Computerized Investing - Spreadsheet Collection/Spreadsheet Collection.iso/pc/ibm11/bondrrci.doc

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Estimating the Realized Return on a Bond
by Fred Shipley
May/June 1991, Computerized Investing

An important, and often ignored, factor of investing in bonds is
the dependence of realized returns on market interest rates. Most
investors know that bond prices change inversely with changes in
market interest rates. Many though, forget that the return they ac-
tually earn depends on the reinvestment of the interest cash flows
the bond generates. The consequences of reinvesting cash interest
payments can be easily examined with the aid of a spreadsheet.

The issue is not limited to fixed-income investments, however. The
realized return and the wealth accumulation for any investment
depends on the reinvestment of the cash the investment generates.
And the impact over time can be substantial.  Dividend reinvestment
generates a wealth level nearly 20 times greater than that of pure
capital appreciation ($518 vs. $26 on $1 invested at the end of
1925).

The differences in wealth accumulation arise from different
realized rates of return. The S&P 500 offers a capital gain of
about 5% a year, and dividend reinvestment offers nearly another
5%. That extra 5% a year from dividend reinvestment eventually
results in a substantially greater accumulated wealth. Of course,
this example covers a 65-year time frame, but the results are still
quite noticeable over much shorter horizons. With fixed-income
investments, an initial 30-year horizon is quite common, and even
a 10-year horizon can be long enough to generate a substantial
difference in return. Long-term government bonds have actually
suffered net capital losses over the period shown. Most of the
losses occurred during the highly inflationary period since the
1970s.

Our spreadsheet deals with a bond's realized return, but it can be
modified for any other investment. For other investments, we would
have to enter the periodic cash flows as well as the estimated
returns to determine the realized return. With fixed-income securi-
ties, it is possible to estimate the future returns from current
market data.

The main question that must be answered in determining the return
a bond might generate is what market interest rate is relevant at
any given period of time. After the fact, this rate can be
determined from a number of sources. The Value Line Investment
Survey now has a section on fixed-income securities giving interest
rates for different bond rating categories and for Treasury issues
of different maturities. The Standard & Poor's Bond Guide carries
similar information.

The Yield Curve

For planning purposes, though, an investor will want to examine the
effect of potential interest rate changes before they occur. While
this estimation is not easy, there are some ways to approach the
problem. One of the most common methods of estimating future
interest rates uses the yield curve. The yield curve is a graphical
view of the relationship between maturity and yield to maturity for
Treasury securities. This relationship between yield and maturity
is also referred to as the term structure of interest rates. From
this data, it is possible to make inferences about the market's
perceptions of future interest rates on government securities. To
pass from Treasuries to corporate issues, it is necessary to
estimate the premium that the riskiness of corporates will command
relative to Treasuries.

Risk Premiums and Returns

The risk premium on any security--or for an entire segment of the
market, such as AA-rated corporates or junk bonds--is a measure of
the extra return that security, or that market segment, is offering
in compensation for its risk over and above the risk of a Treasury
issue. We usually presume that Treasuries are free of default risk
(which seems reasonable), but of course, there is interest rate
risk even on Treasuries. Nevertheless, Treasury securities serve as
our benchmark, and the base from which yields on other securities
are estimated. Estimates of these premiums (or yield spreads) can
be derived from the Federal Reserve Bulletin, which most large
libraries should carry.

This yield spread is called a default yield spread since it
measures the extra return (or risk premium) an investor receives
for taking the risk of default from a corporate security. The
default yield spread depends on the creditworthiness of the issuer,
which can change over time. Various investment information services
rate fixed-income securities by credit risk. Perhaps the best known
of these companies are Standard & Poor's and Moody's. The top four
credit ratings (AAA to BBB for S&P; Aaa to Baa for Moody's) are re-
ferred to as investment grade. Securities with these ratings
usually carry relatively low risk of default and therefore have a
small yield spread. One of the major determinants of a high bond
rating is substantial coverage of interest and other fixed outlays.
This is called cash flow coverage and it measures the amount by
which a company's cash flow can shrink before they cannot meet
their fixed obligations.

The 1970s saw a vast growth in the issuance of lesser-rated
securities, so called "junk bonds." Barron's tracks the yield
spread on junk bonds in its Market Laboratory section each week.

The growth in this market was fueled by research that showed a
relatively low rate of default on these securities, especially
considering the substantial yield premiums they offered. However,
these studies did not examine the effect of a general economic
slowdown on this entire segment of the market. Diversification
among a range of junk bonds will indeed reduce the risk of default
from factors that affect only one company. A recession will affect
this entire market segment much more severely than the segment of
investment-grade bonds because the cash flow coverage of low-grade
bonds is much tighter.

Interestingly enough, research on fixed-income securities has shown
that yield spreads on high-grade issues tend to get larger as the
time to maturity increases, while yield spreads on lower-grade
issues tend to shrink as the maturity lengthens. The explanation
for this is that longer maturities open up the possibility for more
bad things to happen to good companies and more good things to
happen to bad companies. In any event, an investor will expect some
extra return for taking default risk. Our approach is to add some
premium to the imputed Treasury return to account for this risk.
The premiums in the spreadsheet are meant to be illustrative and
should not be taken as investment rules.

Using the Yield Curve

Estimating (short-term) interest rates from the yield curve is
relatively simple. We just assume that the return over a two-year
period (which we can determine from the rate on a two-year maturity
Treasury bond) is equal to the rate we can earn for one year (from
a one year maturity bond) times the return that could be earned by
reinvesting after one year in another one-year maturity bond. In
other words, the market is telling us, from the two-year return and
the one-year return, its judgment of the future one-year return.
This analysis can be applied to any future time period.

Economists call this the expectations theory of the term structure,
but as investors we simply want to use this information to estimate
future interest rates for analyzing our bond portfolio. We do this
with the market expectation of future rates. In a formula, the two-
year return is the product of the known one-year return times the
estimated one year return one year from now.

(1 + 0R2)^t = (1 + 0R1)^t (1 + 1R2)^t

where:
0R2= the return on a two-year maturity Treasury note issued today
maturing two years from today

0R1= the return on a one-year maturity Treasury note issued today
maturing one year from today

1R2 = the market's implicit (expected) return on a one-year
maturity investment starting one year from now--in other words, the
return over the year starting one year from now and maturing two
years from now

t = years to maturity for each issue from time of investment

The estimated future one-year return is what we want to know, so we
rearrange this relationship to get:

(1 + 1R2)^1 = (1 + 0R2)^2 (1 + 0R1)^1

This "forward rate" is the market's expected return on a one-year
Treasury investment starting a year from now. As investors, we can
infer other forward rates in the same fashion. The market is giving
us a kind of consensus forecast of future interest rates.

By matching the maturity of the Treasury issue with the maturity of
the bond we own (or are considering investing in), we can infer the
reinvestment return over our entire future holding period of the
bond. In addition, these expected future levels of interest rates
imply market returns that can be used to determine the price at
which our bond investment might trade in the future. This allows us
to determine our realized return at any future date on the
assumption that we might have to sell the bond before maturity.

The expected market yield on a bond with 20 years to maturity in
2001 (10 years from now) is determined by an analogous formula.

10R30 = [(1 + 0R30)^30 / (1 + 0R10)^10]^1/20-1

Duration and the Realized Rate of Return

The duration of the 30-year bond is 11.26 years initially. In
addition, the realized return (total return if sold) on the bond is
around 8.5% after 11 years. This corresponds to our initial yield
to maturity and roughly matches the duration. We know that if we
hold a bond to its duration (rather than to its maturity), our
realized yield is independent of future changes in rates, and the
spreadsheet shows this.

Setting Up the Spreadsheet

The spreadsheet requires only a few major formulas, most of which
can be copied to perform the rest of the calculations you need. The
first part of the spreadsheet contains the basic input data on the
30-year bond. The second part contains the Treasury yield curve
data derived from the implied forward rates. We then enter formulas
to determine the accumulated value, the value of the bond if sold
prior to maturity, and the realized return at that selling date. In
addition, we have added the calculation of the bond's

Duration (in years):
Effect of 1% Market Yield Change:-10.8%
duration (a measure of its price sensitivity to interest rate
changes) and the approximate percentage change in the bond's value
for a one percentage point move in interest rates. For more infor-
mation on duration, see the September/October 1988 issue of CI.

The data entered are the bond's current market price, its initial
yield to maturity, its face value, its coupon rate (to determine
the interest payments), the current date and the maturity date. The
yield curve data are entered in columns J through L. Estimated
default yield spread data is in columns N and T.

To see in detail the changes in future market interest rates, we
have created several columns of information from J26 to L87. This
information shows the expected future interest rate from each date
for a period of time equal to the bond's remaining maturity. Since
we have used Treasury bond data at five year increments to estimate
future rates, we have averaged these rates to prevent large jumps
or declines in the realized return that are simply due to starting
with a new market rate at five-year intervals. The calculations
perform a weighted average between the new rate and the old rate.
These rates are then referred to in calculating the bond's market
price and realized rate of return.

(c) Copyright 1991 by the
American Association of Individual Investors