DURATION:

Duration can be viewed from different angles and thus interpreted differently. We
consider three main views:
o The average of time where different cash flows of the bond occur
o Measure of sensitivity of the bond to changes in its yield
o The derivative of bond price with respect
Average Time of receiving bonds cash flows:

A zero coupon bond is one where there is one payment at maturity. The price is
at a discount, say, $75. We pay $75 up front and at maturity of say 10 years
receive 100. The difference of $ 25 ($100-$25) represent the interim cash flows,
i.e., coupons that we are not receiving in the interim. In this example, the implied
annual coupon (ignoring compounding) can be calculated in a fairly straight
forward manner. The $25 is for 10 years(maturity) of the bond. This translates
into $2.5 = $25/10 per year. Since the bond originally costs us $75 the annual
interest rate is $2.5/$75 = 0.0333 = 3.33%. Going back to the topic of cash flow if
we ask the question of what is the average time where we receive the cash flows
of the zero coupon bond the answer is the maturity on the bond. Let us call this
concept of the average time to receive the cash flow the duration of the bond.
The previous argument says that the duration of a zero coupon bonds is the
same as its maturity i.e., the time the approximately (in this case exactly) we
receive the bond’s cash flows. Let us consider a coupon bond with maturity of 2
years and semiannual coupon of 4%. Suppose the price o this bond is 98.12.
This implies a yield of 5%. Conversely recall if we give the yield of 5% we can
calculate the price and it will be 98.12. Now here are the cash flows
6m
12m
18m
24m
$2
$2
$2
$102
The above cash flows can each be viewed as a zero coupon bond with following
maturity and pay of
6m
Bond 1
Bond 2
Bond 3
Bond 4
12m
18m
24m
$2
$2
$2
$102
So this bond can be viewed as a portfolio of four zero coupon bonds with face values
(what we get paid at maturity) of $2,$2,$2,$102. For each of these zero coupon bonds it
is easy to determine their duration which is their maturity i.e., 0.5 Yr, 1 Yr, 1.5 Yr, and 2
Yrs. However, it is not clear what is the average time when we receive the cash flows.
Putting it in another way, we add up all the cash flows i.e., 108 and say what would be
the maturity (hence duration) of a zero coupon bond that pays 108 and economically is
almost the same as the coupon bond. i.e., if we want to receive all the cash flows at one
point in time when would that point in time be? The natural answer is the “average” of
the times that we receive these cash flows. Now the challenge is how to calculate this
average. Clearly it should be weighted since $2 cash flow is less important than $102
and this importance depends both on the size of the cash flow and the time that it is
received. Recall that the way price is calculated from the yield is to discount all the cash
flows using the yield and the sum of all these discounted cash flows is the price. For a
yield of 5% the discount factors based on this yield are
Bond Price
98.12
Year
Period
Cash Flow
Discount
Factor
0.5
1
1.5
2
1
2
3
4
2
2
2
102
0.9756
0.9518
0.9286
0.9060
1.9512
1.9036
1.8572
92.4070
Total
100.0000%
Factor
1.9886%
1.9401%
1.8928%
94.1785%
Now when discounting the cash flows by multiplying them by the discount factor we
arrive at column 5 the is the PV of cash flows using the yield and the sum of these
discounted cash flows is the price of the bond, namely 98.12. Now the contribution of
cash flow that occurs at time m to the price of the bond is simply the proportion of the
price that comes from this the pv of this cash flow namely
[(Cash Flow)/(1 +5%/2)m]/98.12 and these proportion contribution should have a sum of
100% which is the case. Look at the last column. Now calculation of duration is easy
and intuitive. Let us look at say, the 3rd cash flow that happens in third period, i.e. 1.5
years. The contribution of this cash flow to the final price when discounting at yield is
$2/(1+5%/2)3 = $2 * 0.9286 = 1.8572. Now as a percent contribution to the final price is
1.8572/98.12 = 1.8928%. So we use 1.5*1.8928% = 0.0284. Continuing in this fashion and
adding all these up gives us the duration of 1.9413 See the table below
Bond Price
98.12
Year
Period
Cash Flow
Discount
Factor
0.5
1
1.5
2
1
2
3
4
2
2
2
102
0.9756
0.9518
0.9286
0.9060
1.9512
1.9036
1.8572
92.4070
Total
Duration
100.0000%
1.9413
Factor
1.9886%
1.9401%
1.8928%
94.1785%
0.0099
0.0194
0.0284
1.8836
Measure of sensitivity of the bond to changes in its yield:



As an investor what is important to you are changes in the value of your assets. If
your assets are bonds , their prices are inversely related to yield. So you are
interested in what happens to the price of your bonds- hence your portfolio- if the
yields go up or down. One can safely assume if the level of interest rates rise say
LIBOR, 2 YR treasury, 5 year Treasury,… then the yield of bonds should also
rise of course the relationship is not one to one and it could happen that say 2
year yield rises but yield of 5 year bond falls. However, as the first estimate one
could assumes that all yield rise parallel i.e., say they all go up say by 5 bps.
Clearly if we are dealing with one bond it is not difficult to calculate the change in
the price of that bond if its yield goes up or down by say 5 bps. However, imaging
having a portfolio of 200 bonds with different maturities and coupon. In this case
it would be good to have a measure of sensitivity of your total portfolio to
changes in the interest rate (i.e., assuming the yield of all the bonds move by the
same amount) just as a quick and dirty way of having an idea of the sensitivity of
your port folio. For example, it will be good to know that approximately for each 1
bps move in yield the value of your portfolio moves by say 4 bps (duration of 4).
This would imply that if interest rates say rise by 25 bps the value of your
portfolio drops by 1%.
Suppose you have 250 of the above bonds with a face value of $5,000. The total
value of your portfolio is
250x $5,000x(98.12/100) = $1,226,500.00
Now suppose the yield goes up by 25 bps = 0.25%. We can calculate the price of the
bond by changing the yield and the new price is 97.66. The value of your portfolio in this
case is 250x $5,000x(97.66/100) = $1,220,750.00. Therefore the change in value of your
portfolio is $1,220,750.00 - $1,226,500.00 = -$5,750.00.
Rather than recalculating the new price of the bond and revaluing the portfolio and
taking the changes in the value, Duration enables us to get an approximation very
quickly namely
Approximate Change in value of portfolio = -(Change in Yield)x Duration x value of
portfolio
Approximate Change in value of portfolio = -(5.25%-5%)x1.94x $1,226,500.00 = - $5,948.53
which is a reasonable approximation.
Derivative of Bond Price with respect to yield:
This part gives a mathematical formulation of duration and illustrates that the reason
we can approximately determine the change in the value of the portfolio by looking at
duration and change in the yield follows from calculus.
Background regarding the definition and properties of derivatives