Chapter 4. Understanding Interest Rates Present Value Yield to Maturity Other Yields Other Measurement Issues Present Value Yield to Maturity Other Yields Other Measurement Issues

I. Measuring Interest Rates A. Credit Market Instruments simple loan  borrower pays back loan and interest in one lump sum A. Credit Market Instruments simple loan  borrower pays back loan and interest in one lump sum

fixed-payment loan  loan is repaid with equal (monthly) payments  each payment is combination of principal and interest fixed-payment loan  loan is repaid with equal (monthly) payments  each payment is combination of principal and interest

coupon bond  purchase price (P)  interest payments (6 months)  face value at maturity (F)  size of interest payments -- coupon rate -- face value coupon bond  purchase price (P)  interest payments (6 months)  face value at maturity (F)  size of interest payments -- coupon rate -- face value

discount bond  zero coupon bond  purchased price less than face value -- F > P  face value at maturity  no interest payments discount bond  zero coupon bond  purchased price less than face value -- F > P  face value at maturity  no interest payments

B. Present & Future Value time value of money $100 today vs. $100 in 1 year  not indifferent!  money earns interest over time,  and we prefer consuming today time value of money $100 today vs. $100 in 1 year  not indifferent!  money earns interest over time,  and we prefer consuming today

example: future value $100 today interest rate 5% annually at end of 1 year: (100 x.05) = 100(1.05) = $105 at end of 2 years: (1.05) 2 = $ $100 today interest rate 5% annually at end of 1 year: (100 x.05) = 100(1.05) = $105 at end of 2 years: (1.05) 2 = $110.25

future value of $100 in n years if interest rate is i: = $100(1 + i) n of $100 in n years if interest rate is i: = $100(1 + i) n

present value work backwards if get $100 in n years, what is that worth today? work backwards if get $100 in n years, what is that worth today? PV= $100 (1+ i) n

exampleexample receive $100 in 3 years i = 5% what is PV? receive $100 in 3 years i = 5% what is PV? PV= $100 (1+.05) 3 =$86.36

n i PV

C. Yield to Maturity (YTM) a measure of interest rate interest rate where a measure of interest rate interest rate where P =PV of cash flows

example 1: simple loan loan = $1500, 1 year, 6% future payment = $1500(1+.06) = $1590 yield to maturity, i loan = $1500, 1 year, 6% future payment = $1500(1+.06) = $1590 yield to maturity, i $1500= $1590 (1+ i) i = 6%

example 2: fixed pmt. loan $15,000 car loan, 5 years monthly pmt. = $300 so $15,000 is price today cash flow is 60 pmts. of $300 $15,000 car loan, 5 years monthly pmt. = $300 so $15,000 is price today cash flow is 60 pmts. of $300

YTM solves i/12 is monthly discount rate i is yield to maturity i/12 is monthly discount rate i is yield to maturity

how to solve for i?  trial-and-error  bond table*  financial calculator  spreadsheet how to solve for i?  trial-and-error  bond table*  financial calculator  spreadsheet

payment between $ & $ YTM is between 7% and 7.5% (7.42%) payment between $ & $ YTM is between 7% and 7.5% (7.42%)

example 3: coupon bond 2 year Tnote, F = $10,000 coupon rate 6% price of $9750 what are interest payments? (.06)($10,000)(.5) = $300  every 6 mos. 2 year Tnote, F = $10,000 coupon rate 6% price of $9750 what are interest payments? (.06)($10,000)(.5) = $300  every 6 mos.

YTM solves the equation i/2 is 6-month discount rate i is yield to maturity i/2 is 6-month discount rate i is yield to maturity

price between $9816 & $9726 YTM is between 7% and 7.5% (7.37%) price between $9816 & $9726 YTM is between 7% and 7.5% (7.37%)

P, F and YTM P = F then YTM = coupon rate P coupon rate  bond sells at a discount P > F then YTM < coupon rate  bond sells at a premium P = F then YTM = coupon rate P coupon rate  bond sells at a discount P > F then YTM < coupon rate  bond sells at a premium

P and YTM move in opposite directions interest rates and value of debt securities move in opposite directions  if rates rise, bond prices fall  if rates fall, bond prices rise P and YTM move in opposite directions interest rates and value of debt securities move in opposite directions  if rates rise, bond prices fall  if rates fall, bond prices rise

example 4: discount bond 90 day Tbill, P = $9850, F = $10,000 YTM solves 90 day Tbill, P = $9850, F = $10,000 YTM solves

= 6.18%

in general,

D. Current Yield approximation of YTM for coupon bonds i c = annual coupon payment bond price

better approximation when  maturity is longer  P is close to F better approximation when  maturity is longer  P is close to F

example 5 2 year Tnotes, F = $10,000 P = $9750, coupon rate = 6% current yield 2 year Tnotes, F = $10,000 P = $9750, coupon rate = 6% current yield i c = = 6.15%

current yield = 6.15% true YTM = 7.37% lousy approximation  only 2 years to maturity  selling 25% below F current yield = 6.15% true YTM = 7.37% lousy approximation  only 2 years to maturity  selling 25% below F

E. Discount Yield yield on a discount basis approximation of YTM yield on a discount basis approximation of YTM i db = F - P F x 360 d

compare w/ YTM i db = F - P F x 360 d i ytm = F - P P x 365 d i ytm > i db

example 6: 90-day Tbill, price $9850 i db = 10, x = 6% true YTM is 6.18%

II. Other measurement issues A. Interest rates vs. return YTM assumes bond is held until maturity if not, resale price is important A. Interest rates vs. return YTM assumes bond is held until maturity if not, resale price is important

B. Maturity & bond price volatility

YTM rises from 6 to 8%  bond prices fall  but 10-year bond price falls the most Prices are more volatile for longer maturities  long-term bonds have greater interest rate risk YTM rises from 6 to 8%  bond prices fall  but 10-year bond price falls the most Prices are more volatile for longer maturities  long-term bonds have greater interest rate risk

Why?  long-term bonds “lock in” a coupon rate for a longer time  if interest rates rise -- stuck with a below-market coupon rate  if interest rates fall -- receiving an above-market coupon rate Why?  long-term bonds “lock in” a coupon rate for a longer time  if interest rates rise -- stuck with a below-market coupon rate  if interest rates fall -- receiving an above-market coupon rate

C. Real vs. Nominal Interest Rates thusfar we have calculated nominal interest rates  ignores effects of rising inflation thusfar we have calculated nominal interest rates  ignores effects of rising inflation

real interest rate, i r nominal interest rate = i expected inflation rate = π e approximately: i = i r + π e The Fisher equation ori r = i - π e nominal interest rate = i expected inflation rate = π e approximately: i = i r + π e The Fisher equation ori r = i - π e

real interest rates measure true cost of borrowing why?  as inflation rises, real value of loan payments falls,  so real cost of borrowing falls real interest rates measure true cost of borrowing why?  as inflation rises, real value of loan payments falls,  so real cost of borrowing falls