ñòð. 37 |

(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)

TLFeBOOK

177

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

TLFeBOOK

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

TLFeBOOK

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-

TLFeBOOK

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.

TLFeBOOK

ñòð. 37 |