Investment Analysis & Portfolio Management
Lecture#
06
BOND YIELDS AND PRICES Contd…
MEASURING BOND PRICE VOLATILITY:
Duration:
In managing a bond portfolio, perhaps the most important consideration is the effects
of yield changes on the prices and rates of return for different bonds. The problem is
that a given change in interest rates can result in very different percentage-price
changes for the various bonds that investors hold. We saw earlier that both maturity
and coupon affect bond price changes for a given change in yields. One of the
problems, however, is that we examined the effects of these two variables separately.
Duration is a measure of a bond's lifetime that accounts for the entire' pattern of cash
flows over the life of the bond
Duration measures the weighted average maturity of a (non-callable) bond's cash
flows on a present value basis. We can also say that duration is the weighted average
of the times until each payment (coupon or principal repayment) from the bond is
received.
Calculating Duration:
To calculate duration, it is necessary to calculate a weighted time period, because
duration is stated in years, the time periods at which the cash flows are received are
expressed in terms of years and denoted by t in this discussion. When all, of these t's
have been weighted and summed, the result is the duration, stated in years.
The present values of the cash flows, as a percentage of the bond's current market
price, serve as the weighting factors to apply to the time periods. Each weighting
factor shows the relative importance of each cash flow to the bond's total present
value, which is simply its current market price. The sum of these weighting factors
will be 1.0, indicating that all cash flows have been accounted for. The sum of all the
discounted cash flows from the bond will equal the bond's price. The equation for
duration is shown as:
n
Macaulay Duration = D = ∑PV (CFt) / market price
*ti=1
Where;
t = the time period at which the cash flow is expected to be
received n = the number of periods to maturity
PV (CFt) = present value of the cash flow in period t, discounted at the yield to
maturity. Market price = the bond's current price or present value of all the cash
flows
Understanding Duration:
How is duration related to the key bond variables previously analyzed? The
calculation of duration depends on three factors:
• The final maturity of the bond
• The coupon payments
• The yield to maturity
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1. Duration expands with time to maturity but at a decreasing rate (holding the
size of coupon payments and the yield to maturity constant particularly beyond
15 years time to maturity). Even between 5 and 10 years time to maturity,
duration is expanding at a significantly lower rate than in the case of a time to
maturity of up to
5 years, where it expands rapidly. Note that for all coupon-paying bonds,
duration is always less than maturity. For a zero-coupon bond, duration is equal
-to time to maturity.
2. Yield to maturity is inversely related to duration (holding coupon
payments and maturity constant).
3. Coupon is inversely related to duration (holding maturity and yield to
maturity constant). This is logical, because higher coupons lead to quicker
recovery of the bond's value, resulting in a shorter duration, relative to lower
coupons.
Why is duration important in bond analysis and management? First, it tells us the
difference, between the effective lives of alternative bonds. Bonds A and B, with the
same duration but different years to maturity, have more in common than bonds C and
.D with the same maturity but different durations. For any particular bond, as maturity
increases, the duration increases at a decreasing rate.
Estimating Price Changes Using Duration:
The real value of the duration mea-sure to bond investors is that it combines coupon
and maturity, the two key variables that investors must consider in response to
expected changes in interest rates, As noted earlier, duration is positively related to
maturity and negatively related to coupon; However, bond-price changes are directly
related to duration; that is, the percentage change in a bond's price, given a change in
interest rates, is proportional to its duration. Therefore, duration can be used to
measure interest rate exposure.
Convexity:
For very small changes in the required yield the approximation is quite close and at
times could be exact. However, as the changes become larger the approximation
becomes poorer. We refer to the curved nature of the price-yield relationship as the
bond's convexity (the relationship is said to be convex because it opens upward). More
formally, convexity is a term used to refer to the degree to which duration changes as
the yield to maturity changes.
The degree of convexity is not the same for all bonds. Calculations of price changes
should properly account for this convexity in order to improve the approximation of a
bond's price change given some yield change.
Convexity is largest for low coupon bonds, long-maturity bonds, and low yields to
maturity. If convexity is large, large changes in duration are implied, with
corresponding inaccuracies in forecasts of price changes. Therefore, when dealing
with securities that have high convexity, the convexity effect on price change must be
considered.
Some Conclusions on Duration:
What does this analysis of price volatility mean to bond investors? The message is
simple to obtain the maximum (minimum) price volatility from a bond; investors
should choose bonds with the longest (shortest) duration. If an investor already owns a
portfolio of bonds, he or she can act to increase the average modified duration of the
portfolio if a decline in interest rates is expected and the investor is attempting to
achieve the largest price appreciation possible. Fortunately, duration is additive, which
means that a bond portfolio's
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modified duration is a (market value) weighted average of each individual bond's
modified duration.
How popular is the duration concept in today's investment world? This concept has
become widely known and referred to in the popular press. Investors can find duration
numbers in a variety of sources, particularly with regard to bond funds.
Although duration is an important measure of bond risk, it is not necessarily always
the most appropriate one. Duration measures volatility, which is important, but is only
one aspect of the risk in bonds. If an investor considers volatility to be an acceptable
proxy for risk, duration is the measure of risk to use along with the correction for
convexity. Duration may not be a complete measure of bond risk, but it does reflect
some of the impact of changes in interest rates.
Zero-Coupon Bonds:
Original issue discount bonds are less common than coupon bonds issued at par. These
are bonds that are issued intentionally with low coupon rates that cause the bond to
sell at a discount from par value. An extreme example of this type of bond is the zerocoupon bond, which carries no coupons and must provide all its return in the form of
price appreciation.
Zeros provide only one cash flow to their owners, and that is on the maturity date of
the bond.
What should happen to prices of zeros as time passes? On their maturity dates, zeros
must sell for par value. Before maturity, however, they should sell at discounts from
par, because of the time value of money. As time passes, price should approach par
value. In fact, if the interest rate is constant, a zero's price will increase at exactly the
rate of interest.