Chapter 5: Interest Rates

Transcription

1 Chapte 5: Inteest Rates -1 Chapte 5: Inteest Rates Note: Read the chapte then look at the following. Fundamental questions: 1) How do we handle non-annual cash flows and inteest ates? ) What factos affect the inteest ate on a secuity? 5.1 Inteest Rate Quotes and Adjustments A. The Effective Annual Rate Effective inteest ate: actual inteest ate fo peiod Ex. 1) Assume effective inteest ate fo yea is 9%. If deposit $100 fo a yea, have $109 at end of the yea ) Assume effective inteest ate fo month is %. If deposit $100 fo a month, have $10 at the end of the month. 1. Adjusting the Discount Rate fo Diffeent Time Peiods => fo any effective inteest ate, can find an equivalent effective inteest ate fo a shote o longe peiod. Geneal Equation fo Discount Rate Peiod Convesion (t) = (1 + ) n 1 (5.1) Notes: 1) (t) is the effective ate fo peiod t. I expess t in tems of yeas. Ex. ( 1 ) = effective semiannual ate (half a yea) ( 1 ) = effective monthly ate (one-twelfth of a yea) ) is effective inteest ate that conveting to a diffeent peiod (length of time)

2 Chapte 5: Inteest Rates - 3) n is adjustment facto: time conveting to/time stating with Examples: If want a ate that is twice as long as the one you stat with, n = If want a ate that is six times as long, n = 6 If want a ate that is one half as long, n = ½ If want a ate that is one-twelfth as long, n = 1/ 4) As noted in chapte 4, must use an effective inteest ate that matches the fequency of the cash flows fo any seies of cash flows (annuities and pepetuities). 5) As noted in chapte 4, can use any effective inteest ate fo taking the pesent o futue value of a single cash flow 6) Footnote 1 is impotant. APY (annual pecentage yield) is the teminology that banks ae equied to use fo the effective annual inteest ate on deposits. This is equied by the Tuth in Savings Act of ) On quizzes and exams, you ae sometimes only equied to set up poblems without solving them. It is impotant in this case to make it clea which effective inteest ate you ae using in a paticula step. B. Annual Pecentage Rates (APR) APR: an inteest ate that ignoes the effect of compounding (t) = APR k (5.) Notes: 1) can t use APRs in time-value-of-money calculations. Only use of an APR is to calculate an effective inteest ate ) k = numbe of compounding peiods pe yea 3) t = time fame of inteest ate in yeas = 1/k 4) any time the inteest ate is an APR, must stat with this equation to convet to an effective inteest ate

3 Chapte 5: Inteest Rates -3 C. Examples 1. Assume an APR of 6% pe yea with semiannual compounding. What is the effective annual inteest ate and the effective monthly inteest ate on this account? t APR k => effective semiannual ate (half a yea) is 3% n t Video Solution, Note:. 03 1, and ae equivalent => end up with same amount of money at the end Ex. If invest $100 fo a yea, then you account balance at the end of the yea equals: V1 = 100(1.03) = 100(1.0609) = 100( ) = Note: esult is same if take futue value at semiannual ate fo two peiods o annual ate fo one peiod o monthly ate fo peiods. => all take value one yea into futue Video Solution. Eight months fom today you want to make the fist of 14 quately withdawals fom a bank account. You fist withdawal will equal $10,000 and each subsequent withdawal will gow by 1% each. How much do you need to deposit today if the account pays an APR of 9% with monthly compounding?

4 Chapte 5: Inteest Rates -4 Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) fill in known vaiables and solve fo unknown vaiable, 6) make sue know whee end up on timeline. t APR k n t V t = C g (1 (1+g 1+ )N ) V 5mo = 10, (1 ( )14 ) = 6,401.7 Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) fill in known vaiables and solve fo unknown vaiable, 6) make sue know whee end up on timeline. V t = C n (1+) n V 0 = 6,401.7 (1.0075) 5 = 1,766,03 Video Solution 3. What if you want to make the fist quately withdawal one month fom today (and nothing else changes)? Q: Will the amount you deposit be lage o smalle if 1 st withdawal is in one month instead of eight months? Lage

5 Chapte 5: Inteest Rates -5 Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) fill in known vaiables and solve fo unknown vaiable, 6) make sue know whee end up on timeline ; ; PV -mo 6, Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) fill in known vaiables and solve fo unknown vaiable, 6) make sue know whee end up on timeline. V t = C (1 + ) n V 0 = 6,401.7(1.0075) = 8, Video Solution 4. A bond matues fo $1000 thee yeas and ten months fom today. The annual coupon on the bond equals $60 but coupons ae paid semiannually. What is the value of the bond if it eans a etun of 8% pe yea?

6 Chapte 5: Inteest Rates -6 Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) fill in known vaiables and solve fo unknown vaiable, 6) make sue know whee end up on timeline. n t Coupons: Pa: V t = C g (1 (1+g 1+ )N ) V mo = (1 ( )8 ) = V t = C (1 + ) n V 0 = 0.657(1.08) = V t = C n (1+) n V 0 = 1000 (1.08) 310 = Pice = = Video Solution Calculato: D. Pacticing N and n V- mo: 30 = PMT, 1000 = FV, 8 = N, 3.93 = I% => PV = V0: = PV, 8 = I%, / = N => FV = Assume you want to take out a loan and can affod to make monthly payments of $100 each stating five months fom today that will continue though thee yeas and thee months fom today. Assume you want to detemine the amount you can boow. Which equation(s) would you use? What would you use fo N o n in each equation?

7 Chapte 5: Inteest Rates -7 Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) figue out N o n. A: 1) pesent value of annuity, N = 35; ) pesent value of single cash flow, n = 4 (o 4 if use futue value of a single cash flow). Assume you lende will allow you to pay off you loan (in numbe 1 above) by making payments of only $10 pe month (still stating five months fom today and continuing though thee yeas and thee months fom today). You would then make a balloon payment thee and a half yeas fom today to pay off the balance of the loan. Assume you want to detemine the size of the balloon payment. Which equation(s) would you use? What would you use fo N o n in each equation? Steps: 1) timeline, ) patten of cash flows, 3) pesent o futue value, 4) equation o equations, 5) figue out N o n. A: 1) futue value of annuity ($90 payments), N = 35; ) futue value of a single cash flow, n = 3 5. Application: Discount Rates and Loans A. Computing Loan Payments An impotant statement you might ovelook: When the compounding inteval fo the APR is not stated explicitly, it is equal to the inteval between payments. B. Computing the Outstanding Loan Balance Concept Check: all

8 Chapte 5: Inteest Rates The Deteminants of Inteest Rates A. Inflation and Real Vesus Nominal Rates = i 1+i (5.5) whee: = nominal inteest ate = ate at which money gows i = inflation ate = ate at which pices change = eal inteest ate = ate at which puchasing powe gows Note: can use expected o ealized ates Ex. Assume that the nominal inteest ate is 6% pe yea and that inflation is 5% pe yea. What is the eal inteest ate? =.0095 = Note: the diffeence between the nominal ate and the inflation ate is a petty good appoximation of the eal ate if inflation is low. Q: Example in book calculates eal inteest ate in 011 was.6%. Why would anyone invest at a negative eal inteest ate? B. Investment and Inteest Rate Policy Key issue: inteest ates affect net pesent values of investment oppotunities => since most investment equie outflow today and inflows in the futue, NPV and inteest ates move in opposite diections 1. Monetay Policy, Deflation, and the 008 Financial Cisis Key issue: Fed uses invese elationship between inteest ates and net pesent values to stimulate o cool the economy

9 Chapte 5: Inteest Rates -9 C. The Yield Cuve and Discount Rates Key issues: V 0 = => inteest ates vay by matuity => tem stuctue => yield cuve C n (1+ n ) n (5.6) V 0 = C 1 (1+ 1 ) + C (1+ ) + + C N (1+ N ) N (5.7a) C n (1+ n ) n V 0 = N n=1 (5.7b) Ex. What is the value today of $100 a yea fom today, $00 two yeas fom today, and $300 thee yeas fom today? Note: To get the most ecent ates, Google teasuy ates and go to the Teasuy s website listing the ates on U.S. Teasuy secuities. Impotant Waning: All of ou shotcuts fo computing pesent values (annuity and pepetuity fomulas) ae based on discounting all cash flows at the same ate. They cannot be used in situations in which cash flows need to be discounted at diffeent ates. D. The Yield Cuve and the Economy 1. Inteest Rate Detemination Key issue: ole of Fed and makets in detemining inteest ates Note: With quantitative easing, the Fed attempted to influence long-tem as well as shot-tem inteest ates. Inteest Rate Expectations Key issue: ole of expectations in detemining long-tem inteest ates Note: The IRR is a complicated weighted aveage of the inteest ates used to calculate the pesent values of the individual cash flows Concept Check: all

10 Chapte 5: Inteest Rates Risk and Taxes A. Risk and Inteest Rates Key issue: => elationship between inteest ates and default isk Notes: 1) Coca-cola ecently issued bonds at lowe ate than the U.S. govenment => suggests that Coca-cola has (o at least had) a lowe chance of default than the U.S. govenment ) long-tem ates tend to exceed shot-tem ates (even fo U.S. Teasuies) => suggests a isk-pemium included in long-tem ates B. Afte-Tax Inteest Rates Key issues: => most inteest income is taxable fo investos and inteest payments ae tax deductible fo copoations => taxes educe the cash that investos get to keep fom inteest they ean => taxes educe the cost of boowing fo fims => both investos and copoate boowes cae about the afte-tax inteest ate Afte-tax inteest ate: AT = (τ ) = (1 τ) (5.8) whee: AT = afte-tax inteest ate = befoe-tax inteest ate = tax ate Ex. Assume a pe-tax inteest ate of 10% and a tax ate of 30%. What is the afte-tax inteest ate? AT =.1 x (1 -.3) =.07 Concept Check: all 5.5 The Oppotunity Cost of Capital Oppotunity cost of capital (o simply cost of capital): best available expected etun offeed in the maket on an investment of compaable isk and tem to the cash flow being discounted. Concept Check: all

11 Chapte 5: Inteest Rates Appendix A. Discount Rates fo a Continuously Compounded APR Key issue: conveting between APR and effective annual inteest ate when inteest is compounded continuously Note: Can t use equation 5. since k = (1) = e APR 1 (5A.1) whee: e = APR = ln(1+(1)) (5A.) Ex. Assume a bank pays an APR of 5% with continuous compounding. What is the effective annual inteest ate? (1) = e.05 1 =.057 Excel: = exp(.05) 1 B. Continuously Aiving Cash Flows Comment: You can skip this section