Recalling that the current tracking error volatility of 3.215% per month (11.14% per year) exceeds the target of 2.887% per month (10% per year), MPT needs to reduce the risk of the portfolio. The risk decomposition above identifies obvious candidates for trades to reduce the risk. These may be either reductions in the positions that contribute to the risk or increases in the sizes of positions that function as hedges. The decomposition also allows MPT to compute the effect of changing the positions on the portfolio risk.
The position in the Swiss market has the largest risk contribution and is an obvious candidate for a trade to reduce the risk. The effect of reducing the posi- tion in the Swiss market from 40% to 35% can be obtained from the equation
yielding a new volatility of approximately 3.215%−0.186% 3.029%, or 10.49% per year. The actual change in risk resulting from this change in the position is −0.174, yielding a new volatility of 3.041% per month. The differ- ence is due to the fact that the risk decomposition is a marginal analysis and is only exactly correct for infinitesimally small changes in the positions. In this example, the exact decrease in risk of 0.174 is smaller than the approximate decrease in risk of 0.186 because the covariance between the Swiss market and the portfolio declines as w3 declines, so that reductions in w3 have decreasing effect on the risk.
This analysis can be reversed to (approximately) determine the trade necessary to have a desired effect on the volatility. For example, to determine the (approximate) change in the Swiss market position necessary to reduce the volatility by 30 basis points, solve the equation
The solution of ⌬w3 8.06% implies that a reduction in the weight in the Swiss market from 40% to about 32% would reduce the monthly volatility by about 30 basis points.
change in risk ∂( )w
∂wi
---wi (wi*–wi)
wi ---
×
≈
risk contribution
( )×(proportional change in volatility)
=
1.488% 35% 40%– ---40%
×
=
0.186%, –
=
=
0.3%
– = (risk contribution)×(proportional change in position) 1.488 ⌬w3
40%---.
×
=
=
The positions in the Spanish and French markets both serve as hedges.
The approximate effect of increasing the short position in the French mar- ket from 5% to 6% is
yielding a new volatility of approximately 3.215%−0.008%=3.207%.
The exact change in volatility is 0.007%, indicating that the estimate of 0.008% is a good approximation of the effect of increasing the hedge (the short position) from w6 −5 to−6%.
The risk decomposition can also be used to compute risk-minimizing trades, or best hedges, for each of the markets. We illustrate this using the position in the Swiss market. Letting ⌬w3 denote the change in the position in the Swiss market, the risk-minimizing trade is the ⌬w3 that minimizes the portfolio variance, or the solution of
.
The first-order condition for the minimum is
which implies that the risk-minimizing trade is
The numerator of the term on the right-hand side, wicov(ri, r3), is the covariance between the return r3 and the active portfolio return wiri, so the right-hand side is the negative of the ratio of the covariance between the
change in risk ∂( )w
∂wi
---wi (wi*–wi)
wi ---
×
≈
risk contribution
( )×(proportional change in volatility)
=
0.041%
– –6%–( )%–5 5%
---–
×
=
0.008%, –
=
=
min⌬w3
wiwjcov(ri,rj) (⌬w3)2var( )r3 ⌬w3wi cov (ri,r3) i=1
∑
12+ +
j=1
∑
12 i=1∑
122⌬w3 var( )r3 2 wi cov(ri,r3) i=1
∑
12+ = 0,
⌬w3
wi cov(ri,r3) i=1
∑
12var( )r3 ---
=
Σ
i=112Σ
i=112portfolio return and r3 to the variance of r3. This implies that the right-hand side can be interpreted as the negative of the regression coefficient, or beta, obtained by regressing the portfolio return on r3. This result is intuitive, for it says that the risk-minimizing trade in the Swiss market is the trade that offsets or hedges the sensitivity of the portfolio return to the Swiss market return. For MPT’s portfolio, the risk-minimizing trade is ⌬w3 −28.7%. If this trade is made, the new position in the Swiss market of 40−28.7 11.3% will be uncorrelated with the returns of the resulting portfolio.
Table 11.4 shows the best hedges for each of the markets, along with the volatility of the resulting portfolio and the percentage reduction in vol- atility. Not surprisingly, the risk-minimizing trades are largest for the Swiss and Canadian markets and smallest for the U.S. market. These are consis- tent with the risk contributions of the portfolio; in particular, the position in the U.S. market was already very nearly a best-hedge position, so the risk-minimizing trade is nearly zero. Also as expected, the best hedges in TABLE 11.4 Risk-minimizing trades or best hedges
Volatility at the Best Hedge (% per month)
Reduction in Volatility (% of total risk)
Portfolio Weight
(%)
Trade Required to
Reach Best Hedge (Change in port. weight)
Australia (SPI) 3.210 0.17 −15.0 3.9
Canada (TSE 300) 2.987 7.09 30.0 −20.0
Switzerland (SMI) 2.628 18.26 40.0 −28.7
Germany (DAX-30) 3.213 0.07 −20.0 1.6
Spain (IBEX 35) 3.214 0.03 −10.0 −1.1
France (CAC-40) 3.193 0.70 −5.0 −5.5
Great Britain
(FT-SE 100) 3.174 1.27 0.0 −10.1
Italy (MIB 30) 3.041 5.42 15.0 −11.9
Japan (Nikkei 225) 3.120 2.95 −20.0 12.3
Netherlands (AEX) 3.056 4.96 15.0 −15.9
New Zealand
(NZSE) 3.196 0.59 0.0 −5.7
United States (S&P 500)
3.215 0.00 −25.0 0.3
=
=
the Spanish and French markets involve increasing the short positions.
More surprisingly, the risk-minimizing trade in the Netherlands market is larger than that in the Italian market, even though the risk contribution of the Italian market is larger. This occurs because the return on the Nether- lands market is more highly correlated with the large Canadian and Swiss market positions, so that its risk contribution decreases less rapidly as the position in the Netherlands market is decreased.
Although we do not show them, it is also possible to compute the risk- minimizing, or best hedge, trade involving two or more markets. If a par- ticular set of H markets has been selected, the risk-minimizing trade in each market is the multiple regression coefficient of the portfolio return on the returns of each of the H markets, and the volatility at the best hedge is the standard deviation of the residual from that regression. By searching over all possible sets of H markets, it is straightforward to compute the risk-minimizing trade involving this number of markets. The H markets that are identified as the best hedges are those that explain the variation in the value of the portfolio, so this analysis also reveals the portfolio’s princi- pal exposures.