Understanding interest rates The Economics of Money, Banking, and Financial Markets Mishkin, 7th ed. Chapter 4 Understanding interest rates

Present value The present value of a given future payment is smaller when: the interest rate is higher, and/or the payment is further in the future.

Present value Simple loan of $1 at 10% interest Year 1 2 3 n $1.10 $1.21 $1.33 $1x(1 + i)n $1 PV of future $1 = (1 + i)n

Types of credit market instruments Simple loan Fixed-payment loan Coupon bond Discount (zero coupon) bond

Yield to maturity Yield to maturity for a debt instrument = the interest rate at which the present value of the payment stream equals the current price of the instrument, i.e., interest rate that equates today’s value with present value of all future payments

Yield to maturity – simple loan (i =10%) $110 – $100 $10 i = = = 0.10 = 10% $100 $100

Yield to maturity – fixed-payment loan (i =12%)

Yield to maturity – coupon bond

Yield to maturity – coupon bond When bond is sold at face value, the yield to maturity = coupon rate Price of bond and yield to maturity are inversely related Yield to maturity is greater than the coupon rate when bond price is below the face value

Consol (perpetuity)

Yield to maturity – discount bond

Overview Current bond prices and interest rates (yields to maturity) are inversely related for all types of bonds.

Bond Page of the Newspaper (a) Current yield=C/P Change on bid price from yesterday Both coupon bonds bond:Maturity>10yrs note: Maturity<10yrs For long-term bonds, the YTM is almost the same as current yield, but this is not true for short-term bonds. Differences represent dealer’s profits

Bond Page of the Newspaper (b), (c) Both discount yields= ((F-P)/F)*(360/days to Mat.) True discount yield= ((F-P)/P)*(365/days to Mat.) discount bonds Maturity<1yr Change on asked yield from yesterday Current yield=C/P Coupon rate 5 5/8 138*$1,000 Mature at 2031

Other measures of interest rates Current yield: ic = C/P ...(7) P=price of the coupon bond, C=yearly payment Only accurate for consols A reasonable approximation to the yield to maturity for bonds with a maturity of 20+ years. Relatively poor approximation to the yield to maturity for short-term bonds. A better approximation when the bond price is close to the face value. Changes in current yield are always in the same direction as changes in the yield to maturity.

Other measures of interest rate idb: yeild on a discount basis, F: face value P: purchase price of the discount bond Always understates the yield to maturity since: this formula uses F instead of P(<F) in denominator a year (=365 days) has more than 360 days Changes in the same direction as the yield to maturity. (larger days to maturity has smaller i-idb )

Interest rates and rate of return Rate of return includes capital gains and losses as well as current interest receipts

One-year returns on 10%-coupon-rate bond

Maturity and the Volatility of Bond Returns Key Findings from Table 2 1. Only bond whose return = yield is one with maturity = holding period 2. For bonds with maturity > holding period, i  P implying capital loss 3. Longer is maturity, greater is % price change associated with interest rate change 4. Longer is maturity, more return changes with change in interest rate 5. Bond with high initial interest rate can still have negative return if i  Conclusion from Table 2 Analysis 1. Prices and returns more volatile for long-term bonds because have higher interest-rate risk 2. No interest-rate risk for any bond whose maturity equals holding period

Bond prices and rate of return Rate of return = initial yield to maturity if a bond is held to maturity Bond prices decline as interest rates rise, leading to capital losses Long-term bonds exhibit more variability in price in response to interest-rate changes Longer-term bonds experience more capital gains and losses when interest rates change (interest-rate risk) A negative return can occur for bonds when the interest rate rise.

Real and nominal interest rates Fisher equation:

U.S. Real and Nominal Interest Rates

Financial innovation In January 1997, the U.S. Treasury introduced indexed bonds. Interest and principal payments are adjusted for changes in the price level.