reduced by increasing the short position in the French market. The failure of the manager to do this indicates a positive implied view. The same analy- sis applies to Great Britain: here the neutral position does not indicate a neutral view, because the best hedge trade in the British market is to estab- lish a short position of −10.1%. Given this, the zero weight in the British market indicates a positive implied view.
In general, the implied views of a portfolio depend on all the other positions and the correlations among them. This is precisely because of the relationship between the implied views and the risk contributions: the implied views depend on all of the other positions in exactly the same way the risk contributions do.
The expected return contribution of the ith position is then just the product of the expected return of the ith market and the position size, or E[ri]wi.
Table 11.6 includes the position sizes, risk sensitivities, and risk contri- butions from Table 11.3, as well as the market expected returns and expected return contributions. In addition, the rightmost column shows the ratio of the expected return contribution to the risk contributions. When these ratios are not all equal, the risk-return tradeoff can by improved by reallocating assets among the markets. For example, the ratio of expected return to risk is 1.090 for the Canadian market and 0.618 for Switzerland. The difference indicates that there is an opportunity to improve the risk-return tradeoff by reallocat- ing assets from the Swiss market to the Canadian market.
TABLE 11.6 Risk decomposition and contributions to expected return of the current portfolio
Portfolio Weight
(%)
Partial Derivative
Contribution to Risk (% per month)
Market Expected
Return (% per month)
Contribution to Portfolio
Expected Return (% per month)
Ratio of Return to Risk Australia (SPI) −15.0 −0.0029 0.043 0.90 −0.1 −3.148 Canada
(TSE 300) 30.0 0.0220 0.661 2.40 0.7 1.090
Switzerland
(SMI) 40.0 0.0372 1.488 2.30 0.9 0.618
Germany
(DAX-30) −20.0 −0.0028 0.056 1.50 −0.3 −5.368
Spain (IBEX 35) −10.0 0.0021 −0.021 1.20 −0.1 5.734 France
(CAC-40) −5.0 0.0081 −0.041 0.70 0.0 0.859
Great Britain
(FT-SE 100) 0.0 0.0080 0.000 0.40 0.0 0.501
Italy (MIB 30) 15.0 0.0285 0.428 2.50 0.4 0.877
Japan
(Nikkei 225) −20.0 −0.0152 0.303 0.40 −0.1 −0.264 Netherlands
(AEX) 15.0 0.0195 0.293 1.80 0.3 0.921
New Zealand
(NZSE) 0.0 0.0066 0.000 1.60 0.0 2.427
United States
(S&P 500) −25.0 −0.0002 0.006 0.10 0.0 −4.461
Total 3.215 1.59
The negative ratios for some of the short positions reveal even greater opportunities for improving the risk-return tradeoff. To interpret them, one must know whether they are negative because increasing the position reduces risk and increases expected return or increases risk and reduces expected return. For the short positions with negative ratios, it turns out that increasing the magnitude of the short position increases risk and reduces expected return, so the risk-return tradeoff is improved by reducing the magnitudes of the short positions. For example, the ratio of −3.148 for the position in the Australian market reflects the fact that reducing the short position from 15%
to 14% (changing w1 from −15% to −14%) reduces risk by approximately 0.043% and increases the expected return by 1% of 0.90%, or 0.009%.
From this discussion, it should be clear that the optimal active portfolio of futures contracts is characterized by equal ratios. (An important excep- tion is when constraints, perhaps self-imposed, on the position size in any market lead to smaller than optimal allocations for some markets.) The actual optimum can be computed using a mean-variance optimizer; in this example, with no constraints on the portfolio, the solution is easy to obtain. Table 11.7 shows the optimal portfolio (maximum expected return), subject to the constraint that the VaR/tracking error volatility be less than or equal to 2.887% per month (10% per year). This Table also shows the risk sensitivities (partial derivatives), risk contributions, expected returns, expected return contributions, and ratios of expected return to risk at the optimal allocations. As expected, the ratios are equal across markets.
Consistent with the preceding discussion of ratios of the expected return to the risk contributions in the Canadian and Swiss markets, the optimal allocation reflects a smaller position in the Swiss market and a larger position in the Canadian market; in fact, the largest position is now in the Canadian market. Also, the optimal short positions in the Spanish and French markets are larger than in the current portfolio, reflecting the fact that, given the current positions, they serve as hedges. The short posi- tion in Australia of −15% has been reduced, and in fact turns into a long position of 2.1%. All of these positions are unsurprising in light of the expected returns and risk decomposition of the existing portfolio in Table 11.6.
The change in the position in the British market can be understood in terms of the best hedge and implied views analyses. The best hedge trade in the British market is to establish a short position of −10.1%; the existing zero weight in the British market indicates an implied view greater than the expected return. In order to be consistent with the expected return forecast, the position in the British market must be decreased.
The reasons for the changes in the weights in the U.S. and New Zealand markets are not obvious from the previous discussion. The best hedges anal- ysis indicated little change in the U.S. market position and that the initial zero position in New Zealand should be changed to a short position. Why does the optimal portfolio reflect an increase in the short position in the United States and a significant long position in New Zealand? The explana- tion lies in the changes in the positions in the other markets. The changes in the portfolio, and in particular the increase in the size of the Canadian posi- tion, make the position in the U.S. market a better hedge of the portfolio, leading to the new optimal U.S. position of w12 −32.4%. This, combined with the optimal large short position in Great Britain, serves to hedge the TABLE 11.7 Risk decomposition and contributions to expected return of the optimal portfolio
Portfolio Weight
(%)
Risk Sensitivity
Risk Contribution (% per month)
Market Expected
Return (% per month)
Contribution to Portfolio
Expected Return (% per month)
Ratio of Return to Risk
Australia (SPI) 2.1 0.01395 0.030 0.90 0.02 0.645
Canada (TSE 300) 35.4 0.03719 1.317 2.40 0.85 0.645 Switzerland
(SMI) 24.2 0.03565 0.862 2.30 0.56 0.645
Germany
(DAX-30) −1.5 0.02325 −0.035 1.50 −0.02 0.645
Spain (IBEX 35) −4.4 0.01860 −0.082 1.20 −0.05 0.645 France
(CAC-40) −13.0 0.01085 −0.141 0.70 −0.09 0.645
Great Britain
(FT-SE 100) −16.9 0.00620 −0.105 0.40 −0.07 0.645
Italy (MIB 30) 11.7 0.03874 0.454 2.50 0.29 0.645
Japan
(Nikkei 225) −6.0 0.00620 −0.037 0.40 −0.02 0.645 Netherlands
(AEX) 14.2 0.02790 0.397 1.80 0.26 0.645
New Zealand
(NZSE) 11.2 0.02480 0.277 1.60 0.18 0.645
United States
(S&P 500) −32.4 0.00155 −0.050 0.10 −0.03 0.645
Total 2.887 1.86
=
New Zealand market. Thus, it is possible to have a significant long position in New Zealand without contributing too greatly to the risk of the portfolio.
NOTES
The implied views analysis is described in Winkelman (2000a; 2000b). This analysis is closely related to mean-variance optimization, the literature on which is too large to attempt to survey here.
12
183
Aggregating and Decomposing the Risks of Large Portfolios
A key feature of MPT’s portfolio discussed in the preceding chapter is that it consists of positions in only 12 instruments, the stock index futures contracts.
For this reason, we were able to work directly with the covariance matrix of changes in the values of the individual instruments. Unfortunately, the MPT portfolio contains too few instruments to be representative of most actual insti- tutional portfolios, which contain hundreds, or even thousands, of instruments.
In particular, this is true of the aggregate portfolios of plan sponsors, which are composed of the sums of the different portfolios controlled by the sponsors’
managers. For such large portfolios, factor models of the sort discussed in Chapters 7 and 8 play a crucial role in simplifying and estimating the risk.
Factor models play a background role in simple value-at-risk computa- tions, in that they are only a means to an end. Typically, the user of simple value-at-risk estimates has little interest in the factor models per se, provided that the resulting VaR estimate is reasonably accurate. In contrast, factor models play an important role in the foreground of risk decomposition. To understand why, imagine taking a portfolio with approximately equal weights in 1000 common stocks and decomposing the portfolio risk into the risk contributions of the 1000 stocks, along the lines of the previous chapter.
With so many stocks, the risk contribution of each common stock will be somewhere between zero and a few tenths of one percent of the total portfo- lio risk. Such risk decompositions reveal little of the important risks in the portfolio. More meaningful risk decompositions are in terms of industry or other groupings, or in terms of the market factors to which the portfolio is exposed. In this latter case, the factor models play a highly visible role because the risk contributions of market risk factors that do not appear in the factor model cannot be measured. Thus, the choice of factors deter- mines the possible risk decompositions of the total portfolio risk.
This chapter describes risk decomposition by both groups of securities and factors. The risk decomposition by groups of securities allows the identification
of undesirable country, regional, or industry concentrations, for example, an unintended overweighting in technology stocks. The risk decomposition by groups of securities also allows the measurement of the risk contribution of particular portfolio managers, because the risk contribution of a manager’s portfolio is just the risk contribution of the group of securities that constitute