Department of Economics
Saint Louis University
Professor: Rapach
Fall 2008
ECON 420
Money and Banking

Chapter Outline for “Chapter 4—Understanding Interest Rates,” Frederic S. Mishkin, The Economics of Money, Banking, and Financial Markets, Eighth Edition (New York: Addison-Wesley, 2006)

MEASURING INTEREST RATES

Present Value

present value: based on the notion that a dollar paid to you one year from now is less valuable to you than a dollar paid to you today

simple loan: lender provides the borrower with an amount of funds (called the principal) that must be repaid to the lender at the maturity date, along with an additional payment for the interest

i = $10/$100 = 0.10 = 10%

$100 x (1 + 0.10) = $110

$110 x (1 + 0.10) = $121

$100 x (1 + 0.10) x (1 + 0.10) = $100 x (1 + 0.10)² = $121

$121 x (1 + 0.10) = $100 x (1 + 0.10)³ = $133

$100 x (1 + i)ⁿ

$100 = $133/(1 + 0.10)³

PV = FV/(1 + i)ⁿ

APPLICATION Simple Present Value

PV = $250/(1 + 0.15)² = $189.04

APPLICATION How Much is That Jackpot Worth?

PV = $1M/(1 + 0.10) + … + $1M/(1 + 0.10)²⁰ = $9.4M

Four Types of Credit Market Instruments

1.      Simple loan

2.      Fixed-payment loan

3.      Coupon bond

4.      Discount bond

Yield to Maturity

yield to maturity: interest rate that equates the present value of payments received from a debt instrument with its value today

Simple Loan

APPLICATION Yield to Maturity on a Simple Loan

PV = CF/(1 + i)ⁿ

$100 = $110/(1 + i)

i = ($110 – $100)/$100 = $10/$100 = 0.10 = 10%

For simple loans, the simple interest rate equals the yield to maturity.

Fixed-Payment Loan

$1,000 = $126/(1 + i) + $126/(1 + i)² + $126/(1 + i)³ + … + $126/(1 + i)²⁵

LV = FP/(1 + i) + FP/(1 + i)² + FP/(1 + i)³ + … + FP/(1 + i)ⁿ

APPLICATION Yield to Maturity and the Yearly Payment on a Fixed-Payment Loan

$100,000 = FP/(1 + 0.07) + FP/(1 + 0.07)² + … + FP/(1 + 0.07)²⁰

Coupon Bond

P = $100/(1 + i) + $100/(1 + i)² + $100/(1 + i)³ + … + $100/(1 + i)¹⁰ + $1,000/(1 + i)¹⁰

P = C/(1 + i) + C/(1 + i)² + C/(1 + i)³ + … + C/(1 + i)ⁿ + F/(1 + i)ⁿ

APPLICATION Yield to Maturity and the Bond Price for a Coupon Bond

See Table 1

1.      When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate.

2.      The price of a coupon bond and the yield to maturity are negatively related; that is, as the yield to maturity rises, the price of the bond falls, As the yield to maturity falls, the price of the bond rises.

3.      The yield to maturity is greater than the coupon rate when the bond price is below its face value.

consol (perpetuity): perpetual bond with no maturity date and no repayment of principal that makes fixed coupon payment of $C forever

P_c = C/i_c

i_c = C/P_c

APPLICATION Perpetuity

i_c = C/P_c = $100/$2,000 = 0.05 = 5%

Discount Bond

$900 = $1,000/(1 + i)

i = ($1,000 – $900)/$900 = 0.111 = 11.1%

i = (F – P)/P

Summary

Current bond prices and interest rates are negatively related: when the interest rate rises, the price of the bond falls, and vice versa.

YIELD ON A DISCOUNT BASIS

i_db = [(F – P)/F] x [360/(days to maturity)]

The greater the difference between the purchase price and the face value of the discount bond, the more the discount yield understates the yield to maturity.

Like the yield to maturity, discount yield is negatively related to the price of a bond.

APPLICATION Reading the Wall Street Journal: The Bond Page

THE DISTINCTION BETWEEN INTEREST RATES AND RETURNS

rate of return: payment to the owner plus the change in its value, expressed as a fraction of its purchase price

The return on a bond will not necessarily equal the interest rate on that bond.

R = (C + P_{t + 1} – P_t)/P_t

R = C/P_t + (P_{t + 1} – P_t )/P_t

C/P_t = i_c

(P_{t + 1} – P_t)/P_t = g

R = i_c + g

See Table 2

·        The only bond whose return equals the initial yield to maturity is one whose time to maturity is the same as the holding period.

·        A rise in interest rates is associated with a fall in bond prices, resulting in capital losses on bonds whose term to maturity are 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 the increase in the interest rate.

·        Even though a bond has a substantial initial interest rate, its return can turn out the be negative if interest rates rise.

Maturity and the Volatility of Bond Returns: Interest-Rate Risk

Prices and returns for long-term bond are more volatile than those for shorter-term bonds.

interest-rate risk: the riskiness of an asset’s returns that results from interest-rate changes

Summary

THE DISTINCTION BETWEEN REAL AND NOMINAL INTEREST RATES

nominal interest rate: makes no allowance for inflation

real interest rate: interest rate that is adjusted by subtracting expected changes in the price level (inflation) so that it more accurately reflects the true cost of borrowing

i = i_r + p^e

i_r = i - p^e

APPLICATION Calculating Real Interest Rates

i_r = 0.08 – 0.10 = –0.02 = –2%

When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend.

See Figure 1

indexed bonds: bonds whose interest and principal payments are adjusted for changes in the price level

QUESTIONS AND PROBLEMS: 2, 4, 6, 8, 12, 14