International Finance and Accounting Handbook

$1,631,483.^7 Today’s yield on these bonds is 7.243% per annum. These bonds are
held as part of the trading portfolio. Thus,

The FI manager wants to know the potential exposure the FI faces should interest
rates move against the FI due to an adverse or reasonably bad market move the next
day. How much the FI will lose depends on the bond’s price volatility. We know that:

(3)

The modified duration (MD) of this bond is:^8

given that the yield on the bond is R= 7.243%. To estimate price volatility, multiply
the bond’sMDby the expected adverse daily yield move.
Suppose we define “bad” yield changes such that there is only a 5% chance that
the yield changes will exceed this amount in either direction—or, since we are con-
cerned only with bad outcomes, and we are long in bonds, that there is 1 chance in
20 (or a 5% chance) that the next day’s yield increase (or shock) will exceed this
given adverse move.
If we assume that yield changes are normally distributed,^9 we can fit a normal dis-
tribution to the histogram of recent past changes in seven-year zero-coupon interest
rates (yields) to get an estimate of the size of this adverse rate move. From statistics,
we know that 90% of the area under the normal distribution is to be found within
±1.65 standard deviations () from the mean—that is, 1.65. Suppose that during the
last year the mean change in daily yields on seven-year zero-coupon bonds was 0%^10

MD

D

1 R

7

1 1.07243 2

6.527

 1 MD 2  1 Adverse daily yield move 2

 1 Adverse daily yield move 2

Daily price volatility
 1 Price sensitivity to a small change in yield 2

Dollar market value of position
$1 million

8 • 6 MARKET RISK

(^7) The face value of the bonds is $1,631,483—that is, $1,631,483/(1.07243) (^7) = $1,000,000 market
value. In the original model prices were determined using a discrete rate of return, Rj. In the 2001 docu-
ment, “Return to RiskMetrics: The Evolution of a Standard,” April 2001, prices are determined using a
continuously compounded return, e–rf. The change was implemented because continuous compounding
has properties that facilitates mathematical treatment. For example, the logarithmic return on a zero-
coupon bond equals the difference of interest rates multiplied by the maturity of the bond. That is:
wherer ̃ is the expected return.
(^8) Assuming annual compounding for simplicity.
(^9) In reality, many asset return distributions—such as exchange rates and interest rates—have “fat
tails.” Thus, the normal distribution will tend to underestimate extreme outcomes. This is a major criti-
cism of the RiskMetrics modeling approach. (See later footnote and references.)
(^10) If the mean were nonzero (e.g., –1 basis point), this could be added to the 16.5 bp to project the
loga
er
~t
ert
b
 1 r~p 2 t