Bond Duration Pavel Ababii//440134
Introduction
Risk Management is one of the most
important aspect of the investment
process. A fundamental idea in finance is
the relationship between risk and return.
Low risk does not necessary imply low
profitability, but low risk in combination
with reasonable profitability is an aspect of
high degree risk management. Even though
the risk by its definition is an uncertain
event in the future that can occur with
certain probability, still it is measurable and
manageable. In this paper, we will try to
look at one of the most basic ideas of risk
measurement, and mainly the concept of
bond duration.
Price Sensitivity
The idea of duration derives from the
relationship between the yield to maturity
and the price of the bond itself. As the price
of the bond is nothing else than the sum of
its future cash flow discounted at the
current yield, the relationship between
these two variables is becoming intuitive. In
the following graph, it is shown the
relationship between the price and yield
curve.
In order to be able to track such aspect of
price sensitivity, the concept of duration
was introduced. In the financial world there
are 2 common ways to estimate this price
sensitivity of a given bond. These are:
1. The Parametric Approach- which is
nothing else than the estimation of
the change in price by using the
derivatives of the price function in
respect to yield.
2. The full Valuation Approach- by
calculating the price of the bond for
every given change in the yield.
We will discuss these two approaches in
the next sections.
Macau Duration
The formula that is usually used to
calculate a bond's basic duration is the
Macaulay duration, which was created by
Frederick Macaulay in 1938, although it
was not commonly used until the 1970s.
Macaulay duration is calculated by adding
the results of multiplying the present value
of each cash flow by the time it is received
and dividing by the total price of the
security.
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Frederick Macaulay’s original 1938
measure of the “length” of life insurance
liabilities. While Macaulay hinted at the
relationship between duration and the
price-yield relationship, no one seemed to
notice. This formula is just a weighted
average of time where the cash flows serve
as weights. Macau duration tells us how
long it will take a security to repay itself
completely.
Modified duration
In the 1960s, Larry Fisher presented a proof
of the relationship between price change
and yield change and duration; it took
another 15 or 20 years for duration to
become common knowledge in the
industry. In order to define the relationship
between price and yield, it was used a very
common approach, and mainly Taylor
Series. We know that the relationship
between price and yield is nothing else than
a function. Taylor Series states that any
function can be approximated through a
polynomial function where each term in
polynomial function is the derivative of the
original function itself. It estimates a
function f(x)’s value at f(x+Δx) as:
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Duration and convexity are linked to this
idea. They can be used to approximate the
behavior of a pricing function. Hence, the
modified duration is just the first two
polynoms of the polynomial function, which
is essentially the derivative of the price
function. These results will end up giving
us a duration, which express the change in
monetary units by a given change in yield.
In order to express it in percentage change,
this result is divided by the respective
security price.
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dP
P
Duration ified
1
mod 
If we do the calculus and re-arrange the
expression, we obtain the formula.
Using the parametric approach we can
derive the modified duration using the
Macau Duration and this expression is
showing the relationship between these
two measures.
Even though modified duration is giving us
a very good insight on our yield risk, there
is still a major dropdown. Depending on the
shape of the price-yield function and the
interval of change in yield (Δy) the modified
duration can give us a slight error, which
can be observed in the graph below.
This error is easily corrected by adding the
next polynom from the Taylor series, often
called convexity.
Effective duration
Effective duration is probably the most
accurate approximation of a bonds
duration. Even though Taylor Series can
predict with a certain accuracy the
movement of the price given the yield curve
change, it still leave some space for errors.
Another reason to use the effective
duration is that the use of modified
duration implies that the cash flows of the
security will not change, which is not true
for securities with embedded options.
These two arguments are enough to move
to the full valuation approach. We have
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already stated that full valuation approach
states that one should calculate the price
for any given yield change. Given this
information, we can easily derive the
formula for this approach.
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This approach gives us a far more accurate
view of the change in price when the yield
curve move, but we have to admit that this
approach implies a higher volume of
calculus.
Characteristics of duration.
It is important to note that duration, as an
indicator is still sensible to several factors.
One of these factors is the coupon rate. The
coupon rate determines the amount of the
future cash-flow that will be received by the
bondholder, hence it has a direct impact on
the duration itself. A bond with higher
coupon rate will have higher coupon
payments and lower duration, and vice
versa when referring to a bond with lower
coupon yield.
Another factor is the yield curve. Since the
yield is one of the components of the
formula for both Macau and modified
duration, it is obvious to consider it as an
important factor.
The third factor is the time. We have
mentioned that Macau duration is a
formula of weighted average of time. The
duration has a tendency to reduce its value
over time as we approach the bond
maturity.
The fourth and the last factor to be
considered is the bond coupon payment.
Coupon represent the future payments for
the bondholder. When the coupon is paid
there is no sense for the coupon holder to
consider it a future cash flow, and it is
eliminated from the formula of duration
calculation. Each time we have a coupon
payment the bond duration increases.
Even though it is very common to depict
duration graphically as a smooth line, we
should know that in essence the function of
duration is not linear. In the graph below we
can find a representation of duration, where
the smooth line is an approximation and
the second function represents the actual
duration function. Each jump in duration is
due to the coupon payment that is set to
increase duration.
Even though duration is, a rather simple
and very handy instrument to use in risk
measurement it feels vital to point out
some of its aspects. The risk measured so
far rely on unrealistic assumptions that
security prices are a function of a single
systematic risk factor – Yield to maturity. If
you have a portfolio of bonds, we are
implicitly assuming each bond’s yield
moves by the same amount, equivalent to
saying that the yield curve moves in a
parallel manner. This is not a bad
assumption, but the yield curve can (and
does) move in different ways. Therefore,
these measures fail to predict accurately
price fluctuations from changes in shape or
slope of yield curve (yield curve risk).
Conclusion
Duration is a handy, quite simple
instrument that is able to give some insight
about the price change risk depending on
the yield change. This is a very widely used
instrument in risk management and in
financial modeling. However, its biggest
disadvantage is that it assumes parallel
shifts in the yield curve, which explains only
85-90% of yield changes, thus leaving
some space for errors.
Bibliography
1. http://www.investopedia.com/unive
rsity/advancedbond/advancedbond
5.asp
2. http://www.investopedia.com/exam
-guide/cfa-level-1/fixed-income-
investments/duration.asp
3. http://www.investopedia.com/term
s/d/duration.asp