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4

Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS regression beta: $$\beta = cov(r_i,r_i^M)/var(r_i^M).$$ Then if you write $r_i = \alpha + \beta r_i^M + \epsilon_i$ on the other hand $$\epsilon_i = r_i - ( ... 0 If R and r are the return on the portfolio after currency hedging and on the currency, if I write V(\cdot) for variance, and a fraction of t of the portfolio is exposed to currency risk, then the return of the unhedged portfolio is R+tr. Then:$$V(R+tr) = V(R) + 2t\mathrm{Cov}(R,r) + t^2 V(r)$$so the marginal contribution (derivative with respect ... 1 Maybe. Certainly you shouldn't use their realized return ("past return") because that does not reflect expectations, it reflects events that became known after the client decided on their asset allocation. On the other hand: with a lot of (unrealistic?) assumptions, you CAN discern the client's risk aversion from their allocation. Suppose for example that ... 1 There is no solution. If w is a solution to the original problem, then consider aw with a>1$$\beta_i(aw) = a(\beta_i w) = 0$$and$$(aw)^T\Sigma(aw) = a^2 (w^T\Sigma w) > w^T\Sigma w so the original solution $w$ was not a maximum.

2

Each of these can be used, but each has serious drawbacks. No. 1 is inaccurate unless you use $N>>10$ years of data. But decades of data may not be available or may no longer be relevant to today's economy. No. 2 is good except that the CAPM has been rejected by empirical tests. More advanced models from Asset Pricing Theory may be helpful (FF3, FF5, ...

0

If a USD based investor owns shares of Toyota Motor in Japan, the variance of USD based returns is approximately equal to the variance of Toyota in yen, plus the variance of USDJPY plus twice the covariance between Toyota and the exchange rate. The last term could be positive or negative; if I had to guess for a big exporter like Toyota it is probably ...

2

How about letting the FX rates remain fixed, and recalculate the portfolio volatility. That seems very obvious - am i missing something?

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