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Chapter 4. UnderstandingChapter 4. Understanding

Interest RatesInterest Rates Present ValuePresent Value

Yield to MaturityYield to Maturity

Other YieldsOther Yields Other Measurement IssuesOther Measurement Issues

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I. Measuring Interest RatesI. Measuring Interest Rates

A. Credit Market InstrumentsA. Credit Market Instruments

simple loansimple loan

borrower pays back loan and interest inborrower pays back loan and interest inone lump sumone lump sum

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fixedfixed--payment loanpayment loan

loan is repaid with equal (monthly)loan is repaid with equal (monthly)paymentspayments

each payment is combination ofeach payment is combination of

principal and interestprincipal and interest

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coupon bondcoupon bond

purchase price (P)purchase price (P)

interest payments (6 months)interest payments (6 months)

face value at maturity (F)face value at maturity (F)

size of interest paymentssize of interest payments

---- coupon ratecoupon rate

---- face valueface value

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discount bonddiscount bond

zero coupon bondzero coupon bond

purchased price less than face valuepurchased price less than face value

---- F > PF > P

face value at maturityface value at maturity no interest paymentsno interest payments

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B. Present & Future ValueB. Present & Future Value

time value ofmoneytime value ofmoney

$100 today vs. $100 in 1 year$100 today vs. $100 in 1 year

not indifferent!not indifferent! money earns interest over time,money earns interest over time,

and we prefer consuming todayand we prefer consuming today

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example: future valueexample: future value

$100 today$100 today

interest rate 5% annuallyinterest rate 5% annually

at end of 1 year:at end of 1 year:100 + (100 x .05)100 + (100 x .05)

= 100(1.05) = $105= 100(1.05) = $105

at end of 2 years:at end of 2 years:100 + (1.05)100 + (1.05)22 = $110.25= $110.25

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future valuefuture value

of$100 inof$100 in nn years if interest rate isyears if interest rate is ii::

= $100(1 + i)= $100(1 + i)nn

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present valuepresent value

work backwardswork backwards

if get $100 inif get $100 in nn years,years,

what is that worth today?what is that worth today?

PV =$100

(1+ i)n

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exampleexample

receive $100 in 3 yearsreceive $100 in 3 years

i = 5%i = 5%

what is PV?what is PV?

PV =$100

(1+ .05)3

= $86.36

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n

i

PV

PV

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C. Yield to Maturity (YTM)C. Yield to Maturity (YTM)

a measure of interest ratea measure of interest rate

interest rate whereinterest rate where

P = PV of cash flows

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example 1: simple loanexample 1: simple loan

loan = $1500, 1 year, 6%loan = $1500, 1 year, 6%

future paymentfuture payment

= $1500(1+.06) = $1590= $1500(1+.06) = $1590 yield to maturity, iyield to maturity, i

$1500 =$1590

(1+ i)i = 6%

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example 2: fixed pmt. loanexample 2: fixed pmt. loan

$15,000 car loan, 5 years$15,000 car loan, 5 years

monthly pmt. = $300monthly pmt. = $300

so $15,000 is price todayso $15,000 is price today cash flow is 60 pmts. of $300cash flow is 60 pmts. of $300

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YTM solvesYTM solves

i/12 is monthly discount ratei/12 is monthly discount rate

i is yield to maturityi is yield to maturity

i/12 is monthly discount ratei/12 is monthly discount rate

i is yield to maturityi is yield to maturity

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how to solve for i?how to solve for i?

trialtrial--andand--errorerror

bond table*bond table*

financial calculatorfinancial calculator

spreadsheetspreadsheet

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payment between $297.02 &payment between $297.02 &

$300.57$300.57 YTM is between 7% and 7.5%YTM is between 7% and 7.5%

(7.42%)(7.42%)

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Example 3: Coupon BondExample 3: Coupon Bond

2 year T2 year T--Note, F = $10,000Note, F = $10,000

coupon rate 6%coupon rate 6%

price of $9750price of $9750 what are interest payments?what are interest payments?

(.06)($10,000)(.5) = $300(.06)($10,000)(.5) = $300

every 6 mos.every 6 mos.

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YTM solves the equationYTM solves the equation

i/2 is 6i/2 is 6--month discount ratemonth discount rate

i is yield to maturityi is yield to maturity

i/2 is 6i/2 is 6--month discount ratemonth discount rate

i is yield to maturityi is yield to maturity

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price between $9816 & $9726price between $9816 & $9726

YTM is between 7% and 7.5%YTM is between 7% and 7.5%(7.37%)(7.37%)

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P, F and YTMP, F and YTM

P = F then YTM = coupon rateP = F then YTM = coupon rate

P < F then YTM > coupon rateP < F then YTM > coupon rate

bond sells at a discountbond sells at a discount P > F then YTM < coupon rateP > F then YTM < coupon rate

bond sells at a premiumbond sells at a premium

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P and YTM move in oppositeP and YTM move in opposite

directionsdirections interest rates and value of debtinterest rates and value of debt

securities move in oppositesecurities move in opposite

directionsdirections if rates rise, bond prices fallif rates rise, bond prices fall

if rates fall, bond prices riseif rates fall, bond prices rise

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example 4: discount bondexample 4: discount bond

90 day Tbill,90 day Tbill,

P = $9850, F = $10,000P = $9850, F = $10,000

YTM solvesYTM solves

!

365901

000109850

i

,$$

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9850

00010

365

901

$

,$i !

19850

00010

365

90!

$

,$i

9850985000010

36590

$,$i

!

90365

9850985000010 v

! $,$i = 6.18%

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in general,in general,

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D. Current YieldD. Current Yield

approximation of YTM for couponapproximation of YTM for couponbondsbonds

ic =annual coupon payment

bond price

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better approximation whenbetter approximation when

maturity is longermaturity is longer

P is close to FP is close to F

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example 5example 5

2 year Tnotes, F = $10,0002 year Tnotes, F = $10,000

P = $9750, coupon rate = 6%P = $9750, coupon rate = 6%

current yieldcurrent yield

ic =600

9750= 6.15%

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current yield = 6.15%current yield = 6.15%

true YTM = 7.37%true YTM = 7.37% lousy approximationlousy approximation

only 2 years to maturityonly 2 years to maturity

selling 25% below Fselling 25% below F

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E. Discount YieldE. Discount Yield

yield on a discount basisyield on a discount basis

approximation of YTMapproximation of YTM

idb =F - P

Fx

360

d

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compare w/ YTMcompare w/ YTM

idb =F - P

Fx

360

d

iytm =F - P

Px

365

d

iytm > idb

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example 6:example 6:

9090--day Tday T--bill, price $9850bill, price $9850

idb = 10,000 - 985010,000

x 36090

= 6%true YTM is 6.18%

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II. Other measurement issuesII. Other measurement issues

A. Interest rates vs. returnA. Interest rates vs. return

YTM assumes bond is held untilYTM assumes bond is held until

maturitymaturity if not, resale price is importantif not, resale price is important

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B. Maturity & bond priceB. Maturity & bond price

volatilityvolatility

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YTM rises from 6 to 8%YTM rises from 6 to 8%

bond prices fallbond prices fall but 10but 10--year bond price falls the mostyear bond price falls the most

Prices are more volatile for longerPrices are more volatile for longer

maturitiesmaturities longlong--term bonds have greater interestterm bonds have greater interest

rate riskrate risk

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Why?Why?

longlong--term bonds lock in a coupon rateterm bonds lock in a coupon ratefor a longer timefor a longer time

if interest rates riseif interest rates rise

---- stuck with a belowstuck with a below--market couponmarket coupon

raterate

if interest rates fallif interest rates fall

---- receiving an abovereceiving an above--market couponmarket coupon

raterate

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C. Real vs. Nominal InterestC. Real vs. Nominal Interest

RatesRates Thus far we have calculated nominalThus far we have calculated nominal

interest ratesinterest rates

ignores effects of rising inflationignores effects of rising inflation

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Real Interest Rate, iReal Interest Rate, irr

nominal interest rate = inominal interest rate = i

expected inflation rate =expected inflation rate = ee

approximately:approximately:i = ii = irr ++

ee

The Fisher equationThe Fisher equation

oror iirr= i= i -- ee

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real interest rates measure true cost ofreal interest rates measure true cost of

borrowingborrowing

why?why?

as inflation rises, real value of loanas inflation rises, real value of loan

payments falls,payments falls, so real cost of borrowing fallsso real cost of borrowing falls

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