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3 NOTES I P (1) P (1) R ate of R etu rn PLUS the capital gain (+) or loss (-) in the market value of A over period T, measured as the market value of A at the end of period T minus the market value of A at the beginning of period T. C C C P (2) C + P (2) Y ield to M a tu rity P V = P (1) The return rate on asset A over the holding period T is then defined to be the return on A over period T divided by the market value of A at the beginning of period T. More precisely, suppose that an asset A is held over a time period that starts at some time t and ends at time t+1. Let the market value of A at time t be denoted by P(t) and the market value of A at time t+1 be denoted by P(t+1). Finally, let V(t,t+1) denote the sum of all payments accruing to the holder of asset A from t to t+1, assumed to be paid out at time t+1. Then, by definition, the return rate on asset A from t to t+1 is given by the following formula: (24) Return Rate on V(t,t+1) + P(t+1) - P(t) Asset A From = time t to t+1 P(t) C V(t,t+1) P(t+1) - P(t) = P(t) P(t) = payments + Capital Gain (if +) received as or Loss (if -) as percentage percentage of P(t) of P(t) Formula (24) holds for any asset A, whether physical or financial. In particular, it holds for bonds. The question then arises: For bonds, what is the connection between C C + F V C + F V the return rate defined by formula (24) and the interest rate on the bond defined by yield to maturity, current yield, or discount yield The return rate on a bond is not necessarily equal to the interest rate on that bond, whether defined by yield to maturity, the current yield, or the discount yield. The reason for this is that the return rate calculated for a particular holding period takes into account any capital gains or losses that occur during this holding period, in addition to payments received during the holding period. In contrast, the current yield ignores capital gains and losses altogether, and the yield to maturity and the discount yield only take into account the overall anticipated capital gain or loss that is incurred when the bond is held to maturity (as measured by the difference between the final face value payment and the initial purchase price). EXAMPLE A: COUPON BONDS Suppose you purchase a coupon bond at time t at a price P(t) with coupon payment C and face value F, you receive a coupon payment C at time t+1, and you also sell the coupon bond in a secondary market at a price P(t+1) at time t+1. By definition, the current yield that you receive on this coupon bond during the holding period from t to t+1 is given by (25) ic(t) = C/ P(t) Also, the percentage capital gain or loss you incur on the coupon bond during the holding period from t to t+1, denoted by g(t,t+1), is given by P(t+1) - P(t) (26) g(t,t+1) = P(t) It then follows from definition (9) that the return rate on the coupon bond from t to t+1 can be expressed as C + P(t+1) - P(t) (27) = ic(t) + g(t,t+1). P(t) Clearly the return rate (24) coincides with the current yield ic(t) if (28) P(t) = P(t+1). Condition (28) implies that there are no capital gains or losses on the coupon bond during the holding period from t to t+1, i.e., g(t,t+1) =. Conversely, if condition (28) fails to hold, then the return rate (27) does not coincide with the current yield ic(t). Thus, condition (28) is both necessary and sufficient for the return rate (27) from t to t+1 to equal the current yield ic(t). That is: if and only if Return Rate = ic(t) < > P(t) = P(t+1) From t To t+1 Now suppose, instead, that Time Period from t to t+1 = Bond Maturity Period: if and only if Return Rate = i(t) < > Period From t to t+1 From t To t+1 = Maturity Period 17

4 EXAMPLE B: DISCOUNT BOND For a discount bond with a purchase price Pd and a face value F, the return rate (9) over any holding period t to t+1 reduces to P(t+1) - Pd (29) Pd Recalling definition for the discount yield idb, it is seen that the return rate will generally differ from idb except in the degenerate case Pd = P(t+1) = F when both are zero. In summary, then, only under special conditions will the return rate for a bond over a given holding period coincide with the yield to maturity, the current yield, or the discount yield. XI. Real vs. Nominal Interest Rates The interest rate measures examined to date have all been "nominal" in the sense that they have not been adjusted for expected changes in prices. What actually concerns a "rational" saver considering the purchase of a debt instrument is not the nominal payment stream he or she expects to earn in future periods but rather the command over purchasing power that this nominal payment stream is expected to entail. This purchasing power depends on the behavior of prices. Let infe(t) denote the expected inflation rate at time t, and let i(t) denote the (nominal) interest rate for some debt instrument at time t. Then the real interest rate associated with i(t) is defined by the following "Fisher equation:" (3) ir(t) = i(t) - inf e (t). That is, the real interest rate is the nominal interest rate minus the expected inflation rate. Note: As explained by Mishkin, the real interest rate defined by (16) is more precisely called the ex ante real interest rate because it adjusts for expected changes in the price level. If the expected inflation rate in (16) is replaced by the actual inflation rate, one obtains the ex post real interest rate. Real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates, and hence provide a better indicator of the incentives to borrow and lend. In particular, for any given nominal interest rate i on a debt instrument D, the incentive to borrow (issue D) will be higher if the real interest rate associated with i is lower (i.e., the expected inflation rate is higher). This is so since a higher expected inflation rate means the borrower (issuer of D) can expect to pay off his future nominal debt obligations using cheaper dollars than he borrowed. For this same reason, the incentive to lend (purchase D) will be lower if the real interest rate associated with i is lower. A similar distinction is made between the (nominal) return rate defined by (9), which has not been adjusted for expected changes in prices, and the "real return rate" which is subject to such adjustment. More precisely, the real return rate on any asset A over any holding period from t to t+1 is defined to be the (nominal) return rate (9) minus the expected inflation rate inf e (t). TABLE One-Year Returns on Bonds When Interest Rates Rise from 1% to (1) (2) (3) (4) (5) (6) Years to Initial Initial Price Rate of Rate of maturity current price next capital Return when bond yield year gain (2+5) is purchased (%) (\$) (\$) (%) (%) XII. Interest Rate Risk As previously seen, at any time t, the yield to maturity i(t) for a bond with a maturity date greater than t moves inversely with its current acquisition price P(t) -- that is, if \$ 1, \$ 1, infinite % \$ 1 + \$1, 1% P(2) \$ 1 + P(2) \$1 one goes up, the other goes down. \$ 1 + \$ 1, \$ 1, \$ P(1) 1, \$ 1 + P(2) A fall in the price of an already held bond signals a capital loss to the holder. Consequently, the net effect of an increase in the yield to maturity for an already held bond can be a decrease in the return rate to its holder. P(2) 1% 1% \$ 1 + P(2) P(2) \$ 1 + \$ 1, \$1 \$1 \$ 1 + \$ 1 18

5 NOTES J The uncertainty regarding return rate that bond holders face due to possible changes in yield to maturity is called interest rate risk. Table illustrates interest rate risk for bonds of different maturities, each with a coupon payment of \$1 and a face value of \$1. This illustration is worth reviewing with some care. First note that, for each bond in the Table, the initial yield to maturity i(1) for year 1 ("this year") is equal to the initial current yield ic(1) = 1 percent for year 1 because the initial price of the bond is set at its face value of \$1. However, by assumption, the yield to maturity i(2) for year 2 ("next year") increases to 2 percent. The coupon bond listed in the Table with a 1-year maturity has a price P(2) in year 2 that is fixed (by contract) at the bond's face value, \$1. For all other listed coupon bonds, however, their maturities exceed one year. Consequently, when i(2) increases, their price P(2) in year 2 decreases to some value smaller than their original price P(1) = \$1 in year 1 and hence smaller than the bond's face value of \$1. For example, for the coupon bond with a 3-year maturity, P(2) = \$53. The return rate from year 1 to year 2 for each of the coupon bonds in Table 2 is given by the sum of the current yield ic(1) = C/P(1) for year 1 and the capital loss g(1,2) = [P(2)-P(1)]/P(1) from year 1 to year 2. Consequently, except for the bond with a one-year maturity, these return rates are smaller than they would have been without the increase in the yield to maturity in year 2. Indeed, for the coupon bond with a 3-year maturity, the capital loss g(1,2) is so large (-49.7 percent) that it overwhelms the 1 percent initial current yield ic(1) = 1 percent, resulting in a negative return rate of percent from year t=1 to t=2. More precisely, examining the return rates in column (6) of the Table as the maturity is decreased from 3 years to 1 year, it is seen that the coupon bonds with longer maturities experience a greater decline in their return rates when the yield to maturity i(2) increases. This is due to the fact that this increase in i(2) results in a smaller decline in the price P(2) for coupon bonds with smaller maturities and hence a smaller capital loss. [To see why, consider the formula Pb = PV(i) from which the yield to maturity i is determined.] Indeed, for coupon bonds with a one-year maturity, P(2) remains fixed at the face value \$1. Increase in yield to maturity from year 1 to year 2 / \ i(1)=ic(1) i(2) (3) /\/\/\-- Year 1 2 N P(1) P(2) Maturity Date (N > 2) \ / Capital loss from t=1 to t=2 An important implication of this illustration is that the return rates of bonds with longer-term maturities respond more dramatically to changes in the yield to maturity than bonds with shorter-term maturities. That is, longer-term bonds are more subject to interest rate risk. This is one reason why investment in longer-term bonds is considered more risky than investment in shorter-term bonds. XIII. More Basic Concepts and Key Issues Current yield Discount yield (Nominal) return rate Real interest rate Real return rate Consol bond (perpentuity) Capital gain or loss Interest rate risk Expected inflation rate Calculating the current yield for a consol bond For a coupon bond, how does its maturity affect the relationship between its current yield and its yield to maturity What is the relationship between its current yield, its coupon rate, its purchase price, and its face value What is the relationship between its current yield, its yield to maturity, its purchase price, and its face value What is the relationship between its current yield and its purchase price, given any fixed level for its coupon payment For a discount bond, all else remaining the same, why do its discount yield and its yield to maturity always move together How to read financial bond pages for information on Treasury bonds and notes, Treasury bills, and corporate bonds traded on stock exchanges. Why is the return rate on a bond not necessarily equal to its interest rate How does the maturity of a bond affect its interest rate risk Why do bonds with long maturities expose bond holders to greater interest rate risk than bonds with shorter maturities What is the relationship between real and nominal interest rates Why do real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates Why do real return rates provide a more accurate measure of the true gains or losses from holding an asset than nominal return rates 19

7 NOTES K Summary of major interest rates on the market Government securities Municipal Issues (notice that some are tax free and so their yield is adjusted by 31% tax bracket) Private Bonds 9

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