CHAPTER 3
MEASURING YIELD
CHAPTER SUMMARY
In Chapter 2 we showed how to determine the price of a bond, and we described the relationship
between price and yield. In this chapter we discuss various yield measures and their meaning for
evaluating the relative attractiveness of a bond. We begin with an explanation of how to compute
the yield on any investment.
COMPUTING THE YIELD OR INTERNAL RATE OF RETURN ON ANY
INVESTMENT
The yield on any investment is the interest rate that will make the present value of the cash flows
from the investment equal to the price (or cost) of the investment.
Mathematically, the yield on any investment, y, is the interest rate that satisfies the equation.
P=
CF 1
CF 2
CF 3
CF N
+
+
+ . . .+
1
2
3
N
1  y  1  y  1  y 
1  y 
where CFt = cash flow in year t, P = price of the investment, N = number of years. The yield
calculated from this relationship is also called the internal rate of return.
Solving for the yield (y) requires a trial-and-error (iterative) procedure. The objective is to find
the yield that will make the present value of the cash flows equal to the price. Keep in mind that
the yield computed is the yield for the period. That is, if the cash flows are semiannual, the yield
is a semiannual yield. If the cash flows are monthly, the yield is a monthly yield. To compute the
simple annual interest rate, the yield for the period is multiplied by the number of periods in the
year.
Special Case: Investment with Only One Future Cash Flow
When the case where there is only one future cash flow, it is not necessary to go through the
time-consuming trial-and-error procedure to determine the yield. We can use the following
equation.
1/ n
 CFn 
y=
 P 
1.
Annualizing Yields
To obtain an effective annual yield associated with a periodic interest rate, the following formula
30
is used:
effective annual yield = (1 + periodic interest rate)m – 1
where m is the frequency of payments per year. To illustrate, if interest is paid quarterly and the
periodic interest rate is 8% / 4 = 2%), then we have: the effective annual yield = (1.02)4 – 1 =
1.0824 – 1 = 0.0824 or 8.24%.
We can also determine the periodic interest rate that will produce a given annual interest rate by
solving the effective annual yield equation for the periodic interest rate. Solving, we find that:
periodic interest rate = (1 + effective annual yield)1/m – 1. To illustrate, if the periodic quarterly
interest rate that would produce an effective annual yield of 12%, then we have: periodic interest
rate = (1.12)1/4 – 1 = 1.0287 – 1 = 0.0287 or 2.87%.
CONVENTIONAL YIELD MEASURES
There are several bond yield measures commonly quoted by dealers and used by portfolio
managers. These are described below.
Current Yield
Current yield relates the annual coupon interest to the market price. The formula for the current
yield is: current yield = annual dollar coupon interest / price. The current yield calculation takes
into account only the coupon interest and no other source of return that will affect an investor’s
yield. The time value of money is also ignored.
Yield to Maturity
The yield to maturity is the interest rate that will make the present value of the cash flows equal
to the price (or initial investment). For a semiannual pay bond, the yield to maturity is found by
first computing the periodic interest rate, y, which satisfies the relationship:
P=
C
1  y 
1
+
C
1  y 
2
+
C
1  y 
3
+ . . .+
C
1  y 
n
+
M
1  y 
n
where P = price of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in
dollars), and n = number of periods (number of years x 2).
For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield
to maturity, which understates the effective annual yield. The yield to maturity computed on the
basis of this market convention is called the bond-equivalent yield.
It is much easier to compute the yield to maturity for a zero-coupon bond because we can use:
1/ n
 M
y= 
 P
31
 1.
The yield-to-maturity calculation takes into account not only the current coupon income but also
any capital gain or loss that the investor will realize by holding the bond to maturity. In addition,
the yield to maturity considers the timing of the cash flows.
Yield to Call
The price at which the bond may be called is referred to as the call price. For some issues, the
call price is the same regardless of when the issue is called. For other callable issues, the call
price depends on when the issue is called. That is, there is a call schedule that specifies a call
price for each call date.
For callable issues, the practice has been to calculate a yield to call as well as a yield to maturity.
The yield to call assumes that the issuer will call the bond at some assumed call date and the call
price is then the call price specified in the call schedule. Typically, investors calculate a yield to
first call or yield to next call, a yield to first par call, and yield to refunding.
Mathematically, the yield to call can be expressed as follows:
P=
C
1  y 
1
+
C
1  y 
2
+
C
1  y 
3
+ . . .+
C
1  y 
n*
+
M*
1  y 
n*
where M* = call price (in dollars) and n* = number of periods until the assumed call date
(number of years times 2). For a semiannual pay bond, doubling the periodic interest rate (y)
gives the yield to call on a bond-equivalent basis.
Yield to Put
If an issue is putable, it means that the bondholder can force the issuer to buy the issue at a
specified price. As with a callable issue, a putable issue can have a put schedule. The schedule
specifies when the issue can be put and the price, called the put price.
When an issue is putable, a yield to put is calculated. The yield to put is the interest rate that
makes the present value of the cash flows to the assumed put date plus the put price on that date
as set forth in the put schedule equal to the bond’s price. The formula is the same as for the yield
to call, but M* is now defined as the put price and n* is the number of periods until the assumed
put date. The procedure is the same as calculating yield to maturity and yield to call.
Yield to Worst
A practice in the industry is for an investor to calculate the yield to maturity, the yield to every
possible call date, and the yield to every possible put date. The minimum of all of these yields is
called the yield to worst.
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Cash Flow Yield
Some fixed income securities involve cash flows that include interest plus principal repayment.
Such securities are called amortizing securities. For amortizing securities, the cash flow each
period consists of three components: (i) coupon interest, (ii) scheduled principal repayment, and
(iii) prepayments. For amortizing securities, market participants calculate a cash flow yield. It is
the interest rate that will make the present value of the projected cash flows equal to the market
price.
Yield (Internal Rate of Return) for a Portfolio
The yield for a portfolio of bonds is not simply the average or weighted average of the yield to
maturity of the individual bond issues in the portfolio. It is computed by determining the cash
flows for the portfolio and determining the interest rate that will make the present value of the
cash flows equal to the market value of the portfolio.
Yield Spread Measures for Floating-Rate Securities
The coupon rate for a floating-rate security changes periodically based on the coupon reset
formula, which has as its components the reference rate and the quoted margin. Since the future
value for the reference rate is unknown, it is not possible to determine the cash flows. This
means that a yield to maturity cannot be calculated. Instead, there are several conventional
measures used as margin or spread measures cited by market participants for floaters. These
include spread for life (or simple margin), adjusted simple margin, adjusted total margin, and
discount margin.
The most popular of these measures is discount margin. This measure estimates the average
margin over the reference rate that the investor can expect to earn over the life of the security.
POTENTIAL SOURCES OF A BOND’S DOLLAR RETURN
An investor who purchases a bond can expect to receive a dollar return from one or more of
these sources: (i) the periodic coupon interest payments made by the issuer, (ii) any capital gain
(or capital loss—negative dollar return) when the bond matures, is called, or is sold, and (iii)
interest income generated from reinvestment of the periodic cash flows
Any measure of a bond’s potential yield should take into consideration each of these three
potential sources of return. The current yield considers only the coupon interest payments. No
consideration is given to any capital gain (or loss) or interest on interest. The yield to maturity
takes into account coupon interest and any capital gain (or loss). It also considers the interest-oninterest component. Implicit in the yield-to-maturity computation is the assumption that the
coupon payments can be reinvested at the computed yield to maturity.
The yield to call also takes into account all three potential sources of return. In this case, the
assumption is that the coupon payments can be reinvested at the yield to call. Therefore, the
yield-to-call measure suffers from the same drawback as the yield to maturity in that it assumes
33
coupon interest payments are reinvested at the computed yield to call. Also, it presupposes that
the issuer will call the bond on some assumed call date.
The cash flow yield also takes into consideration all three sources as is the case with yield to
maturity, but it makes two additional assumptions. First, it assumes that the periodic principal
repayments are reinvested at the computed cash flow yield. Second, it assumes that the
prepayments projected to obtain the cash flows are actually realized.
Determining the Interest-on-Interest Dollar Return
The interest-on-interest component can represent a substantial portion of a bond’s potential
return. The coupon interest plus interest on interest can be found by using the following
equation:
 1  r n  1
C


r

where r denote the semiannual reinvestment rate.
The total dollar amount of coupon interest is found by multiplying the semiannual coupon
interest by the number of periods: total coupon interest = nC.
The interest-on-interest component is then the difference between the coupon interest plus
interest on interest and the total dollar coupon interest, as expressed by the formula
 1  r n  1
interest on interest = C 
  nC .

r

Yield to Maturity and Reinvestment Risk
The investor will realize the yield to maturity at the time of purchase only if the bond is held to
maturity and the coupon payments can be reinvested at the computed yield to maturity. The risk
that the investor faces is that future reinvestment rates will be less than the yield to maturity at
the time the bond is purchased. This risk is referred to as reinvestment risk.
There are two characteristics of a bond that determine the importance of the interest-on-interest
component and therefore the degree of reinvestment risk: maturity and coupon. For a given yield
to maturity and a given coupon rate, the longer the maturity, the more dependent the bond’s total
dollar return is on the interest-on-interest component in order to realize the yield to maturity at
the time of purchase. In other words, the longer the maturity, the greater the reinvestment risk.
For a given maturity and a given yield to maturity, higher coupon rates will make the bond’s
total dollar return more dependent on the reinvestment of the coupon payments in order to
produce the yield to maturity anticipated at the time of purchase.
34
Cash Flow Yield and Reinvestment Risk
For amortizing securities, reinvestment risk is even greater than for nonamortizing securities.
The reason is that the investor must now reinvest the periodic principal repayments in addition to
the periodic coupon interest payments.
TOTAL RETURN
In the preceding section we explain that the yield to maturity is a promised yield. At the time of
purchase an investor is promised a yield, as measured by the yield to maturity, if both of the
following conditions are satisfied: (i) the bond is held to maturity and (ii) all coupon interest
payments are reinvested at the yield to maturity.
The total return is a measure of yield that incorporates an explicit assumption about the
reinvestment rate.
The yield-to-call measure is subject to the same problems as the yield to maturity. First, it
assumes that the bond will be held until the first call date. Second, it assumes that the coupon
interest payments will be reinvested at the yield to call.
Computing the Total Return for a Bond
The idea underlying total return is simple. The objective is first to compute the total future
dollars that will result from investing in a bond assuming a particular reinvestment rate. The total
return is then computed as the interest rate that will make the initial investment in the bond grow
to the computed total future dollars.
APPLICATIONS OF THE TOTAL RETURN (HORIZON ANALYSIS)
Using total return to assess performance over some investment horizon is called horizon
analysis. When a total return is calculated over an investment horizon, it is referred to as a
horizon return.
An often-cited objection to the total return measure is that it requires the portfolio manager to
formulate assumptions about reinvestment rates and future yields as well as to think in terms of
an investment horizon.
CALCULATING YIELD CHANGES
The absolute yield change (or absolute rate change) is measured in basis points and is the
absolute value of the difference between the two yields as given by
absolute yield change = │initial yield – new yield│ X 100.
The percentage change is computed as the natural logarithm of the ratio of the change in yield as
35
shown by
percentage change yield = 100 X ln (new yield / initial yield).
36
ANSWERS TO QUESTIONS FOR CHAPTER 3
(Questions are in bold print followed by answers.)
1. A debt obligation offers the following payments:
Years from Now Cash Flow to Investor
1
$2,000
2
$2,000
3
$2,500
4
$4,000
Suppose that the price of this debt obligation is $7,704. What is the yield or internal rate of
return offered by this debt obligation?
The yield on any investment is the interest rate that will make the present value of the cash flows
from the investment equal to the price (or cost) of the investment.
Mathematically, the yield on any investment, y, is the interest rate that satisfies the equation:
P=
CFN
CF 1
CF 2
CF 3
+
+
+ . . .+
1
2
3
N
1+ y  1+ y  1+ y 
1+ y 
where CFt = cash flow in year t, P = price of the investment, and N = number of years. The yield
calculated from this relationship is also called the internal rate of return. To solve for the yield
(y), we can use a trial-and-error (iterative) procedure. The objective is to find the interest rate that
will make the present value of the cash flows equal to the price. To compute the yield for our
problem, different interest rates must be tried until the present value of the cash flows is equal to
$7,704 (the price of the financial instrument). Trying an annual interest rate of 10% gives the
following present value:
Promised Annual Payments
Present Value
Years from Now
(Cash Flow to Investor)
of Cash Flow at 10%
1
$2,000
$1,818.18
2
$2,000
$1,652.89
3
$2,500
$1,878.29
4
$4,000
$2,732.05
Present value = $8,081.41
Because the present value of $8,081.41 computed using a 10% interest rate exceeds the price of
$7,704, a higher interest rate must be used, to reduce the present value. Trying an annual interest
rate of 13% gives the following present value:
37
Promised Annual Payments
Present Value
Years from Now
(Cash Flow to Investor)
of Cash Flow at 13%
1
$2,000
$1,769.91
2
$2,000
$1,566.29
3
$2,500
$1,732.63
4
$4,000
$2,453.27
Present value = $7,522.10
Because the present value of $7,522.10 computed using a 13% interest rate is below the price of
$7,704, a lower interest rate must be used, to reduce the present value. Thus, to increase the
present value, a lower interest rate must be tried. Trying an annual interest rate of 12% gives the
following present value:
Years from Now
1
2
3
4
Promised Annual Payments
Present Value
(Cash Flow to Investor)
of Cash Flow at 12%
$2,000
$1,785.71
$2,000
$1,594.39
$2,500
$1,779.45
$4,000
$2,542.07
Present value = $7,701.62
Using 12%, the present value of the cash flow is $7,701.62, which is almost equal to the price of
the financial instrument of $7,704. Therefore, the yield is close to 12%. The precise yield using
Excel or a financial calculator is 11.987%.
Although the formula for the yield is based on annual cash flows, it can be generalized to any
number of periodic payments in a year. The generalized formula for determining the yield is
N
P =
CF t
t
t =1 (1 + y )
where CFt = cash flow in period t, and n = number of periods.
Keep in mind that the yield computed is the yield for the period. That is, if the cash flows are
semiannual, the yield is a semiannual yield. If the cash flows are monthly, the yield is a monthly
yield. To compute the simple annual interest rate, the yield for the period is multiplied by the
number of periods in the year.
2. What is the effective annual yield if the semiannual periodic interest rate is 4.3%?
To obtain an effective annual yield associated with a periodic interest rate, the following formula
is used:
effective annual yield = (1 + periodic interest rate)m – 1
38
where m is the frequency of payments per year. In our problem, the periodic interest rate is a
semiannual rate of 4.3% and the frequency of payments is twice per year. Inserting these
numbers, we have:
effective annual yield = (1.043)2 – 1 = 1.087849 – 1 = 0.087849 or about 8.785%.
3. What is the yield to maturity of a bond?
The yield to maturity is the interest rate that will make the present value of the cash flows equal
to the price (or initial investment). For a semiannual pay bond, the yield to maturity is found by
first computing the periodic interest rate, y, which satisfies the relationship:
P=
C
C
C
C
M
+
+
+ . . .+
+
2
3
n
n
1  y  1  y  1  y 
1  y  1  y 
where P = price of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in
dollars), and n = number of periods (number of years times 2).
It is much easier to compute the yield to maturity for a zero-coupon bond because we can use:
1/ n
 M
y= 
 P
 1.
The yield-to-maturity calculation takes into account not only the current coupon income but also
any capital gain or loss that the investor will realize by holding the bond to maturity. In addition,
the yield to maturity considers the timing of the cash flows.
4. What is the yield to maturity calculated on a bond-equivalent basis?
For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield
to maturity, which understates the effective annual yield. The yield to maturity computed on the
basis of this market convention is called the bond-equivalent yield.
5. Answer the following questions.
(a) Show the cash flows for the following four bonds, each of which has a par value of
$1,000 and pays interest semiannually.
Bond Coupon Rate Number of Years
(%)
to Maturity
W
7
5
X
8
7
Y
9
4
Z
0
10
39
Price
$884.20
$948.90
$967.70
$456.39
Bond W has cash flows of 0.07($1,000) / 2 = $35 for semiannual periods from periods 1 to 10.
At the end of period 10, Bond W also pays back the par of $1,000 for a total payment of $1,000
+ $35 = $1,035.
Bond X has cash flows of 0.08($1,000) / 2 = $40 for semiannual periods from periods 1 to 14. At
the end of period 14, Bond X also pays back the par of $1,000 for a total payment of $1,000 +
$40 = $1,040.
Bond Y has cash flows of 0.08($1,000) / 2 = $45 for semiannual periods from periods 1 to 8. At
the end of period 8, Bond Y also pays back the par of $1,000 for a total payment of $1,000 + $45
= $1,045.
Bond Z has cash flows of 0 ($1,000) / 2 = $0 for semiannual periods from periods 1 to 20. At the
end of period 20, Bond Z also pays back the $1,000 for a total payment of $1,000 + $0 = $1,000.
Below we show these cash flows in table format.
Cash Flow Cash Flow Cash Flow Cash Flow
Period for Bond W for Bond X for Bond Y for Bond Z
1
$35
$40
$45
$0
2
$35
$40
$45
$0
3
$35
$40
$45
$0
4
$35
$40
$45
$0
5
$35
$40
$45
$0
6
$35
$40
$45
$0
7
$35
$40
$45
$0
8
$35
$40
$1,045
$0
9
$35
$40
$0
10
$1,035
$40
$0
11
$40
$0
12
$40
$0
13
$40
$0
14
$1,040
$0
15
$0
16
$0
17
$0
18
$0
19
$0
20
$1,000
(b) Calculate the yield to maturity for the four bonds.
The yield to maturity is computed in the same way as the internal rate of return; the cash flows
are those that the investor would realize by holding the bond to maturity. For a semiannual pay
bond, the yield to maturity is found by first computing the periodic interest rate, y, which
satisfies the relationship
40
P=
C
C
C
C
M
+
+
+ . . .+
+
2
3
n
n
1  y  1  y  1  y 
1  y  1  y 
where P = price of the bond, C = semiannual coupon interest (in dollars), M = maturity value (in
dollars), and n = number of periods (number of years times 2).
For a semiannual pay bond, doubling the periodic interest rate or discount rate (y) gives the yield
to maturity. However, annualizing the yield by doubling the periodic interest rate understates the
effective annual yield. Despite this, the market convention is to compute the yield to maturity by
doubling the periodic interest rate, y, that satisfies our equation. The yield to maturity computed
on the basis of this market convention is called the bond-equivalent yield.
The computation of the yield to maturity requires a trial-and-error procedure. To illustrate the
computation, we first look at bond W. The cash flows for this bond are ten coupon payments of
$35 every six months and the principal of $1,000 to be paid in ten six-month periods from now.
To get y using our equation given above, different interest rates must be tried until the present
value of the cash flows is equal to the price. In doing this, we get the following yield to
maturities for the four bonds.
For bond W, we get a periodic interest rate real close to 5%. This is seen below.
Years from Promised Annual Payments
Present Value
Now
(Cash Flow to Investor)
of Cash Flow at 5%
1
$35
$33.33
2
$35
$31.75
3
$35
$30.23
4
$35
$28.79
5
$35
$27.42
6
$35
$26.12
7
$35
$24.87
8
$35
$23.68
9
$35
$22.56
10
$1,035
$635.40
Present value = $884.17
Using 5%, the present value of the cash flow is $884.17, which is almost equal to the price of the
financial instrument of $884.20. Therefore, the periodic interest rate is close to 5%. The precise
yield using Excel or a financial calculator is 4.99964%. Doubling the periodic interest rate of 5%
gives a yield to maturity of 10% (doubling 4.99964% gives 9.99928%).
For bond X, we get an interest rate real close to 4.50%. Using this rate, the value of the cash flow
is $951.59, which is almost equal to the price of the financial instrument of $948.90. Therefore,
the yield is close to 4.5%. The precise periodic interest rate using Excel or a financial calculator
41
is 4.5271%. Doubling the periodic interest rate of 4.5% gives a yield to maturity of 9% (doubling
4.5271% gives 9.0542%).
For bond Y, we get an interest rate close to 5%. Using this rate, the value of the cash flow is
$967.68, which is almost equal to the price of the financial instrument of $967.70. Therefore, the
yield is close to 5%. The precise periodic interest rate using Excel or a financial calculator is
5.11078%. Doubling the periodic interest rate of 5% gives a yield to maturity of 10% (doubling
5.11083% gives 10.2215%).
For bond Z, we get an interest rate close to 4%. Using this rate, the value of the cash flow is
$456.39, which is equal to the price of the financial instrument of $456.39. Therefore, the yield
is virtually 4%. The precise periodic interest rate using Excel or a financial calculator is
3.99965%. Doubling the periodic interest rate of 4% gives a yield to maturity of 8% (doubling
3.99965% gives 7.9993%).
6. A portfolio manager is considering buying two bonds. Bond A matures in three years
and has a coupon rate of 10% payable semiannually. Bond B, of the same credit quality,
matures in 10 years and has a coupon rate of 12% payable semiannually. Both bonds are
priced at par.
(a) Suppose that the portfolio manager plans to hold the bond that is purchased for three
years. Which would be the best bond for the portfolio manager to purchase?
The shorter term bond will pay a lower coupon rate but it will likely cost less for a given market
rate. Since the bonds are of equal risk in terms of credit quality (the maturity premium for the
longer term bond should be greater), the question when comparing the two bond investments is:
What investment will be expected to give the highest cash flow per dollar invested? In other
words, which investment will be expected to give the highest effective annual rate of return. In
general, holding the longer term bond should compensate the investor in the form of a maturity
premium and a higher expected return. However, as seen in the discussion below, the actual
realized return for either investment is not known with certainty.
To begin with, an investor who purchases a bond can expect to receive a dollar return from (i)
the periodic coupon interest payments made by the issuer; (ii) any capital gain (or capital loss—
negative dollar return) when the bond matures, is called, or is sold; and (iii) interest income
generated from reinvestment of the periodic cash flows. The last component of the potential
dollar return is referred to as reinvestment income. For a standard bond (our situation) that makes
only coupon payments and no periodic principal payments prior to the maturity date, the interim
cash flows are simply the coupon payments. Consequently, for such bonds the reinvestment
income is simply interest earned from reinvesting the coupon interest payments. For these bonds,
the third component of the potential source of dollar return is referred to as the interest-oninterest component.
If we are going to compute a potential yield to make a decision, we should be aware of the fact
that any measure of a bond’s potential yield should take into consideration each of the three
components described above. The current yield considers only the coupon interest payments. No
42
consideration is given to any capital gain (or loss) or interest on interest. The yield to maturity
takes into account coupon interest and any capital gain (or loss). It also considers the interest-oninterest component. Additionally, implicit in the yield-to-maturity computation is the assumption
that the coupon payments can be reinvested at the computed yield to maturity. The yield to
maturity is a promised yield and will be realized only if the bond is held to maturity and the
coupon interest payments are reinvested at the yield to maturity. If the bond is not held to
maturity and the coupon payments are reinvested at the yield to maturity, then the actual yield
realized by an investor can be greater than or less than the yield to maturity.
Given the facts that (i) one bond, if bought, will not be held to maturity, and (ii) the coupon
interest payments will be reinvested at an unknown rate, we cannot determine which bond might
give the highest actual realized rate. Thus, we cannot compare them based upon this criterion.
However, if the portfolio manager is risk inverse in the sense that she or he doesn’t want to buy a
longer term bond, which will likely have more variability in its return, then the manager might
prefer the shorter term bond (bond A) of three years. This bond also matures when the manager
wants to cash in the bond. Thus, the manager would not have to worry about any potential capital
loss in selling the longer term bond (bond B). The manager would know with certainty what the
cash flows are. If these cash flows are spent when received, the manager would know exactly
how much money could be spent at certain points in time.
Finally, a manager can try to project the total return performance of a bond on the basis of the
planned investment horizon and expectations concerning reinvestment rates and future market
yields. This permits the portfolio manager to evaluate which of several potential bonds
considered for acquisition will perform best over the planned investment horizon. As we just
argued, this cannot be done using the yield to maturity as a measure of relative value. Using total
return to assess performance over some investment horizon is called horizon analysis. When a
total return is calculated over an investment horizon, it is referred to as a horizon return. The
horizon analysis framework enables the portfolio manager to analyze the performance of a bond
under different interest-rate scenarios for reinvestment rates and future market yields. Only by
investigating multiple scenarios can the portfolio manager see how sensitive the bond’s
performance will be to each scenario. This can help the manager choose between the two bond
choices.
(b) Suppose that the portfolio manager plans to hold the bond that is purchased for six
years instead of three years. In this case, which would be the best bond for the portfolio
manager to purchase?
Similar to our discussion in part (a), we do not know which investment would give the highest
actual realized return in six years when we consider reinvesting all cash flows. If the manager
buys a three-year bond, then there would be the additional uncertainty of now knowing what
three-year bond rates would be in three years. The purchase of the ten-year bond would be held
longer than previously (six years compared to three years) and render coupon payments for a sixyear period that are known. If these cash flows are spent when received, the manager will know
exactly how much money could be spent at certain points in time. Not knowing which bond
investment would give the highest realized return, the portfolio manager would choose the bond
that fits the firm’s goals in terms of maturity.
43
(c) Suppose that the portfolio manager is managing the assets of a life insurance company
that has issued a five-year guaranteed investment contract (GIC). The interest rate that the
life insurance company has agreed to pay is 9% on a semiannual basis. Which of the two
bonds should the portfolio manager purchase to ensure that the GIC payments will be
satisfied and that a profit will be generated by the life insurance company?
The portfolio manager needs to generate a semiannual cash flow of 9% semiannual basis for five
years. Bond A will only lock in a 10% cash flow per dollar invested for three years. However,
bond B will lock in a 12% cash flow per dollar invested for ten years. Thus, the portfolio
manager would choose bond B and hopefully we able buy as many of these bonds s are needed
to generate the cash flows required to meet its five-year guaranteed investment contract.
7. Consider the following bond:
Coupon rate = 11%
Maturity = 18 years
Par value = $1,000
First par call in 13 years
Only put date in five years and putable at par value
Suppose that the market price for this bond $1,169.
(a) Show that the yield to maturity for this bond is 9.077%.
First of all, we could compute the internal return based upon the cash flows if the bond is held to
maturity. We would get 4.5385%. For a semiannual pay bond, doubling the periodic interest rate
(y) gives the yield to maturity on a bond-equivalent basis. Taking 4.5385% times two gives us a
yield to maturity equal to 9.077%.
We can also verify that the yield to maturity is 9.077% by using this rate to compute the value of
the bond to determine if it is $1,168.99. In doing this, we first compute the present value of the
coupon payments where C is the annuity coupon payment and N is the number of periods. We
have:
1

1  1  y N


C

y

1


1  1 .045385 36
 = $55 
0.045385




 = $55[17.57569] = $966.663.

We next compute the present value of the maturity value where M is the par value of $1,000. We
get:
1


 1 
= $1, 000 
= $1,000[0.2023273] = $202.327.
M
36 
N 


1.045385
1

y






44
When a 9.077% / 2 = 4.5285% semiannual interest rate is used, the present value of the cash
flows is $966.663 + $202.327 = $1,168.99 or about $1,169. Thus, the yield to maturity for this
bond is 9.077%.
(b) Show that the yield to first par call is 8.793%.
First of all, we could compute the internal return based upon the cash flows if the bond is held
for 13 years. We would get 4.39651%. For a semiannual pay bond, doubling the periodic interest
rate (y) gives the yield to call on a bond-equivalent basis. Taking 4.39651% times two gives us a
yield to first par call of 8.793%.
We can also verify the yield to call is 8.793% by using this rate to compute the value of the bond
to determine if it is $1,168.99. Doing this, we first compute the present value of the coupon
payments where N is now the number of periods until the bond is assumed to be called. We get:
1

1

N

1  y 

C

y

1


1  (1 .043965) 26
 = $55 

0.043965



 = $55[15.314173] = $842.280.

We next compute the present value of the maturity value under the assumption it will be called in
13 years where M is now M* (which is the call price in dollars), and n is now n* (which is the
number of periods until the assumed call date, e.g., number of years times 2). We get:




1
1
M *
= $1, 000 
= $1,000[0.3267123] = $326.712.
N* 
26 
 1.043965 
 1  y  
When a semiannual interest rate of 8.793% / 2 = 4.3965% is used, the present value of the cash
flows is $842.280 + $326.712 = $1,168.99 or about $1,169. Thus, the yield to call for this bond
is 8.793%.
(c) Show that the yield to put is 6.942%.
First of all, we could compute the internal return based upon the cash flows if the bond is held
for 5 years. We would get 3.4710%. For a semiannual pay bond, doubling the periodic interest
rate (y) gives the yield to put on a bond-equivalent basis. Taking 3.4710% times two gives us a
yield to first par put of 6.9420%.
We can also verify the yield to put is 6.942% by using this rate to compute the value of the bond
to determine if it is $1,168.99. Doing this, we first compute the present value of the coupon
payments where N is now the number of periods until the bond is assumed to be sold. We get:
1

1  1  y N


C
y


1


1  (1 .03471)10
 = $55 
 0. 03471

45


 = $55[8.328775] = $458.083.

We next compute the present value of the maturity value under the assumption the put will be
exercised five years where M is now M* (which is the put price in dollars), and n is now n*
(which is the number of periods until the put date, e.g., number of years times 2). We get:




1
1
= $1, 000 
= $1,000[0.7109769] = $710.977.
M *
N* 
10 
 1.03471 
 1  y  
When a 6.942% / 2 = 3.471 % semiannual interest rate is used, the present value of the cash
flows is $458.083 + $710.977 = $1,169.06 or about $1,169. Thus, the yield to put for this bond is
6.942%.
(d) Suppose that the call schedule for this bond is as follows:
Can be called in eight years at $1,055.
Can be called in 13 years at $1,000.
And suppose this bond can only be put in five years and assume that the yield to first par
call is 8.535%. What is the yield to worst for this bond?
A practice in the industry is for an investor to calculate the yield to maturity, the yield to every
possible call date, and the yield to every possible put date. The minimum of all of these yields is
called the yield to worst. If the bond is called in eight years at $1,055, then we can compute the
yield to maturity and get 9.986% or about 10%. If the bond is called in thirteen years at $1,000,
then we can compute the yield to maturity and get 11.00%. If the bond is put in five years, the
yield to maturity is 8.535%. Since the latter is the lowest the yield to worse is for this bond is
8.535%.
8. (a) What is meant by an amortizing security?
Amortized securities are fixed income securities whose cash flows include scheduled principal
repayments prior to maturity. That is, the cash flow in each period includes interest plus principal
repayment.
For amortizing securities, reinvestment risk is even greater than for nonamortizing securities.
The reason is that the investor must now reinvest the periodic principal repayments in addition to
the periodic coupon interest payments. Moreover, the cash flows are monthly, not semiannually
as with nonamortizing securities. In brief, the investor must not only reinvest periodic coupon
interest payments and principal, but must do it more often. This increases reinvestment risk.
(b) What are the three components of the cash flow for an amortizing security?
As stated in part (a), an amortizing security includes both interest plus principal repayment.
However, we must also note that the amount the borrower can repay in principal may exceed the
scheduled amount. This excess amount of principal repayment over the amount scheduled is
called a prepayment. Thus, for amortizing securities, the cash flow each period consists of three
components: (1) coupon interest, (2) scheduled principal repayment, and (3) prepayments.
46
(c) What is meant by a cash flow yield?
For amortizing securities, market participants calculate a cash flow yield. It is the interest rate
that will make the present value of the projected cash flows equal to the market price. The
difficulty in computing a cash flow yield is projecting what the prepayment will be in each
period.
9. How is the internal rate of return of a portfolio calculated?
The yield for a portfolio of bonds is not simply the average or weighted average of the yield to
maturity of the individual bond issues in the portfolio. It is computed by determining the cash
flows for the portfolio and determining the interest rate that will make the present value of the
cash flows equal to the market value of the portfolio. Mathematically, the yield, y, on a portfolio
(like any investment) is the interest rate that satisfies the equation:
P=
C1
1  y 
1
+
CN
C2
C3
+
+ . . .+
.
2
3
N
1  y  1  y 
1  y 
This expression can be rewritten in shorthand notation as
N
P =
CF t
t
t =1 (1 + y )
where CFt = cash flows from all investments in the portfolio for year t, P = price of the
investment (or present value of the portfolio’s cash flows), N = number of years, and y is the
yield or internal rate of return.
Solving for the yield (y) requires a trial-and-error (iterative) procedure. The objective is to find
the interest rate that will make the present value of the cash flows equal to the price. An example
demonstrates how this is done. Suppose that all investments in the portfolio selling for $903.10
promises to make the following annual payments:
Years from Promised Annual Payments
Now
(Cash Flow to Investor)
1
$100
2
$100
3
$100
4
$1,000
To compute the portfolio internal rate of return, different interest rates must be tried until the
present value of the cash flows is equal to $903.10 (the price of the financial instrument). Trying
an annual interest rate of 10% gives the following present value:
47
Years from Promised Annual Payments
Present Value
Now
(Cash Flow to Investor)
of Cash Flow at 10%
1
$ 100
$ 90.91
2
$ 100
$ 82.64
3
$ 100
$ 75.13
4
$1,000
$ 683.01
Present value = $ 931.69
Because the present value computed using a 10% interest rate exceeds the price of $903.10, a
higher interest rate must be used, to reduce the present value. If a 12% interest rate is used, the
present value is $875.71, computed as follows:
Years from Promised Annual Payments
Present Value
Now
(Cash Flow to Investor)
of Cash Flow at 12%
1
$ 100
$ 89.29
2
$ 100
$ 79.72
3
$ 100
$ 71.18
4
$1,000
$ 635.52
Present value = $ 875.71
Using 12%, the present value of the cash flow is less than the price of the financial instrument.
Therefore, a lower interest rate must be tried, to increase the present value. Using an 11%
interest rate:
Years from Now
1
2
3
4
Promised Annual Payments
Present Value
(Cash Flow to Investor)
of Cash Flow at 11%
$ 100
$ 90.09
$ 100
$ 81.16
$ 100
$ 73.12
$1,000
$ 658.73
Present value = $ 903.10
Using 11%, the present value of the cash flow is equal to the price of the portfolio. Therefore, the
yield is 11%.
Keep in mind that the yield computed is now the yield for the period. That is, if the cash flows
are semiannual, the yield is a semiannual yield. If the cash flows are monthly, the yield is a
monthly yield. To compute the simple annual interest rate, the yield for the period is multiplied
by the number of periods in the year.
10. What is the limitation of using the internal rate of return of a portfolio as a measure of
the portfolio’s yield?
Implicit in the internal rate of return computation is the assumption that the portfolio cash flows
can be reinvested at the computed internal rate of return. Also, when we compute an internal rate
of return, we annualized interest rates by multiplying by the number of periods in a year (we call
48
the resulting value the simple annual interest rate). For example, multiplying by 2 annualizes a
semiannual yield. Alternatively, an annual interest rate is converted to a semiannual interest rate
by dividing by 2. This simplified procedure for computing the annual interest rate given a
periodic (weekly, monthly, quarterly, semiannually, and so on) interest rate is not accurate. To
obtain an effective annual yield associated with a periodic interest rate, the following formula is
used:
effective annual yield = (1 + periodic interest rate)m – 1
where m is the frequency of payments per year. For example, suppose that the periodic interest
rate is 4% and the frequency of payments is twice per year. Inserting in the values we get:
effective annual yield = (1.04)2 – 1 = 1.0816 – 1 = 0.0816 or 8.16%.
This is different from 8.00%, which we get by multiplying 4.00% times two.
11. Suppose that the coupon rate of a floating-rate security resets every six months at a
spread of 70 basis points over the reference rate. If the bond is trading at below par value,
explain whether the discount margin is greater than or less than 70 basis points.
If the bond is trading below par value, then the discount margin or assumed annual spread (basis
points) will be greater than 70 basis points. This is because the spread must increase to make the
present value of the cash flows less than the par value. This is illustrated in Exhibit 3-1 where the
bond is trading below par and the spread (basis points) had to increase in order for the present
value of the cash flows to fall to a level to equal the current trading value.
12. An investor is considering the purchase of a 20-year 7% coupon bond selling for $816
and a par value of $1,000. The yield to maturity for this bond is 9%.
Answer the below questions.
(a) What would be the total future dollars if this investor invested $816 for 20 years earning
9% compounded semiannually?
To determine the future value of any sum of money invested today, we use the below equation:
Pn = P0 (1 + r)n where n = number of periods, Pn = future value n periods from now (in dollars),
P0 = original principal (in dollars), and r = interest rate per period (in decimal form). Inserting in
our values, we have: Pn = P0(1 + r)n = $816(1.045)40 = $816(5.8163645) = $4,746.15.
(b) What are the total coupon payments over the life of this bond?
The total dollar amount of coupon interest is found by multiplying the semiannual coupon
interest by the number of periods: total coupon interest = nC. Thus, the total coupon payments
are: nC = 40($35) = $1,400.00.
(c) What would be the total future dollars from the coupon payments and the repayment of
principal at the end of 20 years?
49
There are several ways to approach this problem. One method is to compute the present value of
the cash flows and then multiply this by the future value factor for a lump sum. Another method
(which involves less work) is to compute the future value of all the cash flows. For this method,
we would (i) compute the future value of the annuity cash flows which is the coupon interest
plus interest on interest, and (ii) add the par value which occurs at maturity which is M = $1,000.
The equation is:
 1  r n  1
Pn = coupon interest plus interest on interest + M = C 
 +M

r

where Pn is the future value of all cash flows at time N, C is the amount of the semiannual
coupon annuity in dollars, r = annual interest rate / number of times interest paid per year (where
we assume interest in reinvested at r), n = number of times interest paid per year times the
number of years, and M = par value at the end of the period.
Using this formula and inserting our values, we have:
 1  r n  1 
 (1.045) 40  1
Pn = C 
 + M = $35 
 + $1,000 = $35[107.03032] + $1,000 =

r

 0.045 
$3,746.06 + $1,000 = $4,746.06.
(d) For the bond to produce the same total future dollars as in part (a), how much must the
interest on interest be?
We can note that the future value of the interest payment just computed in part (c) is $3,746.06
and the coupon payments over the life of the bond computed in part (b) is $1,400. The different
is the interest on interest, which is $2,346.06.
Another way of computing the interest on interest is to note that it is the difference between the
coupon interest plus interest on interest and the total dollar coupon interest, as expressed by the
formula:
 1  r n  1
interest on interest = C 
 – nC.

r

40
 (1.045 )  1
Inserting in our values gives: $35 
 – 40($35) = $35[107.03032] – $1,400 =
 0.045 
$3,746.06 – $1,400.00 = $2,346.06.
(e) Calculate the interest on interest from the bond assuming that the semiannual coupon
payments can be reinvested at 4.5% every six months and demonstrate that the resulting
amount is the same as in part (d).
50
Since the computation assumes interest on interest is invested at 4.5% we have the same
computation given in part (d) where the yield to maturity of 4.5% was used in computation. Once
again, we have:
 1  r n  1
interest on interest = C 
 – nC

r

 1.045 40  1 
where r is still 4.5%. Inserting in our values gives: $35 
 – 40($35) =
 0.045 
$35[107.03032] – $1,400 = $3,746.06 – $1,400.00 = $2,346.06 which is the same amount as in
part (d).
13. What is the total return for a 20-year zero-coupon bond that is offering a yield to
maturity of 8% if the bond is held to maturity?
For zero-coupon bonds, none of the bond’s total dollar return is dependent on the interest-oninterest component, so a zero-coupon bond has zero reinvestment risk if held to maturity. The
yield earned on a zero-coupon bond held to maturity is equal to the promised yield to maturity.
This is because whenever one can reinvest the coupon payments at the yield to maturity, then the
total return will be the same as the yield to maturity. Thus, the total return is 8%.
14. Explain why the total return from holding a bond to maturity will be between the yield
to maturity and the reinvestment rate.
The yield to maturity is based upon the coupon payments and the current market value of the
bond. The yield to maturity is below (above) the coupon rate if the current market value is above
(below) the par value. If one could reinvest the coupon payments at the yield to maturity, then
the total return would be the same as the yield to maturity. If it cannot reinvest the coupon
payment at the yield to maturity then it will be earning a rate below the yield to maturity. To
illustrate assume the yield to maturity is 9% and you reinvest at 8%. Then your total return
would have to lie between 9% and 8%. Similarly, if you are able to invest above the yield to
maturity of 9%, say 10%, your total return will have to lie between 9% and 10%. In either case,
it is true to say that your total return from holding a bond to maturity will be between the yield to
maturity and the reinvestment rate.
15. For a long-term high-yield coupon bond, do you think that the total return from
holding a bond to maturity will be closer to the yield to maturity or the reinvestment rate?
For a longer term bond the future value of the coupon payments will be greater than the future
value of the par value (which is simply the par value). For example, consider a 20-year bond
paying 14% and selling at par. The future value of the $70 semiannual interest payments for 40
periods will be $13,974 and the future value of the par value is $1,000. If the reinvestment rate
falls to 10%, the future value of the $70 semiannual interest payments for 40 periods will fall
39.5% to $8,456 while the future value of the par value remains unchanged at $1,000. The total
return will be:
51
1/ h
 total future dollars 
 purchase price of bonds 


1 / 40
 $9,456 
1 = 

 $1,000 
 1 = [9.456]0.025 – 1 = 1.05777 – 1 = 0.05777.
Taking this semiannual rate times two and converting to percentage renders a total return of
about 11.55%. This is closer to the reinvestment rate of 10% than the yield to maturity of 14%.
If the reinvestment rate increases to 18%, the future value of the interest payments will rise
108.74% to $15,196. The total return will be:
1/ h
 total future dollars 
 purchase price of bonds 


1 / 40
 $24,652 
1 = 

 $1,000 
 1 = [24.652]0.025 – 1 = 1.0834 – 1 = 0.0834.
Taking this semiannual rate time two and converting to percentage renders a total return of
16.68%. This is closer to the reinvestment rate of 18% and the yield to maturity of 14%.
We conclude that for a long-term high-yield coupon bond, that the total return from holding a
bond to maturity will be closer to the reinvestment rate than the yield to maturity.
16. Suppose that an investor with a five-year investment horizon is considering purchasing
a seven-year 9% coupon bond selling at par. The investor expects that he can reinvest the
coupon payments at an annual interest rate of 9.4% and that at the end of the investment
horizon two-year bonds will be selling to offer a yield to maturity of 11.2%. What is the
total return for this bond?
The investor has a five-year investment horizon to purchase a seven-year 9% coupon bond for
$1,000. The yield to maturity for this bond is 9% since it is selling at par. The investor expects to
be able to reinvest the coupon interest payments at an annual interest rate of 9.4% and that at the
end of the planned investment horizon the then-two-year bond will be selling to offer a yield to
maturity of 11.2%. The total return for this bond is found as follows:
Step 1: Compute the total coupon payments plus the interest on interest, assuming an annual
reinvestment rate of 9.4%, or 4.7% every six months. The coupon payments are $45 every six
months for five years or ten periods (the planned investment horizon). Applying equation (3.7),
the total coupon interest plus interest on interest is
 (1 + r )n  1
 (1 .047 )10  1
coupon interest plus interest on interest = C 
 = $45 
 = $45[12.40162]
r


 0.047 
= $558.14.
Step 2: Determining the projected sale price at the end of five years, assuming that the required
yield to maturity for two-year bonds is 11.2%, is accomplished by calculating the present value
of four coupon payments of $45 plus the present value of the maturity value of $1,000,
52
discounted at 5.6%. As seen below, the projected sale price is $961.53.
projected sale price = present value of coupon payments + present value of par value =
1

1  1  r n
C

r
1


1

 (1.056) 4
  M 
= $45 
+ 
  ( 1 + r )n 
 0.056

  $1, 000 
 + 
=
4
  (1.056) 
$45[3.4970813] + $1,000[0.8041634] = $157.37 + $804.16 = $961.53.
Step 3: Adding the amounts in steps 1 and 2 gives total future dollars of $558.14 + $961.53 =
$1,519.67.
Step 4: To obtain the semiannual total return, compute the following:
1/ h
 total future dollars 
 purchase price of bonds 


1 / 10
 $1,519.67 
1 = 

 $1,000 
 1 = [1.63840]0.16667 – 1 =
1.042738 – 1 = 0.042738 or 4.2738%.
Step 5: Double 4.2738%, for a total return of about 8.55%.
17. Two portfolio managers are discussing the investment characteristics of amortizing
securities. Manager A believes that the advantage of these securities relative to
nonamortizing securities is that because the periodic cash flows include principal
repayments as well as coupon payments, the manager can generate greater reinvestment
income. In addition, the payments are typically monthly so even greater reinvestment
income can be generated. Manager B believes that the need to re-invest monthly and the
need to invest larger amounts than just coupon interest payments make amortizing
securities less attractive. Whom do you agree with and why?
For amortizing securities, reinvestment risk is even greater than for nonamortizing securities.
The reason is that the investor must now reinvest the periodic principal repayments in addition to
the periodic coupon interest payments. Moreover, the cash flows are monthly, not semiannually
as with nonamortizing securities. Consequently, the investor must not only reinvest periodic
coupon interest payments and principal, but must do it more often. This increases reinvestment
risk. Thus, one would tend to agree with manager B. However, manager A may feel that interest
rates will increase so as to make reinvestment income greater. However, if the borrower prepays
early (by accelerating the periodic principal repayment) then manager A would never realize the
opportunity to reinvest at a greater rate. Also, if rates fall then manager A would be stuck with
investing greater cash flows with amortizing securities relative to nonamortizing securities.
In regard to accelerating the periodic principal repayment, nonamortizing securities typically
allow for a greater acceleration of the periodic principal repayment for the borrower. But a
borrower will typically prepay when interest rates decline. Consequently, if a borrower prepays
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when interest rates decline, the investor faces greater reinvestment risk because he or she must
reinvest the prepaid principal at a lower interest rate. If this is case, then manager A’s choice of
amortizing securities may do a better job of avoiding reinvestment risk.
18. Assuming the following yields:
Week 1: 3.84%
Week 2: 3.51%
Week 3: 3.95%
Answer the following questions.
(a) Compute the absolute yield change and percentage yield change from week 1 to week 2.
The absolute yield change (or absolute rate change) is measured in basis points and is the
absolute value of the difference between the two yields as given by
absolute yield change = │initial yield – new yield│ X 100.
Inserting in our yields where Week 1’s yield is the initial yield and Week 2’s yield is the new
yield, we get:
absolute yield change = │3.84% – 3.51%│X 100 = 33 basis points.
The percentage change is computed as the natural logarithm of the ratio of the change in yield as
shown by
percentage change yield = 100 X ln (new yield / initial yield).
Inserting in our yields where Week 1’s yield is the initial yield and Week 2’s yield is the new
yield, we get:
percentage change yield = 100 X ln (3.51% / 3.84%) = –8.99%.
(b) Compute the absolute yield change and percentage yield change from week 2 to week 3.
The absolute yield change (or absolute rate change) is measured in basis points and is the
absolute value of the difference between the two yields as given by
absolute yield change = │initial yield – new yield│ X 100.
Inserting in our yields where Week 2’s yield is the initial yield and Week 3’s yield is the new
yield, we get:
absolute yield change = │3.51% – 3.95%│ X 100 = 44 basis points.
We see that there has been a greater change in basis point compared to (a).
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The percentage change is computed as the natural logarithm of the ratio of the change in yield as
shown by
percentage change yield = 100 X ln (new yield / initial yield).
Inserting in our yields where Week 2’s yield is the initial yield and Week 3’s yield is the new
yield, we get:
percentage change yield = 100 X ln (3.95% / 3.51%) = 11.81%.
We see that unlike part (a), the percentage change yield has now increased. Keep in mind that
these are weekly percentage changes and were they annualized they would be extraordinarily
large.
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