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Lecture 5

Department
Course Code
Professor
Chen Guo
Lecture
5

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CHAPTER 3
MEASURING YIELD
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.
)
y + (1
C
+. . . +
)
y + (1
C
+
)
y + (1
C
+
)
y + (1
C
= P N
N
2
2
1
1
3
3
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 (i.e., the APR), 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 solve for yield, y, using the following equation:
1 =
/1
P
CF
yn
n
.
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Annualizing Yields
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. 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
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:
)
y + (1
F
+
)
y + (1
C
+. . . +
)
y + (1
C
+
)
y + (1
C
+
y + (1
C
= P nn2 3
)
where P = price of the bond, C = semiannual coupon interest (in dollars), F
= 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 (BEY).
It is much easier to compute the yield to maturity for a zero-coupon bond
because we can use:
1 =
/1
P
F
y
n
. Remember that n should be counted as the
2