ADV ENGINEER ECONOMY
ADV ENGINEER ECONOMY EIN 6357
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This 23 page Class Notes was uploaded by Tod Murray on Friday September 18, 2015. The Class Notes belongs to EIN 6357 at University of Florida taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/206887/ein-6357-university-of-florida in Industrial Engineering at University of Florida.
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Date Created: 09/18/15
Retirement Accounts Government and employer sponsored retirement accounts such as 401 ks and IRAs are the best ways to save for your retirement goals Depending on your situation you can choose from For almost everyone IRAs Regular Spousal Roth For employees at larger organizations 401k 403b For small businesses and the selfemployed SEP IRAs Keoghs SIMPLEs In this section we describe Why retirement accounts are such powerful investment vehicles for retirement savings The primary types of retirement accounts including 401ks the various types of IRAs and options for the selfemployed Why Retirement Accounts Are So Powerful Government and employer sponsored retirement savings accounts are the most powerful retirement vehicles available because they combine the benefits of 4 of the most effective principles of saving Compounding interest Consistent saving Reduction in taxable income Taxdeferred growth Compounding interest THE POWER OF COMPOUNDING Earnings Beyond investment 19 25 Age as mum I m r mm Au 5 Klm 19 Ennn zs zhnn 2nnn s5 2nnn Yolal lnvhslmhnl tshnn saunnn By Investing early agzs 1 925 to 122 ME paw2r o f compounding workfm her 342 mvzsts 15 as much as Ktm but ms 25 mmz than Ktm 39 39 nu Udll OVe39l mp L L 39 your money 4 the mme nu will enjoy the bene ts WI nnn invpdpd a I 3000 This may sound ne but with that same investment compounding at 10 interest you would end up with 5730 almost 7 times your original investment Consistent saving Steady progress towards retirement savings even if small and steady has proven to be the most successful approach to saving for retirement Why The reason isn t financialpeople simply tend to put off saving for retirement and thereby let valuable time pass when their money could be compounding Compounding is just as effective on small amounts of money as it is on large amounts Reducing taxable income One of the best reasons to use retirement accounts some but not all offer this benefit is that those contributions reduce your taxable income In other words you don t have to pay taxes on the amount you invest This may not sound all that significant but if you consider that you don t have to pay tax on a 2000 contribution to your retirement account and your effective tax rate is 28 you simultaneously save 560 in taxes and have that much more money compounding for you Taxdeferred growth Much like reducing your taxable income taxdeferred investments are given special tax status With retirement investments you have to pay capital gains tax when you sell investments for more than their purchase price You also pay tax on any dividends received In taxdeferred retirement accounts however your money is allowed to grow taxfree so you only have to pay taxes when you withdraw funds from your account Since taxes don t take a bite while your money grows the acceleration of compounding is maximized Retirement Vehicles Individual retirement accounts IRAs were created by Congress to help individuals supplement their social security and pension benefits during retirement There are several types of IRAs traditional Roth Spousal that have significant differences but they all allow for taxdeferred compounding have an annual contribution limit of 2000 and have certain income limits How do these IRAs differ Roth and Spousal IRAs were created in 1998 to offer more exible alternatives to the regular IRA Spousal IRAs These allow spouses to take full advantage of IRAs regardless ofthe other spouse s retirement arrangements A fulltime homemaker is now eligible for an IRA contribution of up to 3000 even if he or she doesn39t generate income Roth IRAs These IRAs were designed primarily for people with qualified company retirement plans or whose income exceeds the deductibility limits of traditional IRAs Contributions made to Roth IRAs are not taxdeductible but when you withdraw your money in retirement you will not have to pay taxes You can withdraw the amount invested at any time without penalty or tax The earnings however are subject to penalty and tax This flexibility is very attractive to many savers Unlike traditional IRAs Roth IRAs don t have a mandatory withdrawal in retirement This makes Roth IRAs attractive for passing taxfree wealth to heirs Employee retirement accounts 401k 403b What exactly are these In a sense these are personal pensions In the past large corporations administered defined 401 k and 403b benefit plans or in other words pensions that guaranteed a certain level of income after you retire Now 401k plans are becoming more common because pensions are costly to administer and people are changing jobs more often Instead of defined benefits 401k plans have defined contributions Employers define how much they will contribute upfront but make no promises about what you will have when you retire You administer the funds in your 401 k plan and are responsible for making sure that they are invested appropriately to provide enough money for you during your retirement 403b plans are similar to the 401ks but are designed for publicsector employees Certain 401k plans are better than others but they are all sound retirement vehicles Why Contributions are taxdeductible Investments grow taxdeferred The annual contribution limit is high 10000 Companies that offer 401k plans usually have a direct draw system so your contributions to your 401k are automatically withdrawn from each paycheck so you can effortlessly and steadily save Employers make matching contributions The typical match is between 50100 for between 36 of your salary The best reason of all to take advantage of a 401 k plan is the matching contributions from your employer See this example and the tremendous savings that are created Situation annual salary 60000 401k saving rate 10 6000yr Company match 50 3000yr Investment return rate 8 our For Small Businesses and the SelfEmployed The retirement savings plans that big companies offer like the 401k are wonderful but their complexity and cost make them inaccessible for small businesses not to mention the selfemployed To help both groups the government created special retirement savings plans such as Keoghs SEP IRAs SIMPLEs In essence these retirement accounts allow you to set up a personal pension Like IRAs the contributions you make to these plans are taxdeductible and cannot be withdrawn before retirement age without penalty What distinguishes these retirement vehicles are their much higher contribution limits39 depending on the type of account you can invest between 6000 and 30000 per year The drawback to these plans at least if you have employees is that employers must contribute the same percentage of their salaries to their employee s accounts as they do to their own accounts Table 4S 1 FINANCIAL FUNCTIONS IN EXCEL Financial Function Irregular Cash Flow Analysis Net Present Value NPVRateValues Internal Rate of Return IRRValuesGuess Single Payment Compounding Future Value FPin FVratenperpmtPVtype Present Value PFin PVratenperpmtFVtype Effective Interest Rate RATEnperpmtPVFVtypeguess Periods to reach FV NPERratepmtPVFVtype Uniform Series Annuities Annuity Payment APin PMTratenperPVFVtype AF in PMT ratenperPVFVtype Present Value PAin PVratenperpmtFVtype Future Value FAin FVratenperpmtPVtype Annuity Rate RATEnperpmtPVFVtype guess Yield Bond YIELDSee Excel Help screen Annuity Term NPERratepmtPVFVtype Periods to reach FV NPERratepmtPVFVtype Uniform Loan Payments Uniform Loan Payment PMTratenperPVFVtype Remaining Balance PVrateRemainjeriodspmtFVtype Interest in Period IPMTratepernperPVFVtype Principal in Period PPMTratepernperPVFVtype Depreciation Double Declining Bal DDBCostSalvageLifePeriodfactor Straight Line SLNCost Salv ageLife SumofYears Digits SYDCostSalvageLifePeriod Logic and Miscellaneous Functions Conditional IfThen IFlogicalitestvalueiifitruevalueiififalse Horiz Lookup Table HLOOKUPsee Excel Help Screen Vert Lookup Table VLOOKUPsee Excel Help Screen Hints on Spreadsheet Development Whenever possible input all costs revenues interest rates and so forth as separate cells rather than imbedding them in formulas Enter data and analysis formulas in compact areas preferably organized vertically for efficient recalculation Build your worksheet to minimize the number of active columns since each column activates up to 512 rows Use Page Down to access additional rows Do not format blank cells because this procedure makes them active Avoid using the button in the upper left hand corner of the spreadsheet to select and format an entire worksheet in order to change some property Use boxes andor text and cell colors to highlight input data cells rather than using white space to isolate these sections This technique makes efficient use of the Home screen Values use less memory than formulas If the output of a formula is not expected to change the formula can be deleted by using a values only command Summary FixedIncome Securities From the viewpoint of the M you provide them with a type of fixedincome annuity when you take out a mortgage This chapter is about various types of fixedincome annuities the most important of which are bonds Financial Instrument a promise of payment Security a financial instrument that can be traded freely and easily in a welldeveloped market Fixedincome security a financial instrument that is traded in a welldeveloped market and promises a fixed definite income to the holder over a span oftime Such a security represents ownership of a definite CFS Remark Bonds are wellknown fixedincome securities They represent the major investment alternative to stocks for individuals US Government F l Securities to Index ofthe Site U S Treasury bills are issued in denominations of 10K or more have terms to maturity of 13 26 and 52 weeks are sold on a discount basis bill with face value of 10 K may sell for 95K can be redeemed for the full face value at maturity Comments U S Treasury bills El are offered weekly El are highly liquid can be sold easily El can be sold prior to maturity date U S Treasury notes El have maturities of1 to 10 years El El are sold in denominations of 1 K or more can be purchased by individuals dealing directly with a FRB Federal Reserve Bank eg Jacksonville El can be sold at a charge Owner of note receives a coupon payment every 6 months until maturity Origin of expression clipping coupons Coupon payment represents an interest payment of fixed magnitude throughout the life ofthe note At maturity the note holder receives the last coupon payment and the face value ofthe note Purchaser has option to renew the note at maturity at the rate prevailing at that time U S Treasury bonds 1 are issued with maturities of more than 10 years 2 are similar to Treasury notes they make coupon payments 3 are not similar to Treasury notes in that they may be callable A note is callable if at some scheduled coupon payment date the Treasury can force the bondholder to redeem the bond at that time for its face value U S Treasury strips El are bonds issued in stripped form Each ofthe coupons is issued separately as is the principal Example 10year stripped bond has 20 semiannual coupon securities each with its own ID number CUSIP and also the principal security Each security generates a single cash flow There are no zero intermediate coupon payments This type of security is called a zerocoupon bond Note the strips could be traded individually whereas coupons of a note could not be so traded Meaning of F Security Originally this was a security paying a fixed welldefined CFS to the owner The only uncertainty about the CFS payment was default by the payer Nowadays fl securities promise CFS s whose magnitudes are tied to various contingencies or fluctuating indices Examples ARM Adjustablerate mortgage Payment levels may be tied to an interest rate indi Corporate bond payments may be in part governed by a stock price General Idea Fl Security a security with a CFS that is fixed except for variations due to well defined contingent circumstances Money Market the market for shortterm 1 year or less loans by corporations and financial intermediaries such as banks This is a wellorganized market designed for large amounts of money Commercial Paper Unsecured loans loans without collateral in the money market made to corporations Example Banker s Acceptance Company A sells goods to company B Company B sends a written promise to A to pay for the goods within a fixed time A bank accepts the promise ifthe bank promises to pay the bill on behalf of company B Company A can then sell the acceptance to someone else at a discount before the fixed time has expired ACOMA Visualization of Banker s Acceptance A gtB goods promise of payment in 3 months accepts promise sell acceptance at discount before 3 mo s C A gets most of before 3 mo s A gets money from C bankthen pays C B pays bank at some point The acceptance amounts to a promise of payment and so has value Would you pay somebody 900 today if a bank promises to pay you 1000 in a year Eurodollar deposits Deposits denominated in dollars but held in a bank outside the US Eurodollar CBS CBS denominated in dollars and issued by banks outside the US Banking regulations and insurance may be different from in the US A U S company might prefer to trade in Eurodollars to avoid problems with currency fluctuations Other Bonds Bonds are issued by agencies ofthe federal government by state and local governments and by corporations Municipal bonds issued by agencies of state and local governments 1 general obligation bonds backed by a governing body eg the state 2 revenue bonds backed either by the revenue to be generated by the project funded by the bond issue or by the agency responsible Municipal bonds and related mutual funds based on them are popular with very wealthy investors The interest income is exempt from federal income tax and from state and local taxes in the issuing state Usually this means a lower interest rate compared to other securities of similar quality If you had a taxable bond paying 6 and were in the 31 tax bracket a municipal bond paying 1031 X 6 414 would be competitive Corporate bonds issued by corporations to raise capital for operations and new ventures They vary in quality depending on the strength of the corporation and features of the bonds Most corporate bonds are traded overthecounter in a network of bond dealers They are often not traded on an exchange and so may not be as liquid Indenture the contract of terms that comes with a bond Callable bonds A bond is callable if the issuer has the right to repurchase the bond at a specified price Usually this call price falls with time Often there is an initial call protection period when the bond cannot be called Sinking funds Rather than incur the obligation to pay the entire face value of a bond issue at maturity an issuer may establish a sinking fund to spread this obligation out over time Under such an arrangement the issuer may repurchase a certain fraction of the outstanding bonds each year at a specified price Debt Subordination To protect bond holders limits may be set on the amount of additional borrowing by the issuer Also the bondholders may be guaranteed that in the event of bankruptcy payment to them takes priority over payments of other debt the other debt is subordinated Mortgages Mortgages are bonds A home ownerwho takes out a mortgage sells it to bank or lending agency the mortgage holder to generate immediate cash to pay for the home The owner is obligated to make periodic payments to the mortgage holder Standard mortgage structure equal monthly payments throughout the term Early repayment is often allowed by comparison Standard bond structure final payment is equal to the face value at maturity earlier payments are for interest Because of allowing early repayment of a mortgage it is not completely fixed income to the mortgage holder Balloon payment mortgages modest sized periodic payments and then a final balloon payment to complete the contract Adjustablerate mortgages ARMs the interest rate is adjusted periodically according to an interest rate index Such mortgages do not generate fixed income Remark Mortgages are often bundled into large packages and traded among financial institutions In this sense they are securities even though they are contracts between two parties Mortgagebacked securities are quite liquid Annuities An annuity is a contract that pays the holder the annuitant money periodically according to a predetermined schedule or formula over a period of time Example Pension for the life of the annuitant Purchase price would depend on age of annuitant when purchased and number of years until payments begin Remark Sometimes interest can be earned until the payments begin lnterest accumulates tax free until the payments begin Postponing tax payments can make an annuity more attractive as an investment Annuities are not traded However they are considered to be investment opportunities available at standardized rates To the investor they serve the same role as other Fl instruments Perpetual Annuity An annuity that pays a fixed sum A periodically forever Useful for analytical purposes Actually exist in the UK Given A amount paid per period r the perperiod interest rate P the present value 2 3 4 forever P A1r A1r A1r A1r Note P A1r 11r A1r1 A lr2 A lr3 A1rP1rP AP1rPrPAP Perpetual annuity formula The present value P of a perpetual annuity that pays an amount A every period beginning one period from the present when ris the onegeriod interest rate is given by Annuity Formula Consider an annuity that begins payment one period from the present paying an amount A each period for a total of n periods Suppose the oneperiod interest rate is r and the number of periods is n The present value P ofthe annuity is given by P Ar Ar 1r 3 Equivalently A r 1rquot P1rn 1 Remark An easy way to handle the computations is to first compute the following term say knr knr 1lr 1r 1 rquot It then follows that P A X knr A Plknr P A1l1r1l1r2 11r3 11r4 1l1rquot A X knr Given n and r compute knr Then given A compute P Or given P compute A As a computational check knr should be gt 1 for r gt 0 Amortization The process of substituting periodic payments for a current obligation of amount A Given A knr compute P A X knr to get the periodic payment Example Take out a loan at 6 for 5 years to pay off a car purchases We amortize the cost of the car over 5 years Bonds Features of bonds Bonds 1 have the greatest monetary value among Fl securities 2 are the most liquid of the Fl securities 3 are very important as investment vehicles 4 are of theoretical usefulness Chapter 4 5 are very complicated in the details but basically simple Example of a 1000 9 Bond annual coupons 5 years You pay 1000 the face value or par value to the issuer or the seller At the end of years 12 3 4 and 5 you get coupon payments of 90 Also at the end of year 5 you get the face value 1000 CFS 1000909090901090 Bonds in the US often have a period of 6 months between payments In this case this 9 bond would have coupon payments of 45 The issuer sells the bonds to raise capital immediately and then is obligated to make the prescribed payments Usually bonds are issued with the coupon rates close to the prevailing general rate of interest As time passes bonds may trade at prices different from their face value due to exterior changes in interest rates or changes in perceived risk of default on payments The vast majority of bonds are sold either at auction or through an exchange organization Prices are determined by a market and thus may vary Table 33 illustrates US Treasury Bonds Prices are quoted as a percent offace value A price of 100 for a 1000 bond means a price of 1000 95 means 950 The indicated coupon rate is the annual The bid price is the price dealers are willing to pay for the bond The ask price is the price at which dealers are willing to sell the bond Prices are quoted in 32quotd s of a point An asked price of 103 3032 is shown in the table as 10330 For a bond of 1000 face value this means its price would be 103938 The yield is based on the ask price in a manner to be discussed Bond quotations ignore accrued interest which must be added to the price quoted to find the actual amount you pay for a bond Bonds are essentially Fl securities However they may default if issuer has financial difficulties or declares bankruptcy Bonds are rated by rating organizations to characterize the risk associated with them US Treasury bonds are considered risk free and are not rated A bond rating organization considers the following aspects among others in assigning a bond rating x ratio of debt to equity 2 ratio of current assets to current liabilities 3 ratio of cash flow to outstanding debt 4 various other ratios 5 trend in ratios over time Bonds with low ratings must have lower prices than bonds with high ratings Buyers will only accept more risk ifthey can expect to increase their earnings Yield A bond s yield is the interest rate implied by the payment structure The yield is the interest rate at which the present value ofthe stream of payments all the coupon payments plus the final facevalue redemption is equal to the current price The yield is termed yield to maturity to distinguish it from other yield measures Yields are quoted on an annual basis Example of a 1000 9 Bond annual coupons 5 years The yield is the number r r 009 so that 1000 10001r5 901r 90 lr2 90 lr5 The yield to maturity is just the internal rate of return ofthe bond at the current price With 2 payments per year for 5 years n 10 periods each of 45 the yield would be the number A 009 for which 1000 10001 11210 451A2 451A22 451A21 Generalization F face value of bond C total of yearly coupon payments m number of coupon payments per year P current price ofthe bond A the yield to maturity A is the number that satisfies P F1Amquot Cm1Am Cm lAm2 Cm1Amquot Bond Price Formula Assuming A gt 0 we can use the general value formula for annuities to collapse the equation P F1Amquot Cm1Am Cm lAm2 Cm1Amquot to P F1Amquot cm 1 11 Am Notes 1 A is not given but must be computed as in IRR computations 2 Excel has a Yield function Sensitivity Analysis PriceYield Curve How does price depend on yield ie how does P depend on A Reasons to study priceyield curve 1 improves qualitative understanding price yield coupon time to maturity 2 motivates ideas underlying bond portfolio construction 3 provides understanding of interest rate risk properties of bonds Boundary Case A 0 First write C as a percent of F then divide bond price formula by F and express in percent P F1Amquot Cm1Am Cm1Am2 Cm1Amquot F nCm add all the coupon payments to the face value to get the price Alternative Percentage Form 100 X PF 100 nm100 CF Price expressed as a percent of face value 100 nm Coupon payment expressed as percent of face value If say Coupon payment expressed as percent of face value 10 the bond would be called a 10 bond with a 10 coupon rate Remaining Case A gt 0 P F1Amquot CA1 11 Amquot coupon Divide by F multiply by 100 rate 100 PF 1001Amquot 100 CFA 1 11 Amquot Questions of interest x What happens to price as of face value as yield increases 2 At what rate does price as of face value change as yield increases 3 What is the effect ofthe value ofthe coupon as of face value on the priceyield curve 4 What is the effect ofthe value ofthe maturity n on the price yield curve 5 How does sensitivity of price to yield depend on the coupon rate Consider the 10 graph for a bond with a 10 coupon payment Price decreases as yield goes up The bond market went down means that yields went up Price decreases at a decreasing rate The priceyield curve is convex For A 0 to 100 we add 30 X 10 300 to get 400 There are 3010 payments added to the 100 face value What is the price if A 10 100 PF 10017mquot 100 CFA1 11 Mmquot lfthe bond is 10 this means 100 CF 10 Thus RHS 100 So price 100 Value of bond par value A bond with yield rate coupon rate is called a par bond Bond Duration Bonds with long maturities have steeper priceyield curves than bonds with short maturities The prices of long bonds are more sensitive to interest rate changes than those of short bonds Because maturity affects sensitivity to yield changes it is useful to consider various measures of time length Example Duration after Figure 35 A 7 bond with 3 years to maturity Bond sells at 8 yield 1 For each period k find the present value of the payment received at period k say PVK 2 Compute the total present value say PV 3 Let tk denote the time of payment k BOTTOM 4 For each k compute the product PVkPV gtlt tk LINE 5 Add these products to get the duration D Question What is the duration of a zerocoupon bond Answer lts maturity Conclusion Duration can be viewed as a generalized maturity measure It is an average of the maturities of all the individual payments Macaulay Duration When the PV is calculated using the bonds yield as the interest rate the general duration formula becomes the Macaulay duration A financial instrument makes payments m times per year there are n periods ck is the payment in period t1 tn A is the YTM PM ck1 7lmk PVPV1 P n tk km D PV1PV1m PVzPV2m PVnPVnm Example Figure 35 n 6 m 2 A 008 Column A gives the km the time values12 22 52 62 Column E gives the PVkPV the weights Column F gives the products of the weights and the time values D is the total ofthe entries in column F Example Duration after Figure 35 A 7 bond with 3 years to maturity Bond sells at 8 yield The text gives formulas for the duration Our approach is equivalent but based on an algorithm to compute the formula values Insights About Duration Table 36 p 59 Duration of a Bond Yielding 5 as Function of Maturity and Coupon Rate Observations 1 Durations do not increase appreciably with maturity 2 With a fixed yield duration increases only to a finite limit as maturity is increased 3 Durations do not vary rapidly with respect to the coupon rate 4 Very long durations 20 years or more are achieved only by bonds that have both very long maturities and very low coupon rates Duration and Sensitivity Key Conclusion Price sensitivity formula The rate of change of the price of a bond is inversely proportional to its duration If D denotes the duration and P the price then dPdA 11m D P DM P ltgt 1P dPdA DM where DM 11Am D is called the modified duration Note for large m small A DM e D We can use the equation dPdA DM P to estimate the change in price due to a small change in yield by using the approximation APAA e dPdA DM P APzDMPAA We now have an explicit approximate equation for the impact of yield variations on prices Example 38 A 30year 10 coupon bond price P 100 at par point A 010 duration D 994 for A 010 DM D1Am D1 012 994105 947 dPdA DM P 947 APe DM PAA 947AA lf yield changes from 10 to 11 AA 001 AP e 947 so price drops from 100 to about 9053 See Figure 36 for more insight Using this approximation amounts to constructing a tangent line to the priceyield curve ofthe bond at A 010 and estimating the function locally with the linear approximation the tangent line provides Because of convexity of the priceyield curve this linear approximation will always overestimate the change due to an increase in A and underestimate the change due to a decrease in A An exact calculation shows price 913062 for A 011 The estimate is 9053 lfYTM decreased from 10 to 9 we would estimate price is now 100 947 10947 An exact calculation shows price 1102737 for A 009 Duration of a Portfolio You have a portfolio of several bonds say A and B of different maturities The portfolio acts like a master F l security The payments for various bonds in portfolio may differ All the bonds in the portfolio have the same yield bond yields tend to track each other closely Basic Question What is the duration ofthe portfolio Bond Durations DA onAPA to PVnAPA tn DB PVoBPB to PVnBPB tn Portfolio PV P PA PB By analogy a way to define the duration D for A B is D onA onB P to PVnA PVnBP tn Now note the following PA DA onAto PVnAtn PB DB onBto PVnB tn PA DA PB DB PV0APVOBt0 PVnAPVoBtn PA P DA PB P DB onA PVOB P to PVnA PVnBP tn D Conclusion If we know the durations of DA and DB ofthe bonds A and B all with a common yield we can compute the duration D of the portfolio AB given P PA PB as follows D PA P DA PB P DB a wei hted avera e of DA DB 9 9 Important Implications The duration of a portfolio measures the interest rate sensitivity ofthe portfolio just as normal duration does for a single bond lfthe yield changes by a small amount the total value ofthe portfolio will change approximately by the amount predicted by the equation relating prices to modified duration dPdA 11m D P DM P AP m DM P AA lfthe bonds in the portfolio have different yields the composite duration can still be used as an approximation In this case a single yield perhaps the average is chosen We calculate PV s with this single yield value so they will only be approximations We then compute the weighted average duration as above
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