Understanding Interest Rates

Fundamentals of Finance – Lecture 3

Measuring Interest Rates

• Present Value:• A dollar paid to you one year from now is less

valuable than a dollar paid to you today• Why?

– A dollar deposited today can earn interest and become 1 x (1+i) one year from today.

Time Line

$100 $100

Year 0 1

FV 100

2

$100 $100

n

110 121 100/(1+i)n

It’s impossible to directly compare payments scheduled in different points in the time line!

Simple Present Value

• PV = today’s (present) value• FV = future cash flow (payment)• i = the interest rate

Yield to Maturity

The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today.

Five Basic Types of Debt Instruments

1. Simple Loan Contracts2. Fixed-Payment Loan Contracts3. Coupon Bond4. (Zero-Coupon) Discount Bond5. Consol (or Perpetuity)

Type 1: Simple Loan

• Borrower issues to lender a contract stating a loan value (principal) LV ($) and interest payment I ($).

• Today the borrower receives LV from lender.• One year from now the lender receives back from

the borrower an amount LV+I.Example: One-Year Deposit Account• Deposit LV = $100; Interest payment I = $10• Borrower’s end-of-year payment = $100 + $10 .

Simple Loan cont’d

1

PV = amount borrowed = $100CF = cash flow in one year = $110

= number of years = 1$110$100 =

(1 + )(1 + ) $100 = $110

$110(1 + ) = $100

= 0.10 = 10%For simple loans, the simple interest rate equ

n

ii

i

ials the

yield to maturity

Type 2: Fixed Payment Loan

• Today a borrower issues to a lender a contract with a stated loan value LV ($), an annual fixed payment FP ($/Yr), and a maturity of N years.

• Today the borrower receives LV from the lender.• For the next N successive years, the lender

receives from borrower the fixed payment FP.• FP includes principal and interest payments.Example: 30-year fixed-rate home mortgage

Fixed Payment Loan cont’d

2 3

The same cash flow payment every period throughout the life of the loanLV = loan value

FP = fixed yearly payment = number of years until maturityFP FP FP FPLV = . . . +

1 + (1 + ) (1 + ) (1 + )n

n

i i i i

Type 3: Coupon Bond• Today a seller offers for sale in a bond market a bond with

stated annual coupon payment C ($/yr), face (or par) value F ($), and a remaining maturity of N years.

• Today the bond seller receives from a buyer a price P ($/bond) as determined in the bond market.

• For next N successive years, the bond holder receives the fixed annual payment C from original bond issuer.

• At maturity, the bond holder also receives the face value F from the original bond issuer.

Examples: 30-year corporate bond, Central Government Treasury notes (1-10yrs) and bonds (≥ 10yrs)

Coupon Bond cont’d

2 3

Using the same strategy used for the fixed-payment loan:P = price of coupon bond

C = yearly coupon paymentF = face value of the bond

= years to maturity dateC C C C FP = . . . +

1+ (1+ ) (1+ ) (1+ ) (1n

n

i i i i

+ )ni

Type 4: Discount Bond

• Today a seller offers for sale in a bond market a bond with a stated face value F ($) and remaining maturity of N years.

• Today the bond seller receives from a buyer a price P ($/bond) as determined in the bond market.

• At the end of N years the bond holder receives the face value F from the original bond issuer.

Example: Treasury Bills Maturity < 1yr., typically offered in 1mo., 3mo., & 6 mo. Maturities.

Discount Bond cont’d

For any one year discount bond

i = F - PP

F = Face value of the discount bondP = current price of the discount bond

The yield to maturity equals the increasein price over the year divided by the initial price.As with a coupon bond, the yield to maturity is

negatively related to the current bond price.

Type 5. Consol (or Perpetuity)

• Today a seller offers for sale in a bond market a bond with a stated annual coupon payment C ($/Yr) and no maturity date (i.e., bond exists “in perpetuity”).

• Today the bond seller receives from a buyer a price P ($/bond) as determined in the bond market.

• In each future year the bond holder receives the coupon payment C from the original bond issuer.

Example: Consols were originally issued by UK in 1751, and remain a small part of UK’s debt portfolio.

Consol (or Perpetuity) cont’d

• A bond with no maturity date that does not repay principal but pays fixed coupon payments forever

consol theofmaturity toyieldpaymentinterest yearly

consol theof price/

c

c

c

iCP

iCP

cc PCi /: thisasequation above rewritecan For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity

Other Measures of Interest Rate

•Current yield :where is the current yield, P is the price of the coupon bond, and C is the yearly coupon payment.

•Yield on a Discount Basis: Where F is the face value of the bond, P is the price of the bond, and d stands for the days to maturity=

The Distinction Between Interest Rates and Returns

The payments to the owner plus the change in valueexpressed as a fraction of the purchase price

RET = CPt

+ Pt1 - Pt

Pt

RET = return from holding the bond from time t to time t + 1Pt = price of bond at time t

Pt1 = price of the bond at time t + 1

C = coupon payment

CPt

= current yield = ic

Pt1 - Pt

Pt

= rate of capital gain = g

• Rate of Return:

The Distinction Between Interest Rates and Returns (cont’d)

• The return equals the yield to maturity only if the holding period equals the time to maturity.

• A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period.

• The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change.

• The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rate.

• Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise.

The Distinction Between Real and Nominal Interest Rates

• Nominal interest rate makes no allowance for inflation.

• Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing.

• Ex ante real interest rate is adjusted for expected changes in the price level.

• Ex post real interest rate is adjusted for actual changes in the price level.

Fisher Equation

= nominal interest rate = real interest rate

= expected inflation rateWhen the real interest rate is low,

there are greater incentives to borrow and fewer incentives to lend.The real inter

er

r

e

i ii

i

est rate is a better indicator of the incentives toborrow and lend.