Financial Mathematics Problems
Financial Mathematics Problems MATH 3615
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Derivatives info 7 942008 FORWARD CONTRACTS A forward contract is an agreement between two parties a buyer and a seller stating that on a specified future date the buyer commits to buy from the seller and the seller commits to sell to the buyer a specified amount number of shares of a particular security at a specified price The forward price of a security is typically equal to the security s current price increased by the amount of interest that can be earned on that price between the date that the forward contract is agreed to and the forward contract s expiration date ie the date when the buyer will buy the security from the seller The following two examples demonstrate thatthe forward price for a security will be approximately equal to the security s current price increased by interest for the time period until the expiration purchase date Suppose that the current price of a stock is 80 per share and that two individuals both believe that the price will increase One thinks the price will be 120 in one year The other thinks it will be 150 They agree to enter into a forward contract at a forward price of 135 the person who expects the price to be 135 being the seller and the person who expects a price of 150 being the buyer under the forward contract The seller reasons If I m right and I believe I am I will get to sell a 120 stock for 135 and make a 15 profit I can buy the shares at the market price of 120 and sell them to the seller for 135 Correspondingly the buyer reasons If I m right about the future price of this stock and I believe I am I will get to buy a 150 stock for only 135 so I ll make a 15 profit I can buy the shares from the seller at 135 and sell them at the market price of 150 Q What s wrong with this scenario A The seller might then have a second thought Hey wait a minute I can just borrow 80 per share to buy the stock now and it won t matter what the price is in one year At that time I ll sell the stock to the buyer for 135 and repay the loan 80 plus one year s interest and make a profit of about 50 per share And I don t even care what the market price is in a year My profit is locked in This is an example of arbitrage which is making money in a financial transaction without taking a risk It can be done only when prices are somehow out of balance as in this case where the agreed on forward price was inappropriately high The buyer may also have a second thought and say to himself Hey why should I pay 135 to buy the stock in a year I could borrow 80 now and buy it I won t have to pay anything immediately And a year from now when I repay the loan with interest it will cost me 85 or so rather than 135 So for a cost of 85 one year from now I can own the stock Why would I want to pay 135 one year from now If both the buyer and the seller in a forward contract think that the stock s price will go up then they should both buy the stock now at its current price They should not wager with each other over who can more accurately estimate how much money they could have made if they d had the good sense to buy the stock immediately Now consider the opposite situation Suppose that the agreed on forward price is inappropriately low For example the two investors agree that in one year the buyer will buy the stock which today trades at 80 from the seller for 50 In that instance the buyer should sell the stock short at 80 now Here s the process for the buyer Borrow shares from another investor for a year sell those shares and invest the proceeds of the sale in a risk free interest bearing security e g a Treasury bill At the end of the year receive the maturity value of the interest bearing security 80 plus interest and buy the stock from the seller at the forward price of 50 with 30 plus interest left over as profit an arbitrage profit The stock that was purchased for 50 is then given to the investor who lent the shares for the short sale It is left for the student to reason through what the seller s reasoning is and what action the seller should take instead of entering into the forward contract to sell the stock for 50 in one year The point of the above analysis is this The forward price for a security will tend to be approximately equal to the security s current price plus interest for the period between the current date and the expiration date transaction date of the forward contract ie interest for the term of the contract If the forward price is significantly different from this price then there will be an opportunity for arbitrage As clever traders arbitrageurs take advantage of the mispricing of the forward contract it will affect the supply and demand for the contract and the forward price will adjust until it is approximately equal to the current price plus interest for the term of the contract Payoff Diagrams and Pro t Diagrams A payoff diagram for a derivative is a graph of the payoff from that derivative as a function of the price of the underlying security Similarly a profit diagram is a function of the profit from the derivative as a function of the security s price In the case of a forward contract the payoff and the profit are equal because the only cash ows occur at the expiration date delivery date of the contract Consequently the two types of diagrams are identical They buyer makes a profit ie receives a positive payoff if the market price is higher than the agreed on forward price on the expiration date of the forward contract Conversely the buyer loses money if the market price is less than the forward price The payoff or profit diagram plots the payoff profit as a function of the security s price on the expiration date It shows the possible payoffs if the buyer buys the security at the forward price on the expiration date and immediately sells it at the market price Buyer39s Payoff amp Profit 10000 7500 5000 2500 000 0 1700 3400 5100 8500 10200 11900 1360015300 17000 25001 Payoff at Delivery 5000 7500 10000 Spot Price at Delivery The same concepts apply to the seller under the forward contract except that the payoff profit is positive if the market price is lei than the forward price Here is the payoffprofit diagram for the seller Selle r s Payoff 81 Profit 10000 7500 5000 2500 000 00 1700 3400 5100 6800 8500 2500 0 11900 13600 15300 17000 Payoff at Delivery 5000 7500 10000 Spot Price at Delivery University of Connecticut Math 3615 Financial Mathematics Problems Fall 2008 Summary Module 4 BONDS De nitions F 2 face value or par value r 2 coupon rate per coupon payment period Fr 2 coupon amount n number of coupon payment periods remaining until redemption date 139 2 effective interest rate yield per coupon payment period based on the bond s price v 1 1i RV redemption value ACTEX manual uses C for redemption value g 2 coupon rate based on Redemption Value g CPN 2 coupon Formulas for the price of a bond on a coupon date in all cases if RV F Concept PVRV PVCPNs Basic formula RVvquot Fro RVvquot g oil Fvquot Froil Premium Discount Formula F RVRVrW lonj FFr zog RVCPN RVioa FCPN Fioa 1 r F r Makeham Formula K F K K F K 139 RV i where K Fv PVRV For a bond purchased between coupon payment dates Total price P0 11t 2 Price on prior coupon date accumulated to settlement date with compound interest P0 2 price on prior coupon date 139 2 effective interest rate per coupon period t 2 fraction of coupon period between prior coupon date and settlement date 2 days between prior coupon date and settlement date days in coupon period Note that this calculation may be based on actual days or on a 360 day year 1 Makeham s formula is not normally used where F RV i Bond prices are typically quoted excluding the accrued coupon Thus the quoted price is equal to the above calculated Total price less the amount of the accrued coupon The amount of the accrued coupon equals the coupon payable at the next coupon date times the fraction of the current coupon period that has elapsed prior to the transaction date ie the accrued coupon is calculated using simple interest methods Price excluding accrued coupon P0 1it 7 Coupon t Terminology for bond prices Term for the Price Corresponding Term for Price including accrued coupon excluding accrued coupon Total sale price Price Flat price Market price Premium plus accrued True price Dirty price Clean price Amortization of Premium or Discount2 in a Bond s Price Amount of premium or discount amortized in kth period 2 Fri Wk Note that the amount by which the bond s price changes during a period if the interest rate remains constant is equal to the change during the prior period times 1i3 If the bond s market price exceeds its par value because the bond s coupon rate exceeds the market interest rate then it will be called at the earliest possible call date unless market interest rates rise Exception If there is a call premium that exceeds the bond s current market premium ie if the issuer would have to pay more than the market price for the bond then it will not be called If the bond s market price is less than its par value because the bond s coupon rate is less than the market interest rate then it will not be called before maturity unless market interest rates fall Technically the premium in a bond s price is said to be amortized over the life of the bond but the discount in a bond s price is said to be accrued over the bonds life However the ACTEX manual uses the term amortized for both premium and discount In either case premium or discount the difference between the bonds current value and its par value decreases over time reaching 0 at maturity and the amount by which the premium or discount changes in a given period equals li times the change in the prior period The practical use of this concept of amortizing a bond s premium or discount lies in determining the bonds book value for accounting purposes The reported value must be consistent with the purchase price on the purchase date and it must be consistent with the maturity value on the maturity date Book values calculated in this way ie at an unchanging interest rate do not represent market values and the bonds value on the company s balance sheet the bonds book value will generally not match the market value of the bond except on the date of purchase and the maturity date unless the market interest rate for the bond happens to match the interest rate that the bondowner is using to determine the book value Interest Functions d 12 1 ltamp i di amp 6 1n1 i Accumulation function an 1iquot eaquot for constant force of interest fame an e for variable force of interest Definition of force of interest a t dat dt 0 d0 6t Nominal interest rates 1 1 m1i 1 ion quot H J 1i m i d m1 v m quot l d J1dV m m m l d J 1i m Level Annuities n 1iquot 1 s a 1l m m gt 1 l vquot am d 1ld S 1iquot 11i 1 11 i m d 1ism1iquot m1iquot1 am 1 vquot m m m m m m amt iv ammmm Slrmlarly for da sa sa usez 0rd 1n denomlnator 1 a Slrmlarly for SE use 6 1n denomlnator Perpetuities a 1 a 1 E 1 Q i Q d 3 6 1 1 am 39 am Increasing and Decreasing Annuities Arithmetically increasing and decreasing annuities 39a39 nv maz f mm 1 1 1quot amh9 Dom Duh 1 i nli sm 1 For annuities due 1mm 13 1mm and 19m multiply the corresponding annuity immediate formulas by 1i or change the denominator from i to d For increasing or decreasing annuities payable more than once a year luff etc change the denominator from i to 1quot For arithmetically increasing or decreasing annuities with terms P PQ P2Q P3Q Pn 1Q i i i i aT wv 1fnisf1mte PamQ z i l l l l P if n is infimte fg z z Geometrically increasing or decreasing annuities Each payment changes by a factor of 1r 11r 11r 7 00 r 00 i r d vr 1i 1r 1i 1r 1 7 d vr w 1 D z r d vr Loan Repayment Loans with n level payments Amount of principal repaid in kth payment Pmtvquot1 k Note If the borrower made an n1St payment the principal portion would equal the entire payment Pmtvquot1Hquot1 Pmt the loan balance would go from 0 to Pmt Thus to solve for the term of a loan determine the number of the payment for which the principal portion equals the entire payment this represents the n1St payment so subtract 1 from the number of this payment to find n eg if it is the 26 h payment then it is a 25 payment loan Loan Balance immediately after kth pmt assuming level payments 0 by retrospective method 2 PVRemaining Payments Pmt um by prospective method 2 Accum d value of initial principal less accum d value of payments to date IMO E Balo 1ik fl m Balo 1ik Pmts Balo 1ik a m Loans with sinking funds Sinking Fund DepOSit where j is the interest rate on the Sinking fund s 5 1 Total payments during a period i Li 2 sinking fund deposit plus interest on loan s 5 1 SH SF Balk SFDs J L S a Principal paid in kth payment 2 growth in sinking fund kil smug sm SFD1 Net interest 2 interest accrued on loan principal during current period less interest earned on sinking fund during the period Li SF Bank1 j This also equals the total amount paid less growth in the sinking fund du1ing the year SFD Li SF Bank SF Balk71 SFD Li SFD 1 jk 1 Bonds Assuming the bond s maturity value 2 face value F Price F vquot F rum 2 PV of maturity value PV of coupons F F r ium 2 face value plus premium or minus discount if r lt i KF K where KFvquot 1 Amount of premium or discount amortized with kth coupon payment F r i 11 Price between payment dates t no days since last couponno days in coupon period i 2 yield rate effective interest rate per coupon period Total price including accrued interest 2 Price on prior coupon date 1i B 1 i Note that total price accumulates with compound interest at the yield rate Accrued interest 2 t F r t times amount of coupon Actually this is the accrued portion of the next coupon payment but it is referred to as accrued interest Note that coupons accrue at simple interest at the coupon rate Market Price1 Total price 7 Accrued interest B 1 i t Fr If the market price exceeds the bond s par value because the bond s coupon rate exceeds the market interest rate the yield rate the bond will be called at the earliest possible call date unless market interest rates rise Exception If the call premium exceeds the bonds current premium calculated based on the bonds maturity date and the current market interest rate it will not be called If the market price is less than the bond s par value because its coupon rate is less than the market interest rate it will not be called before maturity unless market interest rates fall 1 Price or market price means the price of the bond excluding accrued interest The buyer must pay the seller the total price market price plus accrued coupon but the bond s price is quoted without the accrued interest Buyer beware Spot rates and Forward rates The n year spot rate is the accumulation rate expressed as an annual rate for an n year investment an n year zero coupon bond The symbol for this rate is s in Module 6 and is 1P0n in Module 14 of the ACTEX manual The n year accumulation factor based on the spot rate is an 1 squot quot The n t year forward t year rate is the rate that will apply during the t year period from nt to n It can be calculated by comparing the accumulation factors for n t years and for n years 1 im n laF S lim or 1 squot quot 1 sm39H 1 inf n Sm The latter formula states that the accumulation factor for n years can be regarded as representing accumulation at s for n years or accumulation at SW for n t years followed by accumulation at in for t years Forward rates are commonly quoted as n year forward rates which means the n year forward one year rate which is the one year rate that currently applies for the 1 s 1 1 period between n and n1 1in1 1 Sn quot Duration Convexitv and 39 i Z t v CE Macaulay duration D t 2 v 02 Modified Duration DM dP dl i P 1 i lam Duration of a level payment loan D a E Duration of n year zero coupon bond D n Duration of coupon bond with face F coupon Fr and redemption value C FrIamnCv FrIamnCv FramCvquot BondPrice D Approximations of a security s change in price due to a change Ai in interest rate 1St approximation AP DM PiAi where Pi 2 price at yield rate 139 v v 2 2quot 1 approximation AP DM Pi Ai Convexity W Criteria for immunization2 A portfolio is immunized if the assets and liabilities are equal in value and in duration but the assets have greater convexity PVassets PVliabilities at i0 2 A 11 Z L 11 Duration assets 2 Duration liabilities ZtAt 11 Z L 11 Assets have greater Convexity th At 11 gt th L 11 2 Immunization describes the situation Where if the value of a company s assets equals the value of its liabilities then after a small change in the interest rate either up or down the value of the company s assets Will exceed the value of its liabilities This Will be the case if the assets and liabilities have the same duration but the assets have a greater convexityi University of Connecticut Math 3615 Financial Mathematics Problems Fall 2008 Summary Module 7 ASSETLIABILITY MANAGEMENT DURATION AND IMMUNIZATION Duration or Macaulay Duration D MacD or 6 1 The duration of a stream of payments is the weighted average of the number of years until each future payment The weight applied to the number of years for each payment is a fraction equal to the present value of that payment divided by the total present value of all the payments Note that duration depends on not only the amounts and dates of the payments but also the interest rate at which the present values are calculated Alternate description Duration is the rst moment about t0 of the present values of a set of future payments divided by the total present value of the payments Modi ed Duration DM ModD or 17 The modified duration of a stream of payments is the duration as defined above divided by 1i The significance of modified duration is that it represents the proportionate percentage change in the payments present value as the result of a change in the interest rate used to value the payments a duration of 1 implies that the percentage change in value equals the number of percentage points by which i changes Note that the sign of the change in value is opposite to that of the change in i DM i i0 1 i P Where P is the price or value of the stream of payments at interest rate 1 Convexity For a bond Convexity modified convexity actually measures the change in an investment s duration due to a change in interest rate ConV 129 i For any single payment occurring in n periods eg an nyear zerocoupon bond Dn DMn1i ConVnn11i2 For a series of level payments an n year level annuity immediate Ia Ia D DM ail 1 i a FrIaa nCvquot FrIam nCvquot D DM 1 i Bond Price Bond Price Note If C redemption value 2 Face and if r coupon rate i then D quota and DM ail For a portfolio of investments the Macaulay or modified duration of the portfolio equals a weighted average of the Macaulay or modified durations of all of the investments in the portfolio with each investment s weight equal to its proportion of the total present value of the portfolio1 Approximating the change in Price for a given change in interest rate PiAi2 AP z DM PiAi or With convex1ty AP z DMPlAlC0n1T Note that we multiply the negative of the Modified Duration by the change in i but we multiply the Convexity not its negative by the square of the change in i and divide by 2 Immunization Consider a portfolio of investments that supports a set of liabilities where the present value of the assets equals the present value of the liabilities The combination of the invested assets and the liabilities is said to be immunized if a small change a parallel shift in the interest rates for both assets and liabilities results in a positive change in the present value of assets minus liabilities This means that the current set of interest rates represented by i0 produces a relative minimum for the value of PVA PVL ie dPVA di liq dPVL di liq and dzPVA ldi2 gt dzPVL ldi2 We can test for this condition by examining the present values of the assets and the liabilities and the first and second moments of their present values Then the requirements are 2 Av 2 Av 2 21 th Av u gtZt2 Lv u Technical note The second expression here lSt moments is the Macaulay not modified duration if the Macaulay durations are equal then the modified durations are also equal The third expression 2 moments is not the convexity convexity uses coefficients of ttl not t2 but if asset and liability durations are equal and the assets 2 moment is greater than the liabilities 2nd moment then asset convexity is greater than liability convexity A portfolio is said to be fully immunized if PVAgtPVL for any positive interest rate i 2 i0 Stocks A common method on exams for valuing stocks is to calculate the present value of future dividends at an appropriate interest rate Two conceivable cases are as follows and there can be many variationscombinations of these Level dividends Stock Value D a D i Geometrically increasing dividends Stock Value D as D i g Note that in each case D is the amount of the next dividend which is due in 1 year CDs Certificates of Deposit are offered by banks and similar institutions They are equivalent to a zero coupon bond and are federally insured 1 Note that the investments may be valued at different interest rates If we use modified duration and convexity to estimate the change in a portfolio s value due to a given interest rate change we are assuming that all interest rates will change by the same amount ie that there is a parallel shift in the yield curve University of Connecticut Math 3615 Financial Mathematics Problems Fall 2008 Summary Module 5 YIELD RATE OF AN INVESTMENT To find the IRR for a series of cash ows solve the following equation for v C0 C1vC2v2 Cnv 0 Then ERR UV 7 1 The BA 11 Plus calculator has 3 functions for calculating IRR TVM keys 7 CFs PMT must be level except for first and last CFs PV amp FV Bond workbook 7 CFs must follow the pattern for a bond CF workbook 7 works for any pattern of CFs spaced at equal time intervals A calculated IRR is unique if the series of CFs has only one sign change If there is more than one sign change in the CFs then an IRR is unique if the accumulated value of the CFs at all times using the calculated IRR has the same sign as the initial CF The accumulated value on any date is equal to the accumulation to that date of all prior CFs The accumulated value of the first k CFs will be positive if the sum of the first k terms in the above series is positive See the series shown in the first sentence on this page Net Present Value or simply present value is the value of a series of CFs as of a specified date on or prior to the date of the first CF calculated at a specified interest rate Assuming that NPV is calculated on the date of the initial CF and that the CFs are equally spaced the following formula applies NPVC0 Clv szz Cnv Note NPV 0 if v ERR is the internal rate of return for the cash ows C0 to Cquot 1 1 in Timeweighted return is calculated as follows Assume there are n consecutive time periods and the rate of growth during the kth period is jk ie the amount invested grows by a factor of 1jk Then 1 H1jk k1 Notes regarding time weighted return 1 The time periods do not have to be of equal length 2 jk is the effective interest rate for the kth time period not the effective rate per year but the effective rate for the length of time in the kth period 3 As calculated here im is an effective interest rate for all n time periods combined If the total time call it tTOT is exactly one year in most problems it will be exactly one year then im is an effective rate per year Otherwise the effective annual rate can be calculated as i1iTW1 TOT 1 Dollarweighted returns are calculated using the following steps Step 1 Calculate interest earned ending balance 7 beginning balance 7 net deposits where net deposits 2 total deposits less total withdrawals Step 2 Calculate exposure in dollar years To determine the dollar years of exposure contributed by each cash ow multiply the amount positive or negative of the cash ow by the length of time from its occurrence until the end of the period Add up these exposures to determine the total exposure Examples where the total period is one year in length Initial balance at time 0 X 1 Deposit at time t1 X 1 7 t1 Withdrawal at time t2 X 1 7 t2 Step 3 The dollar weighted return 2 interest earned total exposure Note that the dollar weighted return is automatically on a peryear basis since we divided dollars of interest by dollaryears of exposure It is not however an effective rate since we used a simple interest methodology to calculate it The Investment Year and Portfolio methods are alternative approaches for determining the amount of interest to credit to each individual who has invested money in an intermediary s e g an insurance company s portfolio of fixed income investments Portfolio method Divide the total interest earned during the year by the total assets that were owned ie were in the portfolio at the beginning of the year in order to determine the portfolio interest rate for the year Apply this portfolio rate to the amount of each individual s balance at the beginning of the year Note that adjustments must be made for deposits and withdrawals during the period probably using the dollar weighted return method Investment Year method Determine the interest rate earned during the current calendar year on the assets owned at the beginning of the year But do a separate calculation for the assets that were acquired in each prior calendar year determining a different investment year rate for the investments of each calendar year Then the investment income credited to each individual investor is determined by applying the appropriate investment year rate to the amount invested by that individual in each prior year