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Understanding Interest Rates

**Chapter 4**

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# Chapter Definitions

Present Value Yield to maturity Simple Loan Fixed payment loan Coupon bond Discount bond Rate of return and interest rate risk Real vs nominal interest rates 2

**Yield to Maturity**

Yield to maturity is the most accurate measure of interest rates.

Sometimes it is called internal rate of return.

It is the interest rate that relates all the future returns to the present value of a debt/investment.

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**Future Value**

Suppose you put a $1000 in the bank at 5% interest rate. What would be the

*future value*

three years?

in 1000 + 1000*(0.05) = 1000 (1.05) at the end of the first year: $1050.

1000 (1.05) + 1000 (1.05)(0.05) = 1000 (1.05)(1+.05) = 1000(1.05)(1.05) at the end of the second year: $1102.50.

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) at the end of the third year: $1157.625.

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**Present Value**

If you expect to be paid $1157.625 three years from today, the present value of this future payment is $1000 if the funds earn 5% interest.

Would the present value of this future payment be greater or less than $1000 if the funds were to earn 10% interest?

FV=1000(1+i) 3 PV=1000/(1+i) 3 5

**Present Value**

If you only want to earn 5% interest for the next two years on the $900 you have, and someone offers you $1102.50 in two years, would you take the offer?

PV 900 r 0.05

n 2 FV 992.25

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**Debt Types**

Simple loan: you borrow today and pay the principal plus the interest in the future.

Fixed payment loan: you borrow today but make equal payments per time period until your loan is all paid.

Coupon bond: firm sells the bond (borrows the funds) pays interest per period and at the maturity date pays the principal, too.

Discount bond: firm sells the bond at less than face value pays the face value at the maturity date.

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**Yield to Maturity: Simple Loan**

If you borrow $1000 today and asked to pay $1100 a year from now, what is the interest rate?

1000 = 1100/(1+i) 1+i = 1100/1000 i = 1100/1000 - 1 i = 1.1 - 1 i = .1 = 10% 8

**Yield to Maturity: Fixed Payment Loan**

Suppose you got a $50,000 mortgage. The bank wants you to pay $550.60 per month for 20 years. What is the interest rate you are charged?

50,000 = [550.60/(1+i)] +[550.60/(1+i )2 ] + [550.60/(1+i )3 ] + … + [550.60/(1+i) 240 ] i = .01 or 1% per month or 12% per year.

If the interest rate were lower than 12%, would your monthly payments increase or decrease?

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**Yield to Maturity: Coupon Bond**

You buy a ten-year corporate bond with a face value of $1000, coupon rate of 10% for $885.30. What is the interest rate you are earning?

885.30 = [100/(1+i)] + [100/(1+i) +1100/(1+i) 10 2 + … i = 0.12 or 12%.

What is the interest rate you are earning if you bought it for $1000?

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**Yield to Maturity: Consol**

You bought a consol for $1250. It pays $100 per year. What is the interest rate you are getting?

i = 100/1250 i = 0.08 or 8%.

What if you bought the consol for $1000; what would be the interest rate?

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**Yield to Maturity: Discount Bond**

You bought a discount bond for $800 which will pay $1000 a year from now. What is the interest rate?

i = (1000 - 800)/800 i = 200/800 i = 0.25 or 25%.

Would the interest rate be higher or lower than 25% if you had bought the bond for $900?

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**Other Measures of Interest Rates**

Current Yield: calculating the interest rate on a coupon bond as if it were a consol.

Current yield approaches yield to maturity when the price of the bond is close to the face value and the maturity date is far away.

Yield on a discount basis: the price of Treasury Bills is quoted in this fashion. It is always less than the yield to maturity.

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**An Example of Treasury Bill**

A T-Bill that has 91 days to maturity sells for 5.4% discount. Find the price and yield to maturity.

.054 (91/360) = [10,000 - P]/10,000 P = $9863.50

i = [(10,000 - 9863.50)/9863.50][360/91] i = 0.0547 or 5.47%.

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**Debt 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 9700 0.00333

60 ($178.62) Check

**DISCOUNT BOND < 1 year**

Price 995 Face Value 1000 Days 91 Interest Rate Check 0.0202

**DISCOUNT BOND > 1 year**

Price Face Value Months Interest Rate Check 950 1000 18 0.0348

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**Interest Rate vs. Rate of Return**

Rate of return on a bond includes the capital gains/losses.

Capital gain is the increase in the price of the bond. Capital loss is the decrease in the price of the bond.

If you bought a bond for $1000, held it for a year, received $100 interest payment and sold it for $1100, the return is R = [100 + 1100]/1000 = .20 or 20%.

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**Why Bonds Are Risky**

If you hold the bond until maturity, return and interest rate will be the same.

If interest rates rise while you own a bond, the price of the bond falls yielding capital losses.

The farther the maturity date the larger is the capital loss (or capital gain when interest rates fall). Prices and returns for long-term bonds are more volatile than those for shorter term bonds.

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**Real and Nominal Interest Rates**

If people expect inflation, they will not lend money at low rates. They would want to preserve the value of their money by including the inflation rate in the nominal interest rate.

Suppose you lend $1. You want to receive a real interest rate of r and want to compensate the erosion of the value of the money by the expected inflation rate, p.

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**Real and Nominal Interest **

1+i = (1+r)(1+p)

**Rates**

= 1 + r + p + rp i = r + p + rp Nominal interest rate is equal to real interest rate plus the expected inflation plus the product of the two.

This is called the Fisher equation.

For low values of real interest rates and expected inflation the last term can be ignored.

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