Having computed the value-at-risk, it is natural to ask to what extent the different positions contribute to it. For example, how much of the risk is due to the S&P 500 position, and how much to the FT-SE 100 position?
How does the S&P 500 futures hedge affect the risk? The process of answering such questions is termed risk decomposition.
At the beginning of this chapter, the portfolio was described as a cash position in the S&P 500, hedged with a position in the S&P 500 index futures contract and then overlaid with a FT-SE 100 futures contract to provide exposure to the U.K. market. This description suggests decompos- ing the risk by computing the VaRs of three portfolios: (i) the cash S&P 500 position; (ii) a portfolio consisting of the cash S&P 500 position, com- bined with the S&P futures hedge; and (iii) the aggregate portfolio of all three positions. The risk contribution of the cash S&P 500 position would be computed as the VaR of portfolio (i); the contribution of the S&P futures position would be the incremental VaR resulting from adding on the futures hedge, that is, the difference between the VaRs of portfolios (ii) and (i); and the risk contribution of the FT-SE 100 index futures position would be the difference between the VaRs of portfolios (iii) and (ii).
However, equally natural descriptions of the portfolio list the positions in different orders. For example, one might think of the portfolio as a cash position in the S&P 500 (portfolio i), overlaid with a FT-SE 100 futures contract to provide exposure to the U.K. market (portfolio iv), and then hedged with a position in the S&P 500 index futures contract (portfolio iii).
In this case, one might measure the risk contribution of the FT-SE 100 index futures position as the difference between the VaRs of portfolios (iv) and (i), and the contribution of the S&P futures position is the difference between the VaRs of portfolios (iii) and (iv). Unfortunately, different order- ings of positions will produce different measures of their risk contributions, a limitation of the incremental risk decomposition. For example, risk decomposition based on the second ordering of the positions would indi- cate a greater risk-reducing effect for the short S&P 500 futures position, because it is considered after the FT-SE 100 overlay, as a result of which there is more risk to reduce. In fact, different starting points can yield
extreme differences in the risk contributions. If one thinks of the portfolio as a short S&P 500 futures position, hedged with the cash S&P 500 posi- tion, and then overlaid with the FT-SE 100 futures position, the risk contri- butions of the S&P cash and futures positions will change sign.
This dependence of the risk contributions on the ordering of the posi- tions is problematic, because for most portfolios there is no natural order- ing. Even for this simple example, it is unclear whether the S&P futures position should be interpreted as hedging the cash position or vice versa and whether one should measure the risk contribution of the FT-SE 100 futures overlay before or after measuring the risk contribution of the S&P hedge.
(Or one could think of the S&P positions as overlays on a core FT-SE 100 position, in which case one would obtain yet another risk decomposition.) A further feature is that each position’s risk contribution measures the incre- mental effect of the entire position, not the marginal effect of changing it.
Thus, the incremental risk contributions do not indicate the effects of mar- ginal changes in the position sizes; for example, a negative risk contribution for the cash S&P 500 does not mean that increasing the position will reduce the VaR. These problems limit the utility of this incremental decomposition.
Marginal risk decomposition overcomes these problems. The starting point in marginal risk decomposition is the expression for the value-at-risk,
where the second equality uses the expressions for the expected value and stan- dard deviation of ⌬V. To carry out the marginal risk decomposition, it is neces- sary to disaggregate the S&P 500 position of X1 54.357 million into its two components, cash and futures; here 110 million dollars and
55.643 million dollars are used to denote these two components, so that X1 . Also, it is necessary to recognize that the dividend D depends on the magnitude of the cash position, D (0.014 12). Using this expres- sion and letting X ( , , X2)′ represent the portfolio, one obtains
From this formula one can see that VaR has the property that, if one multi- plies each position by a constant k, that is, if one considers the portfolio
VaR=–(E[∆V]–1.645×s.d.[∆V])
E X[ 11+X22+D]–1.645 X1212+2X1X212+X2222
,
–
=
=
X1c = X1f =
–
= X1c +X1f
= X1c ⁄
= X1c X1f
VaR( )X = –E X[ 1c(1+0.014 12⁄ )+X1f1+X22] 1.645 X1212+X1X212+X2222
– .
kX (k , k , kX2)′, the value-at-risk is multiplied by k. Carrying out this computation, the value-at-risk is
As we will see in chapter 10, this property of value-at-risk implies that it can be decomposed as
(2.1)
This is known as the marginal risk decomposition. Each of the three terms on the right-hand side is called the risk contribution of one of the positions, for example, the term (∂VaR ∂ ) is the risk contribution of the cash S&P 500 position. The partial derivative (∂VaR ∂ ) gives the effect on risk of increasing by one unit; changing by a small amount from to *, changes the risk by approximately (∂VaR ∂ )( * ). The risk contri- bution (∂VaR ∂ ) can then be interpreted as measuring the effect of per- centage changes in the position size . The change from to * is a percentage change of ( * ) , and the change in value-at-risk result- ing from this change in the position size is approximated by
the product of the risk contribution and the percentage change in the posi- tion. The second and third terms, (∂VaR ∂ ) and (∂VaR ∂X2)X2, of course, have similar interpretations.
A key feature of the risk contributions is that they sum to the portfolio risk, permitting the portfolio risk to be decomposed into the risk contributions of the three positions , , and X2. Alternatively, if one divides both sides of (2.1) by the value-at-risk VaR(X), then the percentage risk contri- butions of the form [(∂VaR ∂ ) ] VaR(X) sum to one, or 100%.
= X1c X1f
VaR(kX) = –
(
[kX1c(1+0.014 12⁄ )+kX1f1+kX22] 1.645 k2X1212+k2X1X212+k2X2222)
–
k
(
[X1c(1+0.014 12⁄ )]+X1f1+X22] –=
1.645 X1212+X1X212+X2222
–
)
kVaR( )X .
=
VaR( )X ∂VaR
∂X1c
---X1c ∂VaR
∂X1f
---X1f ∂VaR
∂X2 ---X2.
+ +
=
⁄ X1c X1c
⁄ X1c
X1c X1c X1c X1c
⁄ X1c X1c – X1c
⁄ X1c X1c
X1c X1c X1c X1c – X1c ⁄X1c
∂VaR
∂X1c
---(X1c*–X1c) ∂VaR
∂X1c
---X1c (X1c*–X1c) X1c ---
× ,
=
⁄ X1f X1f ⁄
X1c X1f
⁄ X1c X1c ⁄
Computing each of the risk contributions, one obtains
(2.2)
The first term on the right-hand side of each equation reflects the effect of changes in the position size on the mean change in value and carries a neg- ative sign, because increases in the mean reduce the value-at-risk. The sec- ond term on the right-hand side of each equation reflects the effect of changes in the position on the standard deviation. The numerator of each of these terms is the covariance of the change in value of a position with the change in value of the portfolio; for example, the term (X1 X212) (X1 X2 12) is the covariance of changes in the value of the cash S&P 500 position with changes in the portfolio value.
This captures a standard intuition in portfolio theory, namely, that the con- tribution of a security or other instrument to the risk of a portfolio depends on that security’s covariance with changes in the value of the portfolio.
Table 2.1 shows the marginal risk contributions of the form (∂VaR
∂ ) and the percentage risk contributions of the form (∂VaR ∂ ) VaR(X), computed using equations (2.2) and the parameters used earlier in this chapter. The S&P 500 cash position makes the largest risk contribution of 8.564 million, or 106% of the portfolio risk, for two reasons. First, the TABLE 2.1 Marginal risk contributions of cash S&P 500 position, S&P 500 futures, and FT-SE 100 futures
Portfolio
Marginal Value-at-Risk ($ million)
Marginal Value-at-Risk (Percent)
Cash Position in S&P 500 8.564 106
S&P 500 Futures −4.397 −54
FT-SE 100 Futures 3.908 48
Total 8.075 100
∂VaR
∂X1c
---X1c = –(1+0.014 12⁄ )X1c
1.645 (X112+X212)X1c X1212+2X1X212+X2222 ---, +
∂VaR
∂X1f
---X1f –1X1f 1.645 (X112+X212)X1f
X1212+2X1X212+X2222 ---, +
=
∂VaR
∂X2
---X2 –2X2 1.645 (X112+X222)X2
X1212+2X1X212+X2222 ---.
+
=
12+ X1c = X1c12+ X1c
⁄
X1c X1c ⁄ X1c X1c⁄
position is large and volatile; second, it is highly correlated with the total portfolio, because the net position in the S&P 500 index is positive, and because this position is positively correlated with the FT-SE 100 index futures position. The risk contribution of the short S&P futures position is negative because it is negatively correlated with the total portfolio, both because the net position in the S&P 500 index is positive and because the short S&P futures position is negatively correlated with the FT-SE 100 index futures position. Finally, the FT-SE 100 index futures position is positively correlated with the portfolio return, leading to a positive risk contribution.
In interpreting the risk decomposition, it is crucial to keep in mind that it is a marginal analysis. For example, a small change in the FT-SE 100 futures posi- tion, from X2 48.319 to X2* 49.319, changes the risk by approximately
million dollars, or from $8.075 million to approximately $8.156 million. This matches the exact calculation of the change in the value-at-risk to four signifi- cant figures. However, the marginal effects cannot be extrapolated to large changes, because the partial derivatives change as the position sizes change.
This occurs because a large change in a position changes the correlation between the portfolio and that position; as the magnitude of a position increases, that position constitutes a larger part of the portfolio, and the corre- lation between the position and the portfolio increases. This affects the value- at-risk through the numerators of the second term on the right-hand side of each of the equations (2.2). Thus, the risk contribution of a position increases as the size of the position is increased. For this reason, the marginal risk contri- butions do not indicate the effect of completely eliminating a position.