Mishkin observes that the yield to maturity is the most accurate measure of interest rates and notes that he will henceforth use the terms "interest rate" and "yield to maturity" interchangeably throughout the remainder of his text.
Nevertheless, since the yield to maturity can be difficult to calculate, other less accurate measures of interest rates are commonly used in the financial pages of newspapers and elsewhere to report the properties of debt instruments. Mishkin discusses two such measures at some length: "current yield" and "discount yield."
The current yield is an approximation to the yield to maturity for coupon bonds. More precisely, letting Pb denote the purchase price of a coupon bond, and C denote its coupon payment, the current yield, denoted below by ic (as in Mishkin), is given by:
C (1) ic = ----- . Pb
In general, for most coupon bonds, the current yield will differ in value from the yield to maturity. However, it can be shown that the current yield equals the yield to maturity for a special type of coupon bond, called a "consol." A consol is a coupon bond that has an infinite maturity and hence never repays its face value F. Rather, the holder of a consol receives a coupon payment C in perpetuity -- that is, in each future payment period without end -- implying that the payment stream to the holder takes the special form (C,C,C,...).
As detailed by Mishkin, the formula Pb = PV(i) for determining the yield to maturity i for a consol reduces to
C C (2) Pb = --- , which implies that i = --- . i Pb
Comparing (1) and (2), it follows that -- for a consol -- the current yield ic equals the yield to maturity i because both are equal to C/Pb.
For coupon bonds with less than infinite maturities, the current yield ic no longer coincides with the yield to maturity i. However, the current yield becomes an increasingly better approximation for the yield to maturity as the maturity of a coupon bond becomes longer and longer (hence closer and closer to the infinite maturity of a consol). That is, all else remaining the same, ic provides an increasingly accurate approximation to i as one considers coupon bonds with successively longer maturities N.
For fixed C, F, and Pb: implies (3) Maturity N increases -------> ic approaches i
Another aspect of a coupon bond that determines how accurate an approximation ic provides to the yield to maturity i is the difference between the bond's purchase price Pb and its face (or par) value F.
From formula (1), it is seen that the current yield ic is equal to the coupon rate C/F for a coupon bond when the purchase price Pb equals the face value F. As previously seen, however, the coupon rate C/F equals the yield to maturity i when the purchase price Pb of a coupon bond equals its face value F.
Consequently, all else remaining the same, the current yield ic provides an increasingly more accurate approximation to the yield to maturity i as one considers coupon bonds whose purchase prices Pb are successively closer to their face values F.
Given C, F, and N: implies (4) Pb approaches F -------> ic approaches i .
Finally, it follows directly from definition (1) for the current yield ic that, given any fixed value for the coupon payment C, ic is inversely related to the bond purchase price Pb. That is,
Given C: if and only if (5) ic increases <--------------> Pb decreases .
Recall from previous discussion that, for given C, F, and N, the yield to maturity i is also inversely related to a bond's purchase price Pb. Consequently, one obtains the following important conclusion:
Note that this positive co-movement between ic and i holds even if ic is a bad approximation to i in level terms in the sense that the difference between ic and i is large.
U.S. Treasury bills are an example of a discount bond. For ease of calculation, interest rates on many discount bonds such as Treasury bills and commercial paper are quoted on a 360-day "discount yield" basis (or "bank discount basis") rather than on a yield-to-maturity basis, as follows.
Let F denote the face value of a discount bond, and let Pd denote the purchase price of the discount bond. Then the discount yield, denoted below by idb (as in Mishkin), is given by:
F - Pd 360 (6) idb = -------- * ----------------- . F Days to Maturity
Let us see how idb compares, for example, to the yield to maturity i for a one-year discount bond. As discussed by Mishkin, in the case of a one-year discount bond the usual formula Pd = PV(i) for determining the yield to maturity takes the form
F - Pd (7) i = ---------- . Pd
Comparing (7) with (6) for the special case of a discount bond with a one year maturity (i.e., days to maturity = 365), it follows that
F 365 (8) i = idb * ----- * ------- . Pd 360
Consequently, recalling that discount bonds are typically priced at a discount (Pd < F), it follows that the yield to maturity i for a discount bond with a one-year maturity is typically greater than the discount yield idb.
Another implication of formula (6) is that, for any given discount bond with a fixed face value F and a fixed maturity N, the discount yield idb is inversely related to the price Pd of the discount bond -- that is, when idb increases, Pd decreases, and vice versa.
Recall from previous notes that the yield to maturity i on a discount bond is also inversely related to the purchase price Pd. This follows directly from the general formula Pd=PV(i) used to determine i for discount bonds -- see, for example, relation (7), which is what the general formula Pd=PV(i) reduces to when the discount bond has a one-year maturity.
Consequently, as for the current yield, one obtains the following important observation:
As for the current yield, this positive co-movement between idb and i holds even if idb is a bad approximation to i in level terms in the sense that the difference between idb and i is large.
This section uses the financial pages from an issue of the New York Times to illustrate and elaborate key points. The class ppt notes use an example from the Wall Street Journal. As will be seen, the form is essentially the same, with only minor differences in notation and ordering of presented information.
Treasury Bonds and Notes:
Treasury bonds (T-bonds) are coupon bonds with a maturity greater than ten years, and Treasury notes (T-notes) are coupon bonds with a maturity of between one and ten years. As in the Wall Street Journal, the New York Times provides a single table reporting on T-bonds and T-notes because both have the same structure.
Below is a sample listing from the T-bonds and T-notes table appearing in the New York Times (February 17, 1999, p. C17) which reports information for the previous trading day, February 16, 1999:
Month Rate Bid Ask Chg Yld Feb 00 p 7 1/8 102.08 102.10 +0.01 4.80 Aug 03-08 8 3/8 112.25 112.27 -0.05 5.14
Remark on Notation: For expositional simplicity, T-bonds and T-notes will hereafter be lumped together and simply referred to as "bonds."
The first column (Month) of the sample listing identifies the month and year that a bond matures. If two maturity dates are shown, the second is the actual maturity date. The first indicates the first date at which the bond might be called. A bond is called when it is retired early at the discretion of the issuer (borrower), an event that generally happens only when prevailing interest rates lie below the coupon rate.
A footnote may next be provided to indicate that a bond has some special feature. For example: the letter "p" denotes "T-note, non-resident aliens exempt from withholding tax"; the letter "k" denotes "T-bond, non-resident aliens exempt from withholding"; and the letters "zr" denote "zero coupon issue".
The second column (Rate) identifies a bond's annual coupon rate, i.e., the annual coupon payment as a percentage of face value. Usually this annual coupon payment is paid in two equal semiannual installments.
The third and fourth columns (Bid, Ask) provide information about a bond's bid and asked prices, which by convention are quoted as a percentage per $100 of face value (so that 100 = face value) with fractions in 32s. Unlike the Wall Street Journal, which uses a colon to indicate fractional values in 32s (e.g., 102:08 = 102 8/32), the New York Times uses a decimal point (e.g., 102.08 = 102 8/32).
The New York Times lists Street Software/Bear Stearns as the source of its price quotations. These quotations are not necessarily prices at which an individual could actually have traded on the trading day in question but rather represent the approximate market prices that prevailed.
More precisely, the bid price quoted for a bond is the approximate market price offered by prospective buyers of the bond on the trading day in question, so it indicates approximately how much you would have received if you had sold the bond on that day. In contrast, the asked price quoted for a bond is the approximate market price demanded by prospective sellers of the bond on the trading day in question, so it indicates approximately how much you would have had to pay to purchase the bond on that day.
The asked price minus the bid price -- referred to as the bid-ask spread -- reflects the gross profit margin of the bond dealers who handle trades in this bond. Hence, for obvious reasons, the asked price always exceeds the bid price.
The fifth column (Chg) indicates the change in the BID price from the previous trading day's quotation.
The sixth and final column (Yld) provides the yield to maturity on the bond using the currently quoted ASKED price as the purchase price.
Treasury bills (T-bills) are discount bonds with a maturity of one year or less. Consequently, they have no coupon rate and are identified solely by their maturity date.
Below is a sample listing from the T-bills table appearing in the New York Times (February 17, 1999, page C17) which reports information for the previous trading day, February 16, 1999:
Date Bid Ask Chg Yield Feb 25 99 4.07 4.05 - 0.04 4.11 Aug 19 99 4.46 4.44 + 0.04 4.61
The first column (Date) gives the month, day, and year of the maturity date.
The second column (Bid) gives the discount yield (6) in percentage terms using as the purchase price Pd the BID price, i.e., the price offered by prospective buyers. The third column (Ask) gives the discount yield (6) in percentage terms using as the purchase price Pd the ASKED price, i.e., the price demanded by prospective sellers.
Recall that the discount yield varies inversely with the purchase price Pd. It follows that, for T-bill issues, the bid discount yield reported in the Bid column is always greater than the asked discount yield reported in the Ask column, indicating that the bid price is less than the asked price.
The fourth column (Chg) reports the change in the asked discount yield from the previous trading day measured in terms of basis points, which are hundredths of a percentage point (e.g., -0.04 means the asked discount yield has fallen in percentage terms by 4 basis points).
The fifth and final column (Yield) provides the yield to maturity using the current ASKED price as the current value.
Corporate Bonds Traded on Stock Exchanges:
As noted by Mishkin, corporate bonds typically take the form of coupon bonds.
A majority of bonds, and all municipal or tax-exempt bonds, are not listed on exchanges; rather, they are traded over-the-counter. However, the New York Stock Exchange (NYSE), and to a much less extent the American Stock Exchange (AMEX), do list various coupon bonds issued by corporations with strong credit ratings.
Below is a sample listing from the NYSE corporate bond table appearing in the New York Times (February 17, 1999, page C17) which reports information for the previous trading day, February 16, 1999:
Company Coupon Mat. Cur.Yld. Vol. Price Chg. Rate ATT 5 1/8 01 5.1 60 99 7/8 - 1/8 ARetire 5 3/4 02 cv 25 89 1/2 + 1 1/2
The first column (Company) shows the issuing company, the second column gives the original coupon rate, and the third column gives the last two digits of the maturity year. The fourth column reports the annual current yield (Cur. Yld.). In some cases, a footnote may instead be inserted to call attention to a special feature of the bond; for example, the letters "cv" in the above table denote "convertible into stock under special conditions".
The remaining three columns report the number of bonds traded for the day measured in $1000 face value (Vol.), the bond's closing price for the day expressed as a percentage of face value with 100 equaling face value (Price), and the difference between the current trading day's closing price and the previous trading day's closing price (Chg).
Given any asset A held over any given time period T, the return to A over the holding period T is, by definition:
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:
(9) Return Rate on V(t,t+1) + P(t+1) - P(t) Asset A From = --------------------------- time t to t+1 P(t) V(t,t+1) P(t+1) - P(t) = --------- + ------------- P(t) P(t) = payments + Capital Gain (if +) received as or Loss (if -) as a percentage percentage of P(t) of P(t)
Formula (9) 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 the return rate defined by formula (9) and the interest rate on the bond defined by yield to maturity, current yield, or discount yield?
As discussed by Mishkin, 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: Coupon Bonds
Suppose you purchase a coupon bond at time t at a purchase price P(t) which has a coupon payment C, a face value F, and a maturity date at time t+k with k GREATER than 1. Suppose you receive a coupon payment C at time t+1, and you also sell the coupon bond in a secondary market at time t+1 at a price P(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
C (10) ic(t) = ------ . 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) (11) 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) (12) --------------------- = ic(t) + g(t,t+1) . P(t)
Clearly the return rate (12) coincides with the current yield ic(t) if
(13) P(t) = P(t+1) .
Condition (13) 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) = 0. Conversely, if condition (13) fails to hold, then the return rate (12) does not coincide with the current yield ic(t).
Thus, condition (13) is both necessary and sufficient for the return rate (12) 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
(14a) Time t+1 = Maturity Date of the Coupon Bond; (14b) Face Value F = Purchase Price P(t+1) of the Coupon Bond at t+1.
Condition (14b) makes sense if the bond is sold at time t+1 AFTER the receipt of the coupon payment C but BEFORE the receipt of the face value payment F. Given conditions (14a,b), the return rate (12) for the coupon bond reduces to the definition of the yield to maturity i(t). To see this, simply apply the usual formula Pb = PV(i) for obtaining the yield to maturity i for a coupon bond held to maturity. Conversely, for a coupon bond for which conditions (14a,b) do NOT hold, the return rate (12) from t to t+1 does NOT coincide with the yield to maturity i(t).
Thus, for a coupon bond purchased at time t, conditions (14a,b) are both necessary and sufficient for the bond's return rate (12) from t to t+1 to equal the bond's yield to maturity i(t). That is:
if and only if Return Rate = i(t) <--------------> Conditions (14a,b) Hold From t To t+1
Example: 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 (15) ----------- . Pd
Recalling definition (6) for the discount yield idb, it is seen that the return rate (15) 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.
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:
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 (i.e. to issue D) will be higher if the real interest rate associated with i is lower (i.e. if the expected inflation rate is higher). This is so since a higher expected inflation rate means the borrower (i.e. the 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 (i.e. to 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 infe(t).
Consider a bond B held at time t whose maturity date exceeds t. Let the yield to maturity on B at time t be denoted by i(t) and let the price of B at time t be denoted by P(t).
As previously seen, the definition of i(t) implies that i(t) must move inversely to P(t). That is, if one goes up, the other goes down.
Suppose holding periods are measured in years, with t denoting the beginning of year t, and i(t) denotes the yield to maturity on B at time t. An increase in i(t) at time t --- equivalently a fall in P(t) at time t --- results in a decrease in the return rate to B over the holding period from t-1 to t. This is because any holder of B at time t who chooses to sell B at time t would receive a smaller payment for B at time t than what he would have received without the price fall at time t, implying a smaller capital gain (or a larger capital loss) over the holding period from t-1 to t.
The uncertainty regarding return rates that bond holders face due to possible changes in yields to maturity is called interest rate risk.
Mishkin (Table 2) illustrates interest rate risk for bonds of different maturities, each with a coupon payment of $10 and a face value of $1000. This illustration is worth reviewing with some care.
First note that, for each bond in Table 2, the initial yield to maturity i(1) for year 1 ("this year") is equal to the initial current yield ic(1) = 10 percent for year 1 because the initial price of the bond is set at its face value of $1000. However, by assumption, the yield to maturity i(2) for year 2 ("next year") increases to 20 percent.
The coupon bond listed in Table 2 with a 1-year maturity has a price P(2) in year 2 that is fixed (by contract) at the bond's face value, $10000. 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) = $1000 in year 1 and hence smaller than the bond's face value of $1000. For example, for the coupon bond with a 30-year maturity, P(2) = $503.
Using representation (12), 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 30-year maturity, the capital loss g(1,2) is so large (-49.7 percent) that it overwhelms the 10 percent initial current yield ic(1) = 10 percent, resulting in a negative return rate of -39.6 percent from year t=1 to t=2.
More precisely, examining the return rates in column (6) of Table 2 as the maturity is decreased from 30 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 $1000.
An important implication of this illustration, then, 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.
Increase in yield to maturity from year 1 to year 2 /|\ | i(1)=ic(1) i(2) (17) |--------------|-----------/\/\/\--| Year 1 2 N P(1) P(2) Maturity Date | (N > 2) \|/ Capital loss from t=1 to t=2