In Goetzmann et al.'s (2007) paper, the authors derive a "Manipulation-Proof Performance Measure" (MPPM), which is a performance measure that is impervious to performance manipulation by fund managers. The authors show that rankings based on MPPM are much more resistant to intentional performance manipulation relative to other popular measures such as the Sharpe ratio, Information ratio, Jensen's alpha etc. Specifically, the MPPM of a portfolio is given by:

$$\widehat{\Theta}(\rho) = \frac{1}{(1-\rho)\Delta t} \ln\left(\frac{1}{T} \sum_{t=1}^T \left(\frac{1+r_t}{1+rf_t}\right)^{1-\rho} \right) \ \ \ \ \cdots \ (1)$$


$T =$ total number of observations

$\Delta t =$ length of time between observations

$r_t =$ portfolio's annualized rate of return at time $t$

$rf_t =$ risk-free rate at time $t$

$\rho =$ can be interpreted as the coefficient of relative risk aversion

$\widehat{\Theta}$ can be interpreted as the portfolio's premium return after adjusting for risk, i.e, the portfolio has the same "score" (ranking) as a risk-free asset whose continuously compounded return exceeds the risk-free rate by $\widehat{\Theta}$. Also, note that $\widehat{\Theta}$ does not require any specific distribution for the portfolio returns, $r_t$.

Question: I am trying to derive the asymptotic properties of a modified version of this MPPM. More specifically, instead of assuming a value for $\rho$ and computing the corresponding MPPM (as was done by Goetzmann et al.), my approach is to set $\widehat{\Theta} = 0$ and compute the implied relative risk aversion coefficient (IRRAC), $\hat{\rho}$. Calculating IRRAC from Eqn.$(1)$ is relatively straightforward using Newton–Raphson method or other linear optimization methods. However, I am not sure where to begin to derive the large sample (asymptotic) properties of IRRAC. More formally, as $T \rightarrow \infty$, what can we say about $\hat{\rho}$ where $\hat{\rho}$ satisfies $\widehat{\Theta}(\hat{\rho}) = 0$?

Attempt: My attempt so far has been related to a 2nd order linear approximation to Eqn.$(1)$ rather than working directly with Eqn.$(1)$. Using Taylor series expansion and properties of the Generalized Mean, we can show that:

$$\widehat{\Theta}(\rho) \approx \frac{1}{\Delta t} \left[\overline{x} + \frac{1-\rho}{2} \left(s_x^*\right)^2\right] \ \ \ \ \ \cdots \ (2)$$


$\overline{x} = \frac{1}{T} \sum_{t=1}^T x_t$ where $x_t = r_t - rf_t$, i.e, the excess return of the portfolio at time $t$.

$\left(s_x^*\right)^2 = \left(\frac{T-1}{T}\right)s_x^2$ where $s_x^2 = \frac{1}{T-1}\sum_{t=1}^T (x_t - \overline{x})^2$, i.e, the sample variance of the excess return of the portfolio at time $t$.

From Eqn.$(2)$, an approximation to $\hat{\rho}$ is given by:

$$\hat{\rho} \approx \frac{2\overline{x}}{\left(s_x^*\right)^2} +1 $$

Can we somehow derive the asymptotic distribution of $\hat{\rho}$ from this approximation?



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