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Understanding Interest Rates
Chapter 4
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Chapter Definitions Chapter Definitions
Present ValuePresent ValueYield to maturityYield to maturitySimple LoanSimple LoanFixed payment loanFixed payment loanCoupon bondCoupon bondDiscount bondDiscount bondRate of return and interest rate riskRate of return and interest rate riskReal vs nominal interest ratesReal vs nominal interest rates
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Yield to MaturityYield to Maturity
Yield to maturity is the most accurate Yield to maturity is the most accurate measure of interest rates.measure of interest rates.
Sometimes it is called internal rate of Sometimes it is called internal rate of return.return.
It is the interest rate that relates all the It is the interest rate that relates all the future returns to the present value of a future returns to the present value of a debt/investment.debt/investment.
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Future ValueFuture Value Suppose you put a $1000 in the bank at 5% Suppose you put a $1000 in the bank at 5%
interest rate. What would be the interest rate. What would be the future valuefuture value in in three years?three years?
1000 + 1000*(0.05) = 1000 (1.05) at the end of the 1000 + 1000*(0.05) = 1000 (1.05) at the end of the first year: $1050.first year: $1050.
1000 (1.05) + 1000 (1.05)(0.05) = 1000 (1.05)1000 (1.05) + 1000 (1.05)(0.05) = 1000 (1.05)(1+.05) = 1000(1.05)(1.05) at the end of the (1+.05) = 1000(1.05)(1.05) at the end of the second year: $1102.50.second year: $1102.50.
1000(1.05)(1.05) + 1000(1.05)(1.05)(0.05) = 1000(1.05)(1.05) + 1000(1.05)(1.05)(0.05) = 1000(1.05)(1.05)(1+.05)= 1000(1.05)(1.05)(1.05) 1000(1.05)(1.05)(1+.05)= 1000(1.05)(1.05)(1.05) at the end of the third year: $1157.625.at the end of the third year: $1157.625.
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Present ValuePresent Value
If you expect to be paid $1157.625 If you expect to be paid $1157.625 three years from today, the present three years from today, the present value of this future payment is $1000 if value of this future payment is $1000 if the funds earn 5% interest.the funds earn 5% interest.
Would the present value of this future Would the present value of this future payment be greater or less than $1000 payment be greater or less than $1000 if the funds were to earn 10% interest?if the funds were to earn 10% interest?
FV=1000(1+i)FV=1000(1+i)33
PV=1000/(1+i)PV=1000/(1+i)33
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Present ValuePresent Value
If you only want to earn 5% interest for If you only want to earn 5% interest for the next two years on the $900 you have, the next two years on the $900 you have, and someone offers you $1102.50 in two and someone offers you $1102.50 in two years, would you take the offer?years, would you take the offer?
PV r n FV
900 0.05 2 992.25
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Debt TypesDebt Types Simple loan: you borrow today and pay the Simple loan: you borrow today and pay the
principal plus the interest in the future.principal plus the interest in the future. Fixed payment loan: you borrow today but Fixed payment loan: you borrow today but
make equal payments per time period until make equal payments per time period until your loan is all paid.your loan is all paid.
Coupon bond: firm sells the bond (borrows Coupon bond: firm sells the bond (borrows the funds) pays interest per period and at the the funds) pays interest per period and at the maturity date pays the principal, too.maturity date pays the principal, too.
Discount bond: firm sells the bond at less Discount bond: firm sells the bond at less than face value pays the face value at the than face value pays the face value at the maturity date.maturity date.
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Yield to Maturity: Simple LoanYield to Maturity: Simple Loan
If you borrow $1000 today and asked to If you borrow $1000 today and asked to pay $1100 a year from now, what is the pay $1100 a year from now, what is the interest rate?interest rate?
1000 = 1100/(1+i)1000 = 1100/(1+i)
1+i = 1100/10001+i = 1100/1000
i = 1100/1000 - 1i = 1100/1000 - 1
i = 1.1 - 1i = 1.1 - 1
i = .1 = 10%i = .1 = 10%
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Yield to Maturity: Fixed Yield to Maturity: Fixed Payment LoanPayment Loan
Suppose you got a $50,000 mortgage. The Suppose you got a $50,000 mortgage. The bank wants you to pay $550.60 per month for bank wants you to pay $550.60 per month for 20 years. What is the interest rate you are 20 years. What is the interest rate you are charged?charged?50,000 = [550.60/(1+i)] +[550.60/(1+i50,000 = [550.60/(1+i)] +[550.60/(1+i)2)2] + ] + [550.60/(1+i[550.60/(1+i)3)3] + … + [550.60/(1+i)] + … + [550.60/(1+i)240240]]i = .01 or 1% per month or 12% per year.i = .01 or 1% per month or 12% per year.
If the interest rate were lower than 12%, would If the interest rate were lower than 12%, would your monthly payments increase or decrease?your monthly payments increase or decrease?
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Yield to Maturity: Coupon BondYield to Maturity: Coupon Bond
You buy a ten-year corporate bond with a You buy a ten-year corporate bond with a face value of $1000, coupon rate of 10% face value of $1000, coupon rate of 10% for $885.30. What is the interest rate you for $885.30. What is the interest rate you are earning?are earning?885.30 = [100/(1+i)] + [100/(1+i)885.30 = [100/(1+i)] + [100/(1+i)22 + … + … +1100/(1+i)+1100/(1+i)1010
i = 0.12 or 12%.i = 0.12 or 12%.What is the interest rate you are earning if What is the interest rate you are earning if
you bought it for $1000?you bought it for $1000?
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Yield to Maturity: ConsolYield to Maturity: Consol
You bought a consol for $1250. It pays You bought a consol for $1250. It pays $100 per year. What is the interest rate $100 per year. What is the interest rate you are getting?you are getting?
i = 100/1250i = 100/1250
i = 0.08 or 8%.i = 0.08 or 8%.What if you bought the consol for $1000; What if you bought the consol for $1000;
what would be the interest rate?what would be the interest rate?
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Yield to Maturity: Discount BondYield to Maturity: Discount Bond
You bought a discount bond for $800 which You bought a discount bond for $800 which will pay $1000 a year from now. What is the will pay $1000 a year from now. What is the interest rate?interest rate?
i = (1000 - 800)/800i = (1000 - 800)/800
i = 200/800i = 200/800
i = 0.25 or 25%.i = 0.25 or 25%. Would the interest rate be higher or lower Would the interest rate be higher or lower
than 25% if you had bought the bond for than 25% if you had bought the bond for $900?$900?
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Other Measures of Interest RatesOther Measures of Interest Rates
Current Yield: calculating the interest rate Current Yield: calculating the interest rate on a coupon bond as if it were a consol.on a coupon bond as if it were a consol.Current yield approaches yield to maturity when Current yield approaches yield to maturity when
the price of the bond is close to the face value the price of the bond is close to the face value and the maturity date is far away.and the maturity date is far away.
Yield on a discount basis: the price of Yield on a discount basis: the price of Treasury Bills is quoted in this fashion. It is Treasury Bills is quoted in this fashion. It is always less than the yield to maturity.always less than the yield to maturity.
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An Example of Treasury BillAn Example of Treasury Bill
A T-Bill that has 91 days to maturity A T-Bill that has 91 days to maturity sells for 5.4% discount. Find the price sells for 5.4% discount. Find the price and yield to maturity.and yield to maturity.
.054 (91/360) = [10,000 - P]/10,000.054 (91/360) = [10,000 - P]/10,000
P = $9863.50P = $9863.50
i = [(10,000 - 9863.50)/9863.50][360/91]i = [(10,000 - 9863.50)/9863.50][360/91]
i = 0.0547 or 5.47%.i = 0.0547 or 5.47%.
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Debt TypesDebt Types
SIMPLE LOAN:Principal Interest Rate # of prd End Payment Check
1000 0.0800 5 $1,469.33
FIXED PAYMENT LOAN:Principal Interest Rate # of prd Payment Check
9700 0.00333 60 ($178.62)
DISCOUNT BOND < 1 yearPrice Face Value Days Interest Rate Check
995 1000 91 0.0202
DISCOUNT BOND > 1 yearPrice Face Value Months Interest Rate Check
950 1000 18 0.0348
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Interest Rate vs. Rate of ReturnInterest Rate vs. Rate of Return
Rate of return on a bond includes the Rate of return on a bond includes the capital gains/losses.capital gains/losses.
Capital gain is the increase in the price of Capital gain is the increase in the price of the bond. Capital loss is the decrease in the bond. Capital loss is the decrease in the price of the bond.the price of the bond.
If you bought a bond for $1000, held it for If you bought a bond for $1000, held it for a year, received $100 interest payment a year, received $100 interest payment and sold it for $1100, the return isand sold it for $1100, the return is
R = [100 + 1100]/1000 = .20 or 20%.R = [100 + 1100]/1000 = .20 or 20%.
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Why Bonds Are RiskyWhy Bonds Are Risky If you hold the bond until maturity, return and If you hold the bond until maturity, return and
interest rate will be the same.interest rate will be the same. If interest rates rise while you own a bond, the If interest rates rise while you own a bond, the
price of the bond falls yielding capital losses.price of the bond falls yielding capital losses. The farther the maturity date the larger is the The farther the maturity date the larger is the
capital loss (or capital gain when interest capital loss (or capital gain when interest rates fall). rates fall).
Prices and returns for long-term bonds are Prices and returns for long-term bonds are more volatile than those for shorter term more volatile than those for shorter term bonds.bonds.
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Real and Nominal Interest RatesReal and Nominal Interest Rates
If people expect inflation, they will not If people expect inflation, they will not lend money at low rates. They would lend money at low rates. They would want to preserve the value of their money want to preserve the value of their money by including the inflation rate in the by including the inflation rate in the nominal interest rate.nominal interest rate.
Suppose you lend $1. You want to Suppose you lend $1. You want to receive a real interest rate of r and want receive a real interest rate of r and want to compensate the erosion of the value of to compensate the erosion of the value of the money by the expected inflation rate, the money by the expected inflation rate, p.p.
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Real and Nominal Interest Real and Nominal Interest RatesRates
1+i = (1+r)(1+p) 1+i = (1+r)(1+p)
= 1 + r + p + rp= 1 + r + p + rp
i = r + p + rpi = r + p + rp Nominal interest rate is equal to real interest Nominal interest rate is equal to real interest
rate plus the expected inflation plus the rate plus the expected inflation plus the product of the two.product of the two.This is called the Fisher equation.This is called the Fisher equation.For low values of real interest rates and For low values of real interest rates and
expected inflation the last term can be ignored.expected inflation the last term can be ignored.
