the discussion so far it should not be surprising that covered interest rate parity holds almost all the time for all of the currencies for which there exist forward markets for currency exchange In fact the standard practice by dealers in these markets ignoring the bidask spread is to quote the forward rate based on knowledge of the interest rates and the spot exchange rate That is given 6 r and r the quote for f is set so that equation 4 holds What about cases where there are no forward contracts Is there profit to be made Uncovered interest rate parig Even in situations where there is no explicit forward market for foreign exchange an quotimplicitquot forward rate can be calculated using equation 4 ie 63 5 f implicit 6 1 rl r What uncovered interest rate parity says is that if investors are unconcemed about risk f implicit should be a forecast of the expected future spot exchange rate call it Eel where E means expected and g1 is the spot exchange rate one year from now The statement that 6f implicit Eeu is a version of the expectations hypothesis for exchange rates So in this case while there is no riskless arbitrage opportunities there are still trading rules that can be developed much like the case of bonds These are risky arbitrage strategies and the general rule is if f implicit gt Eel then your forecast is that the dollar will be stronger at date 1 than that implied by current spot exchange rates and interest rates Using 5 this means that you think that 1 rl r gt Ee e and you W dollars in the spot market next period The analogy to term structure is that dollars play the role of long term bonds and the foreign currency that of short term bonds In this example you want to go long investments and short the pounds just like if the forward rate is greater than the expected future spot interest rate you want to buy the longer term bond and borrow and the short term rate How would you accomplish this Well in the example with pounds earlier 64 f implicit 2386 Suppose your expectation was that the spot exchange rate next year was going to be 225 Eel 225 you would go through a strategy similar to that outlined for the case of covered interest rate parity Step 1 Same as above Step 2 Same as above Step 3 Skip there is no forward market Step 4 Collect 262561 L s from steps 1 and 2 Step 5 Same as above Pay back your loan in pounds from step 1 You owe 11L Notice that if your forecast is correct on average 61 will turn out to 225 and this strategy will end up yielding a pro t in L s However there is risk here For example if 61 turns out to be 25 the dollar appreciated relative to the implicit forward rate In this case your cash in ow in step 4 will be 262525 105L s so you end up losing O5L s The opposite strategy is used when f implicit lt Eel In this case you want to end up buying dollars in the spot market next period You should then reverse steps 1 to 5 What causes exchange rates to fluctuate over time The lntemational Fisher Hypothesis 65 Recall from lecture 3 that the Fisher Hypothesis says that in a given currency it should be the case that the nominal rate re ects the real rate and expected in ation Using the same notation this means that in US dollars it should be the case that 7 1 rf 1 R1 Ei where i is the in ation rate over the coming year in the US R is the real rate index bond rate and as usual E means expectation But if the Fisher hypothesis holds in dollars there is no reason why is should not hold in pounds so it should be true that 8 1 rf 1R1Eil where the s as usual mean the foreign currency The international Fisher Hypothesis asserts that if there is a free ow of nancial capital real rates should be the same across countries In this case R R and therefore nominal interest rates across countries re ect only differences in expected in ation If this is true and the expectations hypothesis for exchange rates holds it must be the case using 5 8 that 9 Ee 1e 1 rf1 rf 1 Ei1 Ei Equation 9 makes perfect economic sense What it says is that relative to the current exchange rate the dollar is expected to depreciate relative to the foreign currency over the next year if investors expect the in ation rate in the US to be higher than that in the foreign country over the coming year and vice versa But this is sensible since what is being said is that the dollars purchasing power over goods is expected to decrease relative to that of the 66 foreign currency and therefore the dollar should fetch fewer units of the foreign currency in the future What detennines the level of e Purchasing Power Paritv The discussion so far has focused on the factors determining the relationship between today s spot rate and forward or expected future spot rates But what determines the level of e Well intuitively it should have to do with how many s vs say pounds it takes to purchase something that you want39 for example shoes Purchasing Power Parity PPP says something more specific even than this PPP says that adjusted for exchange rates the cost of the same commodig should be the same evegwhere So if a pair of shoes costs 50 in the US and the same pair of shoes cost 20L s in the UK the exchange rate for to L s better be 5020 25 More generally if we take a particular price index say the CPI consumer price index in the US and find a similar measure in the foreign country call it CPI then if PPP holds it should be the case that 10 e CPICPI Simply put if goods cost more here than in the UK it should take more than one dollar to get a pound However since the price index can be scaled up or down PPP is usually stated as saying the level of the exchange rate uctuates up or down depending on whether the price index in the US increases more than in the UK In particular if PPP holds next period then 1 l e 1 CPIlCP11 where the subscript 1 denotes the exchange rate and price indices at date 1 But the in ation rate is just defined as either i CP11 CPICPI or i 67 CPI1CPICPI depending on the country So what PPP says is something stronger than the Intemational Fisher Hypothesis equation 9 PPP says dividing l l by 10 that 12 6 16 CPIlCP11CPICPI 1 il i or that the m as opposed to just expected rate fluctuates up or down over time depending on whether the US in ation rate is higher than that in the foreign country so the dollar depreciates and Vice versa The intuition behind PPP is that if there is a free and costless flow of physical goods as opposed to just nancial capital across countries the quotrealquot exchange rate adjusted price of goods will be the same everywhere What is the Evidence Regarding All of These Pang Relationships Well as you might expect those parity relationships that are based on the absence of riskless arbitrage almost always hold That is Fact 1 Cross rates almost always egual actual exchange rates There is seldom the opportunity for triangular arbitrage Fact 2 Between Countries for which there are forward foreign exchange contracts covered interest rate parig almost always holds The is seldom an opportunity for riskless arbitrage These two facts should not be surprising If they were not true most of the time there would ample opportunities for a quotfree lunchquot Fact 3 Uncovered interest rate parity does not always hold 68 Facts 2 and 3 can be viewed graphically in Exhibit 510 of Shapiro Notice that the covered interest rate differentials are almost always zero fact 2 but the uncovered differentials are consistently positive or negative across the four currencies listed Fact 4 The international Fisher Relationship does not generally hold The failure of the international Fisher Hypothesis must be due to things like capital controls or tax differentials across countries Exhibit 78 makes clear that countries with nominal rates rf s also have high real rates R s The intemational Fisher Hypothesis conversely would imply the R s should be the same regardless of rf Fact 5 PPP fails in the short run but tends to hold over long periods of time Governments can of course keep the prices of goods in their country higher or lower than world prices by using things like capital controls import quotas and subsidies for domestic industry Exhibit 74 shows quite clearly that actual exchange rates can differ from PPP predictions for some periods of time However notice that over long periods of time the actual series tends to revert to the theoretical PPP line The moral of the story is that govemments can keep prices quotout of linequot with world prices for awhile but eventually market forces work to eliminate price discrepancies worldwide Derivatives A derivative security as used in the context of this class simply means a security whose value depends on that of another security called the underlying Now since any security can be valued as a function of its future 69 payoffs the value of a derivative security must depend on the underlying because its future payoffs must depend on the payoffs from the underlying The main tool used for purposes of valuation in this section will be one you already know how to value things if there is to be no arbitrage So the real point in this section is to first discuss whL someone might want to purchase or sell derivative securities for purposes of speculation or risk management and then discuss how the securities are actually priced Before proceeding it is worthwhile to first look more specifically at what is meant by a derivative security Almost all derivative claims no matter how complex are basically composed of two sorts of contracts39 socalled forward contracts and option contracts You are already familiar with the notion of forward rates for bonds and foreign exchange and you can think of forward contracts as ones that formally allow you to trade on what you think future bond prices or interest rates or foreign exchange rates will be in the future Options are contracts that give you the right to buy or sell a security over some period of time In both cases the terms of trade are worked out at the beginning of the contract The following definitions may prove helpful De nition of a Forward Contract Terminology A forward contract is one that calls for the transfer of some good between two parties at a future point in time with the delivery price fixed today The individual who obligates him or her self to deliver the good at a price fixed today is called the short side of the contract while the individual who 70 LI39 4 him or her self to take delivery is said to be on the long side of the contract The price for future delivery is called the forward price while naturally the price for immediate delivery is called the spot price The key term here is obligates Next consider the option De nition of an Option Contract Terminology The purchaser of an option contract purchases the to buy a call or sell a put some good at a European option or before an American option a future point in time with the sales price fixed today Conversely the m of a call put option for a fee called the option price obligates him or her self to deliver take delivery of the good at a price fixed today which is called the strike or exercise price The key term here is obligates as well However notice that in this case the purchaser has only rights which they will clearly exercise only when it is in their monetary interests The writer of the option must charge an up front fee since they have only obligations and will be called on to meet their obligations only when it is not in their monetary interests Why Do These Contracts Exist The are two primary reasons why individuals and firms use derivative contracts The first is to speculate on price movements While you could speculate by buying or shorting the underlying derivatives provide a much more cost efficient means of speculating They provide leverage as you will 71 soon see for yourself The second reason that derivatives exist is for purposes of hiiging or trying to cheaply reduce risk Hedging and speculation are just two sides of the same coin so derivatives can be viewed as ef cient ways to engage in risk management de ned as any overt strategy that changes the riskretum pro le of the cash ows from a given underlying position Examples Speculation and leverage Suppose that you are willing to regularly lose small sums of money for the chance of a big payoff Then you might be a candidate to buy or purchase options Someone else willing to collect money but risk potentially large losses would be someone who would be a candidate to write options For example if you thought a stock was going to increase greatly in value then you could buy the security but buying a call option would be a much cheaper way in terms of current cash out ow to engage in this activity This is an example of speculation and the leverage effect discussed here is easy to show Remember that the call option gives you the right to buy the underlying at the strike or exercise price Suppose the current price of a stock is 100 and you think the price will increase to 150 in the near future Ignoring dividends you could i buy the stock and hope the price rises If it does you will have a return on investment of 150lOO100 However if the price should fall to say 50 your return is 50100 100 50 72 Q Buy the call option Just for illustration suppose you buy an option with an exercise price of 100 for say 10 note just take this figure as given for now figuring out the correct price for the option is the point of the section on valuing options Then should the price go to 150 your option is m wretum is 15010010 10 400 However should the price fall to 50 your option expires worthless or out of the money and your return is OlOlO M So by buying the option instead of the stock you have magnified both your potential gains and losses but this is precisely what leverage does Hedging Futures contracts a form of forward contracts actually got started in this country in the llidwest Farmers were perpetually concemed about the prices they would get for their crops Since agricultural prices are very volatile farmers got into the habit of agreeing to sell part of their anticipated crop at a price fixed today This quotlocked inquot revenue for at least a portion of the crop So the farmers are short in this example Conversely a miller or other merchant who uses agricultural commodities as an input and other folks who just might want to speculate on the price movements of agricultural commodities might be willing to take delivery at this fixed price and is said to be on the long side of the contract Clearly the farmer s total risk is going to be lower by locking in a price for part of his or her anticipated crop So the farmer is said to be hedging in this instance Notation and Payoffs T maturitydelivery date of the forward contract or option 73 F0 forward price for delivery at date T determined today SO pot price today ST Spot price at date T random from today s point of viewdetermined at date T X strike or exercise price on the option r1 spot rate for pure discount loans with maturity Tdetermined today C0 Price of the call option today P0 Price of the put option today Payoffs at date T Netgross payo at maturity to short position in forward F 0 S T This makes sense because if you agree to deliver for F 0 and ST turns out to be less than F 0 you have a gain but if ST tums out to be greater than F 0 you have a loss Netgross payo at maturity to gig position in forward ST iFo Notice that in this case the net and gross payoffs are the same except for marginto be discussed later which is just another way of saying that the initial investment in a forward contract is zero Gross payo at maturity from buying a ll option ST X ST gt X 0 ifSTltX and 74 Netpoyo atmaturity om t zguill option STX C0 ST gt X C0 if ST lt X Similarly Gross poyo at maturity from buying aw option X ST ST lt X 0 if ST gt X Netpoyo at maturityfrom buying aw option X ST P0 ST ltX P0 if ST gt X The gross payoffs from writing calls and puts is just the ip side of the coin Gross poyo from writing a ll option X ST ST gt X 0 if ST lt X Netpoyo from writingaill option XST C0 ST gtX C0 if ST lt X and nally Gross poyo from writing aw option ST X ST lt X 0 if ST gt X Netpoyo from writing awoption ST X P0 ST lt X P0 if ST gt X 75 Sometimes the gross payoff at maturity or expiration is called the intrinsic value of the option since you can show that its value will never be less than this amount Relationship Between Prices Case 1 No intermediate dividends or coupon payments Cost of Cam Model An investment banker whose firm is now out of business once said that if you understood the cost of carry model you understood 90 of what was important in pricing derivative securities This seems a bit much but nevertheless the cost of carry model is basic to understanding how derivatives are priced All the cost of carry model says is that if you can costlessly contract in forward markets and spot markets and can borrow and lend at spot rates then it must be the case that C 0st of Carry The present value of the forward price for delivery at date T must equal the spot price today That is there is to be no arbitrage F0 1rTT S0 or F0 so 1 rTT All the cost of carry model says is that if it is costless to enter into a forward contract then the payoff at maturity from entering into a long position on a 76 forward contract must be the same as the payoff from borrowing enough money at rT for T periods to purchase the stock Example Suppose that S0 10 T2 and r2 10 Then F0 10112 1210 in order to prevent arbitrage To see why suppose that F0 13 So the forward contract is quotovervaluedquot Recall that you always want to short the overvalued and buy the undervalued In this case you go through the following steps Step 1 Today you enter the short side of the forward contract Agree to deliver in 2 years at F013 Step 2 Myou borrow 10 at 10 interest per year and purchase the security Step 3 In two years you deliver the security that you own collect 13 pay back 10112 1210 leaving a profit of 90 regardless of what the stock price turns out to be at date 2 The reason step 3 works is that you already own the security that needs to be delivered on the forward contract So you have quotlocked inquot your profit today regardless of what happens to prices over the next two years Notice that holding the underlying and hedging going short in the forward market is the same as selling the security now and investing the proceeds at the riskless rate of interest Likewise holding the underlying and speculating in this case also going long a forward contract is equivalent to a quotTexas 77 Hedgequot or borrowing at the riskless rate and doubling your bet on the underlying Example You are a bond fund and currently hold a two year zero coupon bond promising 100 at delivery and zero otherwise You can enter into a contract that calls for delivery of a one year bond in one year T 1 Suppose that T1 05 and T2 10 We know that SO 100ll2 m and the no arbitrage forward price F0 8264l 05 MFurthermore you think that there is a 5050 chance that one year bond prices one year from now will be either 9523 or 8696 Here are your options Option 1 do nothing just hold the bond for one period Your expected return is 95235 86965 82648264 910958264 1 amp And the standard deviation of your retum is 95239l0952 5 8696 9109525128264 m Option 2 hold the bond and go short the forward contract At date 1 you will receive 8678 for certain the forward price and your return will be 8678 82648264 05 which is the riskless rate in riskexpected return space Option 3 you can hold the underlying and go long the forward contract Your eXpected retum is going to be 91095 91095 F0 82648264 182 198678 82648264 your standard deviation of retums will be amp Check for yourself that if you just borrowed 8264 for one year at 05 and invested this to buy another two year bond your eXpected return and risk would be the same as that outlined in option 3 So the point here is that derivatives may just allow us to do cheaper what we could have done before in terms of moving up and down the eXpected returnrisk line 78 M Cost of carry when the underlying pays a coupon or dividend cash yield Let d the coupon or dividend yield for the underlying we called this the current yield when we did bond pricing To keep things simple let us suppose that the shortterm interest rate is a constant Then since the forward contract does not receive the cash yield it39s price must be reduced by this amount or F0 So 1r dT Example Suppose you have a forward contract to deliver a perpetuity at date 1 so T 1 Let the current short rate r 05 and assume that the perpetuity has a rate of 1 right now with a coupon payment of C 100 per year Then S0 100l 1000 Furthermore d C S0 1001000 1 So F0 must be F0 1000105 11 m Interest Rate Swaps Interest Rate Swaps are nothing more than a series of one period forward contracts settled in arrears They were created in part because there are no exchange traded contracts with maturity greater than a year so hedging interest rate exposure for a long period of time created a challenge Consider a fixed for oating rate swap agreement where one party pays the xed rate r xed and receives an random oating rate call it filmng where the t l means that this is the oating rate at t l to be paid at date t and t runs as usual from 1 to T the maturity of the contract Let N be the quotnotional principalquot on which this contract is written T he of dollars on which to 79 calculate the interest paymentsreceipts Then the total cash ow to pay fixed receive oating is given by Cash ow to pay fixed receive oating party NltI oa ng0 r xed Nr oating1 39 r xed Nr oatinglquot2 39 r xed Nr oating1quotl 39 r xed Notice that if oating rates tum out to be higher on average than fixed rates this party makes a positive cash ow while the reverse would be true if oating rates turn out to be less than fixed on average Since this swap is just a bunch of forwards and forwards require no initial investment swaps should be priced so that the initial cash ow to both parties is zero How can this be achieved Well there are two problems to tackle l The oating rates beyond today s rate date 0 are not known and must be estimated 2 Given estimates in 1 a fixed rate must be found that makes the present value of the expected payments equal to zero for both parties The first problem is solved by using forward rates from today s term structure to estimate the expected oating rates In particular we have that T oa ngm fl today s one year rate known Er oating1 1f1 1 r221 r1 391 Bahama 2f1 1 r3gt3lt1 r92 1 Er oatingr 1 Tlfl 2 1 rTT1 In 391 Where E means what participants are using this as their forecast Keep in mind that if liquidity preference is true f will overstate actual oating rates on average and those who receive fixed and pay oating will tend to gain Given these forecasts it is easy to find the fixed rate that makes the present value when discounted at today s spot rates of this contract equal to zero In particular solve the following for r xed 0 ltNltr1 rem1 ro ltNlt1f1 rhea1 r92 gt NT2f1 rhea1 rm gt ltNltT1f1 rhea1 rmT Example Suppose that T4 I will sometimes call this a quotthreequot year swap since only three payments will be made in this example and the notional amount is N 10000000 The current term structure is r1205 r2075 and T 10 Then r oating0 r1 Bahama 1f1 1 r2gt2lt1 r1 1 1075gt2lt105gt 1 m Bahama 2f1 1 r3gt3lt1 r92 1 11 3lt1075gt2 1 w So we need to solve the following equation for r xed 0 1000000005 rltixedl05 100000001006 rhea10732 100000001518 r Xd113 This gives 0 2487 39 r xed 2569 notice that N 10000000 cancels out so T xed m which we already knew would be less than 1 the three year spot rate recall that quotyieldsquot are less than spot rates if the term structure is upward sloping PutCall Parity Even though figuring out the price of a particular option is covered in the next section these calculations require some assumptions about the price of the underlying as it evolves over time The socalled putcall parity relationship on the other hand simply involves the lack of arbitrage The standard version of putcall parity relates the prices of puts and calls on the same security with the same exercise price and the same maturity date Definition of PutCall Parity If two options a put and call have a common exercise price of X and a common maturity of T then in the absence of arbitrage it must be the case that P0 C0 XI mT So where 17 is the spot rate for date T pure discount bonds To see why this works consider the following two investment altematives altemative 1 Buy a call and write a put at an exercise price of X and a maturity T Initial Cash Flow P0 C0 Payoffs at date T if ST lt X if ST gt X payoff from buying call 0 ST X payoff from writing put ST X 0 total payoff from option 1 ST X ST X at maturity altemative 2 Borrow X1l rTT for T periods at the rate IT and invest in one share of the stock Initial Cash Flow XHmT so Payoffs at date T i ST lt X i ST gt X payoff from buying stock ST ST payoff from loan X11TT11TT X11TT11TT total payoff from option 2 ST X ST X at maturity By now you know that if altematives 1 and 2 have the same total payoff at maturity then if there are no other cash ows the initial cash ows associated with the two alternatives must also be equal or else there are arbitrage possibilities Example Suppose S0 10 T 2 X 1210 r2 10 and P0 2 gt C0 1 Then according to putcall parity the put option is overvalued Then you should follow the following arbitrage strategy 83 Go long alternative 1 Write a put and buy a call Cash ow now 1 Cash ow at date 2 82 1210 Go short alternative 2 Short the stock collect 10 and invest the proceeds at r2 10 for two years Cash Flow now 0 Cash ow at date 2 1210 82 So total cash ow now 1 and total cash ow at date 2 0 regardless of what 82 turns out to be This is a riskless arbitrage Everyone will try to short the stock driving SO down write puts and buy calls until putcall parity holds Notice that putcall parity does not say that for the same X and T P0 C0 However in the special case where X F0 it must be the case that P0 Co as long as the cost of carry model holds But this is nothing more than another way of saying that You can replicate the long position in a forward contract by writing a put and buying a call with an exercise price equal to the forward price Conversely you can replicate the short position on a forward contract by writing a call and buying a put with an exercise price equal to the forward price Now that you are familiar with the basics of options it is worthwhile to pause for a second and ask how investors can use options to achieve some desired objectives The bene ts of leverage when you use derivatives has already been discussed in the context of pure speculation but there are some other interesting strategies that are worth a brief review Scenario You think prices are going to be volatile but you are clueless which way they will move up or down Strategy Buy a quotstraddlequot The easiest way to achieve a straddle position is to buy a put and a call option with the same exercise price and maturi date In this case if prices increase a lot you can exercise the call while if prices fall a lot you can exercise the put If prices don t move too much however you are out both the price of the put and the price of the call Scenario You already have a position in the underlying but want protect yourself against losses over some future period of time Strategy 1 quotProtective Putquot In this case you simply buy a put option with a maturity date equal to your investment horizon If the price of the underlying increases you are out the cost of the put but if it declines your capital losses are offset by the gains from exercising the put Strategy 2 quotShort Forward Positionquot In this case you agree to deliver however many units of the underlying that you own ignoring dividends at the forward price This locks in your return but that return must be the riskless spot rate if there is to be no arbitrage This is an example of a general principle to be discussed later but what it says is essentially common sense If you fully hedge insure risk the rate of return on your investment must egual the riskless rate Scenario You have a position in some underlying but want to generate some current income by trading off future upside potential Strategy quotCovered Call Positionquot In this case you write a call option which generates current income Via the call price If the value of the underlying increases in the future the underlying gets quotcalled awayquot from you and no gain on the underlying is realized but if prices do not increase too much then you have your current income and the call remains unexercised General Properties of Option Prices The following are general properties of option prices that do not depend on specific models Intuition is provided although in some cases the answer is fairly obvious The price of a call C0 is a increasing in the current price of the underlying increasing in SO b increasing in the interest rate increasing in IT c decreasing in the exercise price decreasing in X Fact a makes sense because the higher the current price the greater the chance that the ultimate price will end up above the exercise price Fact b makes sense because an increase in the interest rate reduces the present value of the exercise price which according to fact c makes the call option more valuable Finally fact c makes sense because a lower exercise price means that there is a greater chance that the call will end up in the money The price of a put P0 is a decreasing in the current price of the underlying increasing in P0 b decreasing in the interest rate decreasing in IT c increasing in the exercise price increasing in X The intuition of the put relationships for these three variables is exactly the same as that for the call The results are just the opposite An increase in the underlying price or a decrease in the exercise price means there is less chance that the put will end up being pro table Similarly an increase in the interest rate causes a decrease in the present value of the exercise price resulting in less chance that the put will end up making money for the holder The price of both puts and calls POand C0 are a increasing in the time to maturi increasing in T b increasing in the volatili of the future stock price These two relationships make sense because a longer time to maturity gives the holder of either a put or a call quotmore chancesquot that his or her option will end up being exercised Likewise higher volatility means that there is a greater chance of both very high and very low prices But the option holder is only going to exercise when it is in his or her interest so the holder of a call likes the fact that very high prices are possible and doesn t care about the low prices while the holder of a put likes the fact that very low prices are possible and doesn t care about the prices Valuation of Options You already know from putcall parity that if you know the price of either the put or the call you can determine the price of the other security given the same exercise price and maturity on the two options The lack of riskless arbitrage alone is generally not sufficient however to tell what you what the price of say the call option should be in the market Another way to think of this is that some additional assumptions are needed to tell you exactly l v the five factors discussed above interact to determine the price of the option given that there is to be no riskless arbitrage For purposes of valuation therefore you need a specific model of how stock prices behave The two example models that are discussed in this section are a the so called quotbinomialquot model which assumes that the stock price can go either up or down by some percentage not necessarily the same over any one period of time and b the socalled BlackScholes model which assumes that the stock price changes continuously and over very short intervals of time the price changes have a normal distribution One way to think about this is to realize that if for any fixed length of calendar time think about from now until the option expires the binomial has a lot of periods make the periods short then prices in case a will be very close to those in case b using the BlackScholes model While BlackScholes forms the basis of a lot of the quotsophistic atedquot models used on Wall Street the mathematics is very complex for a class like 4000 So the binomial model will be developed in some detail first This will hopefully provide you with some sense of the important elements that go into more sophisticated formulas like BlackScholes which is covered after the binomial model Finally to keep things simple the examples will generally cover cases of nondividend paying securities Binomial g lption Pricing The most efficient way to leam this approach to option pricing is to look at an example where the maturity of the option is two periods which by convention in these notes is two years Why two you might ask Well once you understand how to solve the two period problem you basically understand the quotalgorithmquot that must be gone through to solve for option values in cases where the maturity is longer than two periods On the ip side one period problems are a breeze since as you will see you will have already solved two quotone periodquot problems in order to find the value of the option with a maturity of two periods The basic approach to solving for the current arbitrage free value of an option involves starting at the end of the problem and working your way back to today39 in this case starting at quotdate 1quot and working your way back to quotdate 0quot today all the time requiring that there be no riskless arbitrage over the next period no matter how the stock price should move Now comes some additional notation quotampO Given a time to maturity of T2 and an interest rate r1 r2 r1 r assume the term structure is flat for now assume that over any one period of time one plus the rate of return on the stock recall that this is the w return will be either quotuquot or quotdquot with d lt lr lt u where lr is of course just the gross rate of return from holding the riskless security for one period This means that since the security pays no dividends the price next period will be the price this period multiplied by either u or d This means that if the price at date 1 is S1 then the price at date 2 will be either S2 S 1u or S2 S 1d But using the same reasoning if S0 is the current price of the security then the price at date 1 must be either S1 S0u or S 1 S0d So for the two period problem you now know what every possible security price and call price will be over the next two periods39 Speci cally they will be Date Price of Stock Price of Call 0 so C0 1 s S0u or s sod C g c 2 s s u S0u2 g s S0du C g C g s 81d Soudg s s0d2 C g C where for example C39 is the value of the call if the stock price rst goes up at date 1 and then goes down at date 2 Notice also that Podu Poud so there are really only three possible stock prices at date 2 Keeping in mind that the price of the option at maturity date 2 in this case is equal to it s 90 intrinsic value the option price at date T will always be either 0 or ST X where X is the exercise price Sometimes this will be written as CT MAXO ST X For this problem T2 so the value of the option at date 2 is C2 MAXO S2 X Below are the general quotstepsquot that you need to go through to solve for C0 the current price of the option One period example Before solving for the more complicated two period case lets look at the problem when there is only one period to maturity Tl Here are the steps A One call option m B quotHOquot shares of the stock H0 will turn out to be less than or equal to l C Borrowing or lending at the risk free rate so that the net investment from AC is zero Next choose HO so that the cash ow from your position is the same whether the stock price goes up or down In general the cash ow will be C1Hosl C0 HOSOHT So if the stock price goes up the cash ow will be C H0S C0 HOS0lr and if the stock price goes down the cash ow will be C39 HoS39 Co HoSo1f Setting these two equations equal to one another and solving for HO yields H0 C1 C39s 839 By choosing the hedge ratio in this way the cash flow at date 1 is riskless Moreover the initial investment as you will recall is equal to zero Therefore the cash ow at date 1 regardless of whether the stock price goes up or down must also be equal to zero This is how we can solve for the option price at date 0 So the option price at date 0 is given by either C0 Cl HOSlr Hos0 91 Or C0 C39 HOS39lr HOSO Either equation will work equally well and give the same answer Example Suppose that S0 10 r 05 X 8 u 15 and d 5 Then S S0u 1015 15 and S39 Sod 105 5 Moreover C s X 15 8 7 and Cquot 0 since the option is not exercised at a price of 5 Then the hedge ratio is given by HO C1 C39S S39 7Ol55 7 Therefore the current price of the all option is given by C0 C HOSlr HOSO 7715l05 710 C39 HOS39lr HOS0 O 75l05 710 367 Two period case It is possible but a little more complicated to extend this example to the two period case Recall that at date 2 there can be one of three prices SH S39 ST or Squot with corresponding option values of CH CT 39 CT or Cquot So now the hedge ratio at date 1 H can take on two values The first is if the stock price goes up and is given by Hf C C39S ST or Hf Ci Cquot S39 Squot So using our earlier results we can find the values of C and C39 and then work our way backwords to find Co In particular we have that C CH H1S1r H1S C39 HimHr His and C39 0 H139S391r HI39S39 Cquot Hl39Squot1r HI39S39 Finally we can use the earlier formulas to calculate C0 Example Let s continue with the example we used earlier In this case we have that S s u 1515 225 st s u 515 75 5 s d 155 75 and Squot 8d 55 25 Similarly we have that C 225 8 145 and CT CT Cquot 0 since the option will not be exercised in any of these 92 cases So Hf 0 C39s 5 145 0225 75 9667 while H139C39 CquotS39 Squot 0 075 25 0 So C 0 H1S1r Hfs 145 9667225105 966715 C39 H1S1r H1s 0 966775105 966715 7595 C39 0 H139S391r HI39S39 OO105 05 0 Finally we have that H0 7595 O155 7595 C0 C HOS1r Hos0 7595759515105 759510 C39 HowHT Hos0 0 75955105 759510 39783 93 94
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