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TABLE 5.2 Calculating Duration
(A) (B) (C) (D)
Câ€™t/(1 Y/2)t
Câ€™t t (B) (C)

3.8125 0.9344 3.6763 3.4352
3.8125 1.9344 3.6763 6.8399
3.8125 2.9344 3.4009 9.9796
3.8125 3.9344 3.2710 12.8694
3.8125 4.9344 3.1461 15.5240
3.8125 5.9344 3.0259 17.9571
3.8125 6.9344 2.9104 20.1817
3.8125 7.9344 2.7992 22.2102
3.8125 8.9344 2.6923 24.0544
103.8125 9.9344 70.5111 700.4868
Totals 98.9690 833.5384
Notes:
Câ€™t Cash flows over the life of the security. Since this Treasury has a coupon of
7.625%, semiannual coupons are equal to 7.625/2 3.8125.
t Time in days defined as the number of days the Treasury is held in a coupon
period divided by the numbers of days from the last coupon paid (or issue date) to
the next coupon payment. Since this Treasury was purchased 11 days after it was
issued, the first coupon is discounted with t 171/183 0.9344.
Câ€™t/(1 Y/2)t Present value of a cash flow.
Y Bond equivalent yield; 7.941%.

The convention is to express duration in years.

Dmod Dmac /(1 Y/2)
4.2111/(1 0.039705)
4.0503

Modified duration values increase as we go from a Treasury bill to a
coupon-bearing Treasury to a Treasury STRIPS, and this is consistent with
our previously performed total returns analysis. That is, if duration is a mea-
sure of risk, it is not surprising that the Treasury bill has the lowest dura-
tion and the better relative performance when yields rise.
Table 5.3 contrasts true price values generated by a standard present
value formula against estimated price values when a modified duration for-
mula is used.

Pe Pd (1 Dmod Y)

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Risk Management

where Pe Price estimate
Pd Dirty Price
Dmod Modified duration
Y Change in yield (100 basis points is written as 1.0)

Price differences widen between present value and modified duration cal-
culations as changes in yield become more pronounced. Modified duration
provides a less accurate price estimate as yield scenarios move farther away
from the current market yield. Figure 5.1 highlights the differences between
true and estimated prices.
While the price/yield relationship traced out by modified duration
appears to be linear, the price/yield relationship traced out by present value
appears to be curvilinear. As shown in Figure 5.1, actual bond prices do not
change by a constant amount as yields change by fixed intervals.
Furthermore, the modified duration line is tangent to the present value
line where there is zero change in yield. Thus modified duration can be
derived from a present value equation by solving for the derivative of price
with respect to yield.
Because modified duration posits a linear price/yield relationship while
the true price/yield relationship for a fixed income security is curvilinear,
modified duration provides an inexact estimate of price for a given change
in yield. This estimate is less accurate as we move farther away from cur-
rent market levels.

TABLE 5.3 True versus Estimated Price Values Generated by Present Value and
Modified Duration, 7.75% 30-year Treasury Bond

Price plus
Change in Accrued Interest; Price plus
Yield Level Present Value Accrued Interest;
(basis points) Equation Duration Equation Difference

400 76.1448 71.5735 4.5713
300 81.0724 78.2050 2.8674
200 86.4398 84.8365 1.6033
100 92.2917 91.4681 0.8236
0 98.0996 98.0996 0.0000
100 105.6525 104.7311 0.9214
200 113.2777 111.3227 1.9550
300 121.6210 117.9942 3.6268
400 130.7582 124.6257 6.1325

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178 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

Price & accrued
interest
140

Present value
120
Modified duration

100

80

500 400 300 200 100 0 +100 +200 +300 +400 +500 Change in yield
(basis points)

FIGURE 5.1 A comparison of price/yield relationships, duration versus present value.

Figure 5.2 shows price/yield relationships implied by modified duration
for two of the three Treasury securities. While the slope of Treasury billâ€™s
modified duration function is relatively flat, the slope of Treasury STRIPS
is relatively steep. An equal change in yield for the Treasury bill and
Treasury STRIPS will suggest very different changes in price. The price of a
Treasury STRIPS will change by more, because the STRIPS has a greater
modified duration. The STRIPS has greater price sensitivity for a given
change in yield.
If modified duration is of limited value, how can we better approximate
a securityâ€™s price? Or, to put it differently, how can we better approximate
the price/yield property of a fixed income security as implied by the present
value formula? With convexity (the curvature of a price/yield relationship
for a bond).
To solve for convexity, we could go a step further with either the Hicks
or the Macaulay methodology. Using the Hicks method, we would solve for
the second derivative of the price/yield equation with respect to yield using
a Taylor series expansion. This is expressed mathematically as (d2Pâ€™ /dY2)Pâ€™,
where Pâ€™ is the dirty price.
To express this in yet another way, we proceed using Macaulayâ€™s method-
ology and solve for

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179
Risk Management

%âˆ†P

âˆ†Y
Treasury bill

Treasury STRIPS

FIGURE 5.2 Price/yield relationships.

1t
C't t
T

11 Y>22 t
12
t 1

11
C't
T

11 Y>22 t
Y>22 2
4
t 1

Table 5.4 calculates convexity for a 7.625 percent 5-year Treasury note
of 5/31/96. We calculate it to be 20.1036.
Estimating price using both modified duration and convexity requires
solving for

Pe Pd Pd (Dmod Y Convexity Y2/2)

Let us now use the formula above to estimate prices. Table 5.5 shows
how true versus estimated price differences are significantly reduced relative
to when we used duration alone. Incorporating derivatives of a higher order
beyond duration and convexity could reduce residual price differences
between true and estimated values even further.
Figure 5.3 provides a graphical representation of how much closer the
combination of duration and convexity can approximate a true present
value.
The figure highlights the difference between estimated price/yield rela-
tionships using modified duration alone and modified duration with con-

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180 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT

TABLE 5.4 Calculating Convexity
(A) (B) (C) (D) (E) (F)
Câ€™/(1 Y/2)t (B)2
Câ€™ t (B) (C) (C) (D)

3.8125 0.9344 3.6763 3.4352 3.2100 6.6452
3.8125 1.9344 3.5359 6.8399 13.2313 20.0712
3.8125 2.9344 3.4009 9.9796 29.2843 39.2638
3.8125 3.9344 3.2710 12.8694 50.6338 63.5033
3.8125 4.9344 3.1461 15.5240 76.6022 92.1263
3.8125 5.9344 3.0259 17.9571 106.5652 124.5223
3.8125 6.9344 2.9104 20.1817 139.9487 160.1304
3.8125 7.9344 2.7992 22.2102 176.2255 198.4357
3.8125 8.9344 2.6923 24.0544 214.9121 238.9665
103.8125 9.9344 70.5111 700.4868 6958.9345 7659.4031
Totals 98.9690 833.5384 7769.5475 8603.0678
Notes:
Câ€™ Cash flows over the life of the security. Since this Treasury has a coupon of
7.625%, semiannual coupons are equal to 7.625/2 3.8125.
t Time in days defined as the number of days the Treasury is held in a coupon
period divided by the number of days from the last coupon paid (or issue date) to
the next coupon payment. Since this Treasury was purchased 11 days after it was
issued, the first coupon is discounted with t 171/183 0.9344.
Câ€™/(1 Y/2) Present value of a cash flow.
Y Bond-equivalent yield; 7.941%.
Columns (A) through (D) are exactly the same as in Table 5.3 where we calculated
this Treasuryâ€™s duration. The summation of column (F) gives us the numerator for
our convexity formula. The denominator of our convexity formula is obtained by
calculating the product of column (C) and 4 (1 Y/2)2. Thus,
Convexity 8603.0678 / (98.9690 4 (1 0.039705)2)
20.1036

vexity; it helps to show that convexity is a desirable property. Convexity means
that prices fall by less than that implied by modified duration when yields
rise and that prices rise by more than that implied by modified duration when
yields fall. We return to the concepts of modified duration and convexity
later in this chapter when we discuss managing risk.

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