Investment Analysis & Portfolio Management
Lecture#
05
BOND YIELDS AND PRICES
Bond Yields:
Bond yields and interest rates are the same concept. Therefore, we begin our
discussion of bond yields with a brief consideration of interest rates. Interest rates
measure the price paid by a borrower to a lender for the use of resources over some
time period—that is, interest rates are the price for loan able funds. The price differs
from case to case, based on the demand and supply for these funds, resulting in a wide
variety of interest rates. The spread between the lowest and highest rates at any point
in time could be as much 10 to 15 percentage points. In bond parlance, this would be
equivalent to 1,000 to 1,500 basis points, since 1 percentage point of a bond yield
consists of 100 basis points.
It is convenient to focus on the one interest rate that provides the foundation for other
rates. This rate is referred to as the short-term riskless rate (designated RF in this 'text)
and is typically proxied by the rate on Treasury bills. All other rates differ from RF
because of two factors:
1.
Maturity differentials
2.
Risk premiums
The Basic Components of Interest Rates:
Explaining interest rates is a complex task that involves substantial economics
reasoning and study. Such a task is not feasible in this text. However, we can analyze
the basic determinants of nominal (current) interest rates with an eye toward
recognizing the factors that affect such rates and cause them to fluctuate. The bond
investor who understands the foundations of market rates can then rely on expert help
for more details, and be in a better position to interpret and evaluate such help.
The basic foundation of market interest rates is the opportunity cost of foregoing
consumption; representing the rate that must be offered to individuals to persuade
them to save rather than consume. This rate is sometimes called the-real risk-free rate
of interest because it is not affected by price changes or risk factors. We will refer to it
simply as the real rate and designate it RR in this discussion.
Thus, for short-term risk-free securities, such as three-month Treasury bills, the
nominal interest rate is a function of the real rate of interest and the expected
inflationary premium. This is expressed as, which is an approximation:
RF ≈RR + EI
Where;
RF = short term Treasury bill rate
RR = the real risk-free rate of interest
El = the expected rate of inflation over the term of the instrument
Measuring Bond Yields:
Several measures of the yield on a bond are used by investors. It is very important for
bond investors to understand which yield measure is being discussed, and what the
underlying assumptions of any particular measure are. To illustrate these measures, we
will use as an
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example a three-year, IC-percent coupon, AAA-rated corporate bond, with interest
payments occurring exactly six months from now, one year from now, and so forth.
Current Yield:
The ratio of the coupon interest to the current market price is the current yield, and,
this is the measure reported daily In the Wall Street Journal for those corporate bonds
shown under the sections "New York Exchange Bonds" and "AMEX Bonds." The
current yield is clearly superior to simply citing the coupon rate on a bond, because it
uses the current market price as opposed to the face amount of a bond (almost always
$1,000). However, current yield is not a true measure of the return to a bond
purchaser, because it does not account for the difference between the bond's purchase
price and its eventual redemption at par value.
Yield to Maturity:
The rate of return on bonds most often quoted for investors is the yield to maturity
(YTM), a promised rate of return that will occur only under certain assumptions. It is
the compound rate of return, an investor will receive from a bond purchased at the
current market price if:
1. The bond is held to maturity, and
2. The coupons received while the bond is held are reinvested at the calculated
yield to maturity.
Barring default, investors will actually cam this promised rate if, and only if,
these two Conditions are met. As we shall see, however, the likelihood of the
second condition actually being met is extremely small.
The yield to maturity, is the periodic interest rate that equates the present value of the
expected future cash flows (both coupons and maturity value) to be received on the
bond to the initial investment in the bond, which is its current price. This means that
the yield to maturity is the internal rate of return (IRR) on-the bond investment,
similar to the IRR used in capital budgeting analysis.
Yield to Call:
Most corporate bonds, as well as some government bonds, are callable by the issuers,
typically after some deferred call period. For bonds likely to be called the yield to
maturity calculation is unrealistic. A better calculation is the yield to call. The end of
the deferred call period, when a bond can first be called, is often used for the yield to
call calculation.
This is particularly appropriate for bonds selling at a premium (i.e. high-coupon bonds
with market prices above par value).
Realized Compound Yield:
After the investment period for a bond is over, an investor can calculate the realized,
compound yield (RCY). This rate measures the compound yield on the bond
investment actually earned over the investment period, taking into account all
intermediate cash flows and reinvestment rates. Defined in this mannerist cannot be
determined until the investment is concluded and all of the cash flows are known.
Thus, if you invest $ 1,000 in a bond for five years, reinvesting the coupons as they are
received, you will have X dollars at the
conclusion of the five years, consisting of the coupons received, the amount earned
from reinvesting the coupons, and the $1,000 par value of the bond payable at,
maturity. You can then calculate your actual realized rate of return on the investment.
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The RCY for a bond can be calculated by dividing the total ending wealth (including
the purchase price) at the bond's maturity by the amount invested; and raising the
result to the 1/n power, where n is the number of compounding periods. Next, subtract
1.0 from the result. Finally, because of the semiannual basis for bonds, multiply by 2
to obtain the bond equivalent rate.
The semi-annual realized compound yield can be calculated using the following
formula:
RCY = [total ending wealth / purchase price of bond] 1/ n – 1.0
BOND PRICES:
The Valuation Principle:
A security's intrinsic value, or estimated value, is the present value of the expected
cash flows from that asset. Any security purchased is expected to provide one or more
cash flows some time in the future. These cash flows could be periodic, such as
interest or dividends, or simply a terminal price or redemption value, or a combination
of these. Since these cash flows occur in the future, they must be discounted at an
appropriate rate to determine their present value. The sum of these discounted cash
flows is the estimated intrinsic, value of the asset. Calculating intrinsic value,
therefore, requires the use of present value techniques.
n
Value t = 0 = ∑cash flows / (1 +
k)t i = 1
Where;
Value t = 0 = the estimated value of the asset now (time period 0)
Cash flows = the future cash flows resulting from ownership of the
asset
k = the appropriate discount rate or rate of return required by an investor for an
investment of this type
n = number of periods over which the cash flows are expected
To solve and derive the “intrinsic value of a security”, it is necessary to determine the
following:
1. The expected cash flows from the security. This includes the size and type of
cash flows, such as dividends, Interest, face value expected to be received at
maturity, or the expected price of the security at some point in the future.
2. The riming of the expected cash flows. Since the returns to be generated from
security occur at various times in the future, they must be properly documented
far discounting back to time period 0 (today). Money has a time value, and the
timing of future cash flows significantly affects the value of the asset today.
3. The discount rate, required rate of return demanded by investors. The discount
rate used will reflect the time value of the money and the risk of the security. It
is an opportunity cost, representing the rate foregone by an investor in the next
best alternative with comparable risk.
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Bond Valuation:
The price of a bond should equal the present value of its expected cash flows. The
coupons and the principal repayment of $1,000 are known, and the present value, or
price, can be determined by discounting these future payments from the issuer at an
appropriate required yield, r, for the issue. To solve for the value of an option-free
coupon bond.
n
P = ∑ct / (1 + r)t + FV / (1 + r)
n
i=1
Where;
P = the present Value or price of the bond today (time period
0) c = the semiannual coupons or interest payments
FV = the face value (or par value) of the bond
n = the number of semiannual periods until the bond
matures r = the appropriate semiannual discount rate or
market yield
In order to conform with the existing payment practice on bonds of paying interest
annually rather than annually, the discount rate being used (r), the coupon (ct on the
bond, and the number of periods are all on a semiannual basis.
For expositional purposes, we will illustrate the calculation of bond prices by referring
to the present value tables at the end of the text; in actuality, a calculator or computer
is used. The present value process for a typical coupon-bearing bond involves three
steps, given the dollar coupon on the bond, the face value, and the current market yield
applicable to a particular bond:
1. Using the present value of an annuity table, determine the present value
of the coupons (interest payments).
2. Using the present value table, determine the present value of the maturity (par)
value of the bond; for our purposes, the maturity value will always be $1,000.
3. Add the present values determined in steps 1 and 2 together.
BOND PRICE CHANGES:
Bond Price Changes Over Time:
We now know how to calculate, the price of a bond, using the cash flows to be
received and the YTM as the discount rate. Assume that we calculate the price of a 20year bond issued five years ago and determine that it is $910. The bond still has 15
years to maturity. What can we say about its price over the next 15 years?
When everything else is held constant, including market interest rates, bond prices that
differ from the bond's face value (assumed to be $1,000) must change over time.
Why? On a bond's specified maturity date, it must be worth its face value or maturity
value. Therefore, over time, holding all other factors constant, a bond's price must
converge to $1,000 on the maturity date because that is the amount the issuer will
repay on the maturity date.
After bonds are issued, they sell at discounts (prices less than $1,000) and premiums
(prices greater than $1,000) during their lifetimes. Therefore, a bond selling at a
discount will experience a rise in price over time, holding all other factors constant,
and a bond selling at
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a premium will experience a decline in price over time, holding all other factors constant,
as the bond's remaining life approaches the maturity date.
Bond Price Changes As a Result of Interest Rate Changes:
Bond prices change because interest rates and required yields change. Understanding
how bond prices change given a change in interest rates is critical to successful bond
management. The basics of bond price movements as a result of interest rate changes
have been known for many years. For example, over 40 years ago, Burton Malkiel
derived five theorems about the relationship between, bond prices and yields. Using the
bond valuation model, he showed the changes that occur in the price of a bond (i.e., its
volatility), given a change in yields, as a result of bond variables such as time to maturity
and coupon. We will use Malkiel's bond theorems to illustrate how bond prices change as
a result of changes in interest rates.
Bond Prices Move Inversely to Interest Rates:
Investors must always keep in mind the fundamental fact about the relationship between
bond prices and bond yields. Bond prices, move inversely to market yields. When the
level of required yields demanded by investors on new issue changes, the required yields
on all bonds already outstanding will also change. For-these yields to change, the prices
of these bonds must change. This inverse relationship is the basis for understanding,
valuing, and managing bonds.
,
Effects of Maturity:
The effect of a change in yields on bond prices depends on the maturity of the bond. An
important principle is that for a given change in market yields, changes in bond prices are
directly related to time to maturity: Therefore, as interest rates change, the prices of
longer term bonds will change more than the prices of shorter term bonds, everything
else being equal.
The Effects of Coupon:
In addition to the maturity effect, the change in the price of a bond as a result of a change
in interest rates depends on the coupon rate of the bond. We can state this principle as
(other, things equal): Bond price fluctuations (volatility) and bond coupon rates are
inversely related. Note that we are talking, about percentage-price fluctuations; this
relationship does not necessarily hold if we measure volatility in terms of dollar price
changes rather than percentage-price changes..