BOND PRICING AND RETURNS

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Investment Analysis & Portfolio Management (FIN630)

Lesson # 28

BOND FUNDAMENTALS Contd...

BOND PRICING AND RETURNS:

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:

Annuities:

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

where;

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,

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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

applications.

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.

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