Chapter 4: Interest RatesObjectives• Define interest and explain its importance.• Write and explain the present value formula. • Write and explain the future value formula. • Calculate present and future value for multiple periods with annual and more

frequent compounding.• Define and price major types of debt instruments including discount bonds,

simple loans, fixed payment loans, coupon bonds, and perpetuities.• Define yield to maturity and identify the types of financial instruments it is

relatively easy to calculate. • Explain why bond prices move inversely to market interest rates.• Explain why some bond prices are more volatile than others.• Define rate of return and explain how it differs from yield to maturity.• Explain the difference between real and nominal interest rates.

1. The Interest of Interest1. The Interest of Interest

• Interest rates are crucial determinants of – Prices of assets, especially financial instruments like

stocks and bonds, and general macroeconomic conditions (economic growth). It is the price of money.

• Ways of measuring interest rates – Yield to maturity (YTM): The most economically

accurate way of measuring interest rates– Present value (PV): The value of money today– Future value (FV): The value of money at some point

in the future

2. Present and Future Value2. Present and Future Value

• Money today is always worth more than money tomorrow

• Nominal interest rates: the one we see• Real interest rate: The rate adjusted for

inflation• Compounding: Earning interest on interest• Compounding period: The amount of time

that passes before interest begins to earn interest

Future Value (FV)Future Value (FV)

• FV = PV(1 + i)n , where– FV = the future value (the value of your

investment in the future)– PV = the present value (the amount of your

investment today)– (1 + i)n = the future value factor – i = interest rate – n = number of terms (years, quarters, months,

days)

Present Value (PV)Present Value (PV)

• PV = FV/(1 + i)n, where– PV = the present value (the amount of your

investment today)– FV = the future value (the value of your

investment in the future)– (1 + i)n = the future value factor – i = interest rate – n = number of terms (years, quarters, months,

days)

3. Compounding Periods

• Interest is more valuable when received sooner than end of the year

• Interest may be compounded• Annually• Quarterly• Monthly• Daily• Continuously

• i - interest paid per period • n - number of periods (rather than the number of years)

4. Pricing Debt InstrumentsA bond, IOU, or other contract (like a discount bond, simple loan, fixed payment loan, or coupon bond) promising the payment of money in the future

4. Pricing Debt InstrumentsA bond, IOU, or other contract (like a discount bond, simple loan, fixed payment loan, or coupon bond) promising the payment of money in the future

• Known as a zero coupon bond, it is a debt instrument that makes only one payment, its face value on its maturity or redemption date

Discount Bond

• A debt instrument where the borrower repays the principal and interest at the end of the loan

Simple Loan

• A debt instrument in which the borrower makes periodic repayments of principal and interest

Fixed-payment Loan

• A debt instrument that makes interest payments periodically until its maturity or redemption date, when the final interest payment and the principal are to be paid

Coupon Bond

• The sum of the present values of each future payment will give you the price

• When the bond pays a rate lower than the going market, people are not willing to pay as much for it, so its price sinks– If the rate is pays is higher, the price of the bond

will be higher than the face value

4. Pricing Debt Instruments

5. What’s the Yield on That?Calculating Yield to Maturity: Zero coupon bond

• i = (FV – PV)/PV

Calculating Yield to Maturity: Perpetuity

• i = FV/PV

• Current yield: A quick but flawed method for calculating interest rates of nonperpetual debt– i = FV/PV

6. Calculating Returns

• Return: A measure of the profitability of an investment that takes into account changes in the value of the bond or other asset

• R = (C + Pt1 – Pt0)/Pt0 where• R: return from holding the asset for some time period, t0 to t1

• Pt0: the price at time t0 (this can also be thought of as the purchase price)

• Pt1: the price at time t1 (this can also be thought of as the sale or going market price)

• C: coupon (or other) payment

6. Calculating Returns• Bond prices and interest rates are inversely

related• Even if there is no default, wealth can be lost

by investing in bonds or other fixed-rate financial instruments, e.g. Is Treasury Bond risk free?

6. Calculating Returns

• The risk that the market price of a bond or other debt instrument will decrease due to increases in the interest rate

Interest Rate Risk

• The risk that a bond or other debt instrument will not make the promised payments

Default Risk

7. Inflation and Interest Rates• If nominal rates do not increase:

– lenders receive more nominal dollars than they lent but actually get back less purchasing power

7. Inflation and Interest Rates7. Inflation and Interest Rates

• The Fisher Equation (approximate!!!):i = ir + π or, ir = i – π or, π = i - ir

• ir: the real interest rate• i: the nominal interest rate• π: inflation (or expected inflation)

7. Inflation and Interest Rates7. Inflation and Interest Rates

• Ex post: After the fact, the nominal interest rate is equal to the real interest rate plus actual inflation

• Ex ante: Before the fact, the nominal interest rate is equal to the real interest rate plus the expectation of inflation