CHAPTER THREE
APPENDIX 3.1 DURATION
Consider first a t-period zero coupon bond. For simplicity we will dis- cuss annual compounding, although the convention is often semi-annual compounding. The price-rate relation can be written as follows
d rt( ) t rt t ( ) .
= +
1 1
For example, compare the value (per $1 of face value) of a one-year vs. a five-year zero, where rates are assumed to be in both cases 5 percent. The value of the one-year zero is $0.9524, and the value of the five-year zero is $0.7835.
In order to discuss the price sensitivity of fixed income securities as a function of changes in interest rates, we first introduce dollar duration, the price sensitivity, and then duration,12 the percentage sensitivity. We define dollar duration as the change in value in the zero for a given change in interest rates of 1 percent. This is approximately the derivative of the zero with respect to interest rates, or:
therefore
The dollar duration of the one-year zero is 1/(1.05)2=0.9070 whereas the dollar duration of the five-year zero is 5/(1.05)6=3.73. What this means is that an increase in rates from 5 percent to 6 percent should generate a loss of $0.00907 in the value of the one-year zero, as com- pared to a loss of $0.0373 in the value of the five-year zero coupon bond. Thus, the five-year zero is more sensitive to interest rate changes. Its sensitivity is close to being five times as large (if interest rates were 0 percent, then this comparison would be precise).
The expression for duration is actually an approximation.13 In contrast, the precise calculation would show that if interest rates increased 1 percent from 5 percent to 6 percent, then the new price of the one-year zero would be 1/(1.06) = $0.9434, and 1/(1.06)5 =
$0.7473 for the five-year. Comparing these new prices to the original prices before the interest rate increase (i.e., $0.9524 for the one-year and $0.7835 for the five-year), we can obtain a precise calculation of the price losses due to the interest rate increase. For the one-year zero, the precise calculation of price decline is $0.9524 −0.9434 =$0.0090 and for the five-year zero, $0.7835 − 0.7472 = $0.0363. Comparing these results to the duration approximation above, we see that the duration approximation overstated price declines for the one-year zero by $0.00007 = 0.00907 − 0.0090. The overstatement was higher for the five-year zero; $0.0010 =0.0373 − 0.0363. Duration is an overly
$ ( )
( ) .
dur d r t
t t t r
t t
= − =
+ +
′ 1 1
d r t
t t r
t t
′( )
( ) ,
= −
+ +
1 1
Table 3.6Duration example td(r=5%)$DurDurD-lossd(r=6%)True lossDuration error 1$0.9524$0.90700.9524$0.0091$0.9434$0.0090−−$0.0001 2$0.9070$1.72771.9048$0.0173$0.8900$0.0170−$0.0002 3$0.8638$2.46812.8571$0.0247$0.8396$0.0242−$0.0005 4$0.8227$3.13413.8095$0.0313$0.7921$0.0306−$0.0007 5$0.7835$3.73114.7619$0.0373$0.7473$0.0363−−$0.0010 6$0.7462$4.26415.7143$0.0426$0.7050$0.0413−$0.0014 7$0.7107$4.73796.6667$0.0474$0.6651$0.0456−$0.0018 8$0.6768$5.15697.6190$0.0516$0.6274$0.0494−$0.0021 9$0.6446$5.52528.5714$0.0553$0.5919$0.0527−$0.0025 10$0.6139$5.84689.5238$0.0585$0.5584$0.0555−$0.0029 1&5$1.7359$4.63812.6719$0.0464$1.6907$0.0453−−$0.0011
pessimistic approximation of price changes resulting from unanticipated interest rate fluctuations. That is, duration overstates the price decline in the event of interest rate increases and understates the price increase in the event of an interest rate decline. Table 3.6 summarizes our example.
It is easy to generalize this price–rate relationship to coupon bonds and all other fixed cash flow portfolios. Assuming all interest rates change by the same amount (a parallel shift), it is easy to show that
portfolio $dur =k1× $durt1+ k2× $durt2+. . . where k1, k2, . . . are the dollar cash flows in periods t1,t2, . . .
Duration is easy to define now as:
duration
≈ [percent change in value] per [100 bp change in rates]
=
/
[1/100]= × 100
= .
Therefore we get
duration = = ,
and for a portfolio we get
duration = = ,
but since
$durt=dt×durt
we get
k1 ×$durt1+ k2× $durt2 +. . . k1× dt1+ k2× dt2+ . . . dollar duration
value
t (1 +rt) t
(1 + rt)t+1 1 (1 +rt)t dollar duration
initial value
Gdollar duration × 0.01J
Gdollar change in value per 100 bpJ
portfolio dur =
⇒portfolio dur =w1 ×durt1 +w2 ×durt2 +. . . .
where wi= is the pv weight of cash flow i.
That is, the duration, or interest rate sensitivity, of a portfolio, under the parallel shift in rates assumption, is just the weighted sum of all the portfolio sensitivities of the portfolio cash flow components, each weighted by its present value (i.e., its contribution to the present value of the entire portfolio).
Going back to our example, consider a portfolio of cash flows con- sisting of $1 in one year and $1 in five years. Assuming 5 percent p.a.
interest rates, the value of this portfolio is the sum of the two bonds,
$0.9524 + 0.7835 = $1.7359, and the sum of the dollar durations,
$0.9070 +3.73 = $4.6370. However, the duration of the portfolio is:
$2.18116 = (0.907) + (3.73).
This tells us that a parallel shift upwards in rates from a flat term struc- ture at 5 percent to a flat term structure at 6 percent would generate a loss of 2.18116 percent. Given a portfolio value of $1.7359, the dollar loss is 2.18116% ×$1.7359 =$0.03786.
The way to incorporate duration into VaR calculations is straight- forward. In particular, duration is a portfolio’s percentage loss for a 1 percent move in interest rates. The percentage VaR of a portfolio is, hence, its duration multiplied by interest rate volatility. For example, suppose we are interested in the one-month VaR of the portfolio of one-year and five-year zeros, whose value is $1.7359 and duration is 2.18116. Suppose further that the volatility of interest rates is 7bp/day, and there are 25 trading days in a month. The monthly volatility using the square root rule is √(25) × 7 = 35bp/month. The
%VaR is therefore 2.18116 × 0.35 = 0.7634%, and the $VaR = 0.007634 × $1.7359 = $0.01325.
There is clearly no question of aggregation. In particular, since we assume a single factor model for interest rates, the assumed VaR shift in rates of 35bp affects all rates along the term structure. The corre- lation between losses of all cash flows of the portfolio is one. It is, therefore, the case that the VaR of the portfolio is just the sum of the VaRs of the two cash flows, the one-year zero and the five-year zero.
0.7835 1.7359 0.9524
1.7359 ki×dti
k1 ×dt1+ k2 ×dt2 +. . .
k1× dt1× durt1+ k2 ×dt2× durt2 +. . . k1× dt1+ k2 ×dt2+ . . .