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Investment Analysis and Portfolio Management

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Investment Analysis & Portfolio Management (FIN630)
Lesson # 28
As with any time-value-of-money application, there is a deterministic relationship between
the current prices of a security, it's promised future cash flows, and the riskiness of those
cash flows. The current price is the market's estimation of what the expected cash flows are
worth in today's dollars.
Valuation Equations:
For an ordinary annuity bond paying interest semiannually and assuming it has just made an
interest payment, the valuation equation is as follows:
Po=Ct/ (1+r/2)t
n = term of the bond in semiannual periods
Ct = cash flow at time t
r = discount rate
Po = current price of the bond
t = time in semiannual periods from the present
The bond pricing relationship is customarily expressed in terms of the number of
semiannual payment periods. An eight-year bond, for example, has 16 semiannual
payments. This procedure also requires dividing the annual discount rate, r, by two to turn it
into a semiannual equivalent.
To illustrate, we split equation into two parts, one for the interest component (the cash flows
Ct and one for the principal:
Po=Ct/ (1+r/2)t + Par/1+r/2)n
Bond price = PV (interest) +PV (principal)
Suppose a bond currently sells for $900, pays $95 per year (interest paid semiannually), and
returns the par value of $1,000 in exactly eight years. What discount rate is implied in these
numbers? To find out, we solve the following valuation equation:
$900=$47.50/(l+r/2)t + $1,000/(l+r/2)16
This equation can be solved using time-value-of-money tables, a finance calculator, or a
spreadsheet package such as Lotus 1-2-5 or Microsoft Excel. We find r = 11.44%.
This bond's return comes from two sources: periodic interest and the return of the bond
principal in eight years. These two components can be valued separately after determining
the appropriate interest rate. Using 11.44%, the value of the interest component is $489.40,
Investment Analysis & Portfolio Management (FIN630)
while the principal value is $410.60 in current dollars. These two values sum to the bond's
current market price of $900.
The bond pricing relationship is customarily expressed in terms of semiannual periods.
Yield to Maturity:
In the preceding valuation equations, investors call the discount rate, r, the yield to maturity.
This concept is precisely the same as internal rate of return in corporate finance
Calculating the Yield to Maturity:
An easy-to-use approximation method usually provides an estimate within a few basis
points of the true yield to maturity.
YTM approximate = (annual interest ­ ((market price-par value)/years until
maturity))/0.6(market price) + 0.4(par value)
Plugging in the values from the previous example, we find an approximate yield to maturity
of 6.32%
In this case, the value from the approximation formula is near the true value from the
complete valuation equation. When the bond sells for near par, the approximation method is
accurate. Some error is introduced when a bond sells for a substantial discount or premium.
Spot Rates:
For a given issuer, all securities of a particular maturity will not necessarily have the same
yield to maturity, even if they have the same default risk. A spot rate is the yield to maturity
of a zero coupon security of the chosen maturity. You can observe spot rates directly from
the U.S. Treasury strips portion of the government bond. A treasury strip is a government
bond or note that has been decomposed into two parts, one for the stream of interest
payments and one for the return of principal at maturity- These are sometimes called
interest only and principal only securities, respectively. The codes in the newspaper listing
are ci for coupon interest, np for note principal, and bp for bond principal- The principal-
only version of a U.S. treasury strip is a manufactured zero coupon security, but one whose
price reflects the prevailing spot rate.
The yield to maturity is the single interest rate that, when applied to the stream of cash
flows associated with a bond, causes the present value of those cash flows to equal the
bond's market price. Yield to maturity is a useful and frequently cited statistic. It is akin to
an average of the various spot rates over the security's life. The market, however, does not
value a bond using the yield to maturity concept. Rather, the yield to maturity is a derived
statistic after the bond value is already known; we need to know the bond price in order to
get the yield to maturity.
For valuation purposes, a bond should be thought of as a package of zero coupon securities,
each providing a single cash flow, and each valued using the appropriate spot rate. In other
words, each component is discounted by a specific rate rather than by some average rate.