IMPLIED VIEWS ANALYSIS

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.

The risk-return tradeoff (mean-variance frontier) can be obtained by maximizing the utility function

where the coefficient ␭ determines the extent to which the utility function penalizes risk. Letting ␭ vary between 0 and ⬁ maps out the usual efficient frontier. For example, as ␭→⬁, the utility function penalizes risk infinitely much, and the optimal portfolio obtained by maximizing the utility func- tion approaches the minimum variance portfolio, while as ␭→0, risk is not penalized and the optimal portfolio involves “plunging” into a levered position in the asset or market with the highest expected return, financed by a short position in the asset with the lowest expected return.

Using our notation, the maximization of the utility function becomes

The first-order conditions are

for i 1, 2, . . . , 12.

The expected returns on the left-hand side are the expected returns, or implied views, that are consistent with the positions (and covariances) on the right-hand side. That is, they are implied by the choice of the w_i. Recall- ing that w_jcov(r_i, r_j) is the covariance between the return on the ith market and the active portfolio return w_j r_j, this result says that the expected returns implicit in a choice of portfolio weights w_i are propor- tional to the covariances. Multiplying both sides of each equation by w_i and 1 in the form of ␴ ␴^,

U = portfolio expected return–␭(portfolio variance),

max

{ }wi w_iE r[ ]_i ␭ ^wiw_j cov(r_i,r_j) j=1

∑

12 i=1

∑

12 – i=1

∑

12 ^.

E r[ ]_i 2␭ ^w_j ^cov(^r_i^,r_j), j=1

∑

12

= =

Σ

^j=112

Σ

^j=112

⁄

w_iE r[ ]_i (2␭␴)

w_j cov (r_i,r_j) j=1

∑

12

---␴ ×w_i

= 2␭␴

( )×(risk contribution of ith position).

=

This says that the expected return contribution w_iE[r_i] is proportional to the risk contribution.

Table 11.5 shows these implied views for MPT’s current portfolio and the expected returns from MPT’s forecasting model, with ␭ ^chosen so that the equally weighted average of the implied views matches the equally weighted average of the expected returns. The most obvious fea- ture of the implied views is that they are generally more extreme than the expected returns from the forecasting model. This follows directly from the risk decomposition in Table 11.3. The dramatic differences in the risk contributions are rationalized only by the conclusion that the port- folio manager has extreme views about expected returns; otherwise the differences in the risk contributions are too large.

A closer examination of Table 11.5 reveals a more interesting feature of the implied views: they do not necessarily correspond to whether the posi- tions are long or short. More generally, the implied views of a portfolio do not necessarily correspond to the deviations from the benchmark.

To see this, consider the French market. The position of w₅ −5%

might be interpreted as indicating that the manager had a negative view on France, but the implied view for the French market is positive, at 1.16%

per month. How can this be? The answer lies back in the risk contributions and best hedge analysis, which indicate that the portfolio volatility can be TABLE 11.5 Implied views of the current portfolio

Implied View (% per month)

Expected Return (% per month)

Portfolio Weight

(%)

Australia (SPI) −0.41 0.90 −15.0

Canada (TSE 300) 3.14 2.40 30.0

Switzerland (SMI) 5.31 2.30 40.0

Germany (DAX-30) −0.40 1.50 −20.0

Spain (IBEX 35) 0.30 1.20 −10.0

France (CAC-40) 1.16 0.70 −5.0

Great Britain (FT-SE 100) 1.14 0.40 0.0

Italy (MIB 30) 4.07 2.50 15.0

Japan (Nikkei 225) −2.17 0.40 −20.0

Netherlands (AEX) 2.79 1.80 15.0

New Zealand (NZSE) 0.94 1.60 0.0

United States (S&P 500) −0.03 0.10 −25.0

=

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.