The value-at-risk calculation carried out above sidesteps the issue of exchange rate risk. Do the parameters 2, 2, 12 and apply to the local cur- rency (i.e., pound-denominated) percentage changes in the FT-SE 100 index, or the dollar-denominated changes? Given that the FT-SE 100 futures con- tract was mapped to a cash position in the portfolio underlying the index, the correct approach seems to be to interpret them as applying to the dollar- denominated returns. If the parameters apply to the pound-denominated returns, the VaR calculation above would not capture the exchange rate risk of the position in the U.K. market.
While the preceding calculation is simple, a drawback of organizing the calculations in terms of the dollar-denominated returns is that the expected returns and covariance matrix depend upon the perspective or base currency of the person or organization performing the calculations. For example, a U.S. investor or organization would use the expected dollar-denominated returns and the covariance matrix of dollar-denominated returns, while a U.K. investor would use a set of parameters describing the distribution of pound-denominated returns. Unfortunately the expected returns and covari- ance matrices are generally different, depending on the base currency. This is inconvenient for companies or other organizations in which different sub- sidiaries or other units use different base currencies, as well as for software vendors whose customers do not all use the same base currency.
More importantly, while the calculation carried out above illustrates the main idea of mapping, it is incorrect. Even though the FT-SE 100 futures and options have the U.K. equity market risk of £10.166 million invested in a portfolio that tracks the FT-SE 100 index, there has not been a cash investment of £10.166 million. Rather, the FT-SE 100 index futures contract has a value of zero at the end of each day, so only the value of the written position in the FT-SE 100 index options is exposed to exchange rate risk. The procedure of mapping the futures and options positions to an investment of £10.166 million in a portfolio that tracks the FT-SE 100 index incorrectly treats this entire amount as if it were exposed to exchange rate risk.
=
– = –
These two difficulties can be overcome by considering the exchange rate to be a separate market factor or source of risk and interpreting the portfolio as having three risk factors, S1, S2, and e, where the new risk factor e is the exchange rate expressed in terms of dollars per pound. To do this, we must be careful about how the exchange rate affects the value of the portfolio.
First, the exchange rate does not affect the value of the positions in the S&P 500 index, so the portfolio delta with respect to the S&P 500 index remains 4863.7 dollars and the positions based on the S&P 500 are mapped to a position consisting of an investment of $4863.7 1097.6
$5.338 million in a portfolio that tracks the S&P 500 index.
The positions based on the FT-SE 100 index are affected by the exchange rate, though in different ways, and we consider them separately.
These positions will be mapped using their deltas, or partial derivatives.
The FT-SE 100 index futures contracts affect the profit or loss on the portfolio through their daily resettlement payments. Even though the values of the futures contracts are zero at the end of each day, the daily resettlement payments must be included in the profit or loss because they are paid or received by the owner of the portfolio. To determine how the daily resettle- ment payment is affected by changes in S2 and e, we use the cost-of-carry for- mula F(S2,t) S2 exp[(r2 d2)(T2 t)], where T2 is the final settlement date of the FT-SE 100 index futures contract, t is the current date, r2 is the £ inter- est rate, and d2 is the dividend yield on the portfolio underlying the FT-SE 100 index. Because the futures contract has value zero following each reset- tlement payment, one need only consider the effects of changes in the market factors on the value of the first daily resettlement payment. This is given by
£500(10){S2exp[(r2 d2)(T2 t)] 5862.3exp[(r2 d2)(T2 t0)]}, where t0 is the initial time (i.e., the end of the previous day) and 10 is the multiplier for the FT-SE contract. In dollar terms, it is
Computing the deltas, or partial derivatives
× =
= – –
– – – – –
dollar value of daily resettlement payment
= 500 10( )e{S2 exp[(r2–d2)(T2–t)] 5862.3 exp[(r2–d2)(T2–t0)]}
–
∂ dollar value of daily resettlement payments
⁄
∂S2and
and then evaluating them at the initial values S1 1097.6, S2 5862.3, e 1.6271, and t0, we obtain
(3.1)
The exchange rate delta of the futures contract is zero because when S2 5862.3 and t t0 the value of the daily resettlement payment is zero, regardless of the exchange rate.
The written position of 600 FT-SE 100 index call options has a pound value of 600C2(S2, t) and a dollar value of 600eC2(S2, t), where the func- tion C2 gives the value (including the effect of the multiplier) of the FT-SE 100 index call option as a function of the index level and time. The partial derivatives are
(3.2)
where again the derivatives are evaluated at the initial values S1 1097.6, S2 5862.3, e 16271, and t t0.
Combining the deltas of the FT-SE 100 futures and options positions, we have
(3.3)
∂ dollar value of daily resettlement payments
⁄
∂e= =
=
∂ dollar value of daily resettlement payment
⁄
∂S2 = 5065.6e = 8242.3,∂ dollar value of daily resettlement payment
⁄
∂e = 0.= =
– –
∂(–600eC2(S2,t))
∂S2
--- –600e∂C2(S2,t)
∂S2
--- –3331.6e –5420.8
= = =
∂(–600eC2(S2,t))
---∂e = –600C S( 2,t) = –2,127,725
=
= = =
∂V∂S1
--- = 4863.7,
∂V∂S2
--- = 8242.3–5420.8 = 2821.5,
∂V
∂e
--- = 0–2,127,725 = –2,127,725,
where for ∂V ∂S2 and ∂V ∂e the right-hand sides consist of the sums of the futures and options deltas. The interpretation of these deltas is that one- unit changes in S1, S2, and e result in changes of 4863.7, 2,821.5, and 2,127,725 in the dollar value of the portfolio, respectively. Thus, the change in the dollar value of the portfolio can be approximated as
change in V 4863.7(change in S1) 2821.5(change in S2) 2,127,725(change in e).
Writing the changes in the risk factors in percentage terms,
(3.4)
or
(3.5)
The upshot of this analysis is that the portfolio is mapped to X1 4863.7 1097.6 5.338 million dollars exposed to the risk of percentage changes in the S&P 500, X2 2821.5 5862.3 16.541 million dollars exposed to the risk of percentage changes in the FT-SE 100, and X3 21,27,725 1.6271 3.462 million dollars exposed to the risk of percentage changes in the exchange rate.
Examining the partial derivatives in (3.1) and (3.2) and following the role they play in equations (3.3) through (3.5), one can see that the written position in the FT-SE 100 index call option with value $600eC2(S2,t) is exposed to the risk of changes in both S2 and e. As a result, it contributes to both of the last two terms on the right-hand sides of (3.4) and (3.5). In contrast, the futures contract is not exposed to the risk of changes in e and does not contribute to the last term on the right-hand side of (3.4) and (3.5). Thus, this mapping cap- tures the fact that the position in the FT-SE 100 index futures contract is not exposed to the risk of changes in the $/£ exchange rate.
⁄ ⁄
–
≈ +
–
change in V 4863.7S1 change in S1 S1 ---
2821.5S2 change in S2 S2 ---
≈ +
2,127,725
– e change in e
---e
change in V 5,338,445 change in S1 1097.6 ---
16,540,531 change in S2 5862.3 ---
≈ +
3,462,022 change in e 1.6271 ---
.
–
=
× =
= × =
= – × = –
–
The mapping amounts to replacing the portfolio with a linear approximation
(3.6)
The constant on the right-hand side is chosen so that, when S1 1097.6, S2 5862.3, and e 1.6271, the right-hand side equals the actual initial value of the actual portfolio, $101,485,220. This linear approximation is shown in Figures 3.4 and 3.5. Figure 3.4 shows the value of the portfolio as a function of the level of the S&P 500 index, holding the FT-SE 100 index and exchange rate fixed at their current levels of 5862.3 and 1.6271, along with the linear approximation of the value of the portfolio. Figure 3.5 shows the value of the portfolio and the linear approximation as a function of the level of the FT-SE 100 index, holding fixed the S&P 500 index and the exchange rate.
FIGURE 3.4 The current value of the portfolio as a function of the level of the S&P 500 index and the linear approximation used in computing delta-normal value-at-risk
V 5,338,445 S1 1097.6 ---
16,540,531 S2
5862.3 ---
+
=
3,462,022 e 1.6271 ---
+83,068,266 –
4863.7S1+2821.5S2–2,127,725e+83,068,266.
=
=
= =
98,000,000 98,500,000 99,000,000 99,500,000 100,000,000 100,500,000 101,000,000 101,500,000 102,000,000 102,500,000
987.8998.81009.81020.81031.71042.71053.71064.71075.61086.61097.61108.61119.61130.51141.51152.51163.51174.41185.41196.41207.4 S&P 500
Portfolio value
Using equation (3.3), the expected change in the value of the portfolio is E[⌬V] X11 X22 X33 D,
where 1 0.01, 2 0.0125, and 3 0 are the expected percentage changes in the three risk factors. The variance of monthly changes in port- folio value is given by the formula
where 1 0.061 is the standard deviation of monthly percentage changes in the S&P 500 index, 2 0.065 is the standard deviation of monthly per- centage changes in the FT-SE 100 index, 3 0.029 is the standard devia- tion of monthly percentage changes in the exchange rate, and 12 0.55,
13 0.05, and 23 0.30 are the correlation coefficients. Using these parameters and the mapping X1 5.338 million, X2 16.541 million, FIGURE 3.5 The current value of the portfolio as a function of the level of the FT-SE 100 index and the linear approximation used in computing delta-normal value-at-risk
96,000,000 97,000,000 98,000,000 99,000,000 100,000,000 101,000,000 102,000,000 103,000,000 104,000,000
5276.15334.75393.35451.95510.65569.25627.85686.45745.15803.75862.35920.95979.56038.26096.86155.46214.06272.76331.36389.96448.5 FT-SE 100
Portfolio value
= + + +
= = =
var[⌬V] = X1212+X2222+X3232+2X1X21212
2X1X31313 2X2X32323,
+ +
=
=
=
=
= = –
= =
and X3 3.462 million, the expected value and standard deviation of the change in value of the portfolio are
The standard deviation is, of course, simply the square root of the variance, or $1.311 million, and the value-at-risk is
As a fraction of the initial value of the portfolio,
or 1.74% of the initial value of the portfolio.
Alternatively, when the time horizon of the value-at-risk estimate is one day, it is common to assume that the expected change in the portfolio value is zero. If this assumption is made, the value-at-risk is then $1.645 mil- lion(1.311) $2.157 million, or 1.645(0.0129) 0.0213 or 2.13% of the value of the portfolio.