In this section we will introduce the estimator that is used for asset return moments. As already pointed out before, we are not interested in finding a particularly good estimator for the asset returns at hand. Ultimately, we rather want to check whether the diversification-aware approach is robust enough to be able to deal with sub-optimal moment estimates, too.
The only requirements that we have for the estimator hence are more of a practical nature.
First, it should not require any data additional to the historic returns of the respective assets.
Second, it should not use any particular knowledge with regards to asset class peculiarities, but simply treat all assets the same. And third, it should not require any complicated numerical optimization during estimation. For all these reasons, we will settle for the EWMA estimator proposed in (“RiskMetrics Technical Document” 1996), an estimator that is quite popular among practitioners.
We will now give a short introduction to the EWMA estimators. Thereby we will closely follow (“RiskMetrics Technical Document” 1996) for all definitions and derivations.
Definition 4.3.1: [EWMA mean estimator]
Let 0< λµ<1 denote a decay factor, and rt,i denote the return of asset i at time t. Then the exponentially weighted moving average model for the mean asset return µt,i is given by
µt,i =Xt
k=1
λk−1µ
Pt
k=1λk−1µ rt−k+1,i
As 0 < λµ < 1, the most recent observations will carry the largest weight in the EWMA formula, while observations far in the past will only enter with tiny weights. Due to convergence of the geometric series, for the limit case t→ ∞ the formula can be further simplified:
∞
X
k=1
λk−1µ = 1 1−λµ
⇒ µt,i = (1−λµ)
∞
X
k=1
λk−1µ rt−k+1,i
Based on this expression, the EWMA mean estimator can be reformulated into recursive form:
µt+1,i =λµµt,i+ (1−λµ)rt+1,i
This way, λµ has a clear interpretation, as new observations will enter with weight (1−λµ) to the existing estimate of µt,i, which itself gets weighted by the decay factor λµ. A decay factor closer to 1 will put less weight on the last observation, and hence represents a less responsive estimator. Still, by putting larger weight on recent observations, the estimator will react faster to data than what a sample mean would do. For variances and covariances we can define EWMA estimators in a similar way.
Definition 4.3.2: [EWMA variance/covariance estimator]
Let σt,ij denote the covariance between asset i and asset j. Let further ¯rt,i denote the sample mean of asset i with observations up to time t:
¯ rt,i = 1
t
t
X
k=1
rk,i
Then the exponentially weighted moving average model for covariance σt,ij is given by:
σt,ij =Xt
k=1
λk−1σ
Pt
k=1λk−1σ (rt−k+1,i−r¯t,i)(rt−k+1,j −r¯t,j)
Again, with infinite history the estimator can be reformulated in recursive form:
σt+1,ij =λσσt,ij+ (1−λσ)(rt+1,i−r¯i)(rt+1,j−¯rj)
According to (“RiskMetrics Technical Document” 1996), there are two advantages over sample moments, which implicitly would use equally weighted observations. First, volatility will react faster to market shocks than with sample moments. And second, volatility declines exponentially, as the weight of observations that did cause a certain shock to volatility will get increasingly less over time.
Despite these potential advantages, however, the value added from EWMA estimators is far from being unanimously acknowledged in academic literature. For example, this is what Andrew Ang writes in (Ang 2014) about backward-looking estimators that are based on historic data only:
“Using short data samples to produce estimates for mean-variance inputs is very dangerous. It leads to pro-cyclicality. When past returns have been high, current prices are high. But current prices are high because future returns tend to be low. While predictability in general is very weak, chapter 8 provides evidence that there is some. Thus, using a past data sample to estimate a mean produces a high estimate right when future returns are likely to be low. These problems are compounded when more recent data are weighted more heavily, which occurs in techniques like exponential smoothing.”
Hence, we have good reasons to believe that EWMA moment estimates will generate large enough estimation errors in order to pose a challenge on the robustness of diversification-aware portfolio selection.
Furthermore, however, we still need to define actual values for decay factorsλµ andλσ. While some optimal values for decay factors ideally would be estimated from data, we will simply fix some reasonable values in the first place. The values that we choose are:
• λµ = 0.99
• λσ = 0.95
This way, moment estimates will quite quickly react to observations for variances and covariances, while EWMA mean estimates are more persistent. Hence, the responsiveness of estimators coincides with two wide-spread beliefs. First, second moments of asset returns are generally easier to estimate. They hence can be made more responsive and be estimated with less data than expected returns. And second, volatility tends to spike quickly as a reaction to market shocks. Hence, historic data tends to get outdated faster, which also requires higher responsiveness of estimators.
To make the responsiveness of different decay factors more meaningful, λ values also can be translated into half-life periods. Half-life periods measure the number of most recent observations required in order to get a cumulative weight larger than 50%. Assuming infinitely many observations, the half-life period can be analytically derived using geometric series. The joint weight of all observations that are at least K periods in the past is given by:
(1−λ) X∞
k=K
λk−1 = (1−λ)λK−1
∞
X
k=1
λk−1 = (1−λ)λK−1 1
1−λ =λK−1
Given this formula, for any given joint probability of the most historic observations we can derive the associated threshold period K:
λK−1 =! α (K −1) log(λ) = log (α)
K−1 = log (α) log (λ)
Solving for K, with α= 0.5 andλ equal to λµ = 0.99 and λσ = 0.95 respectively, we get:
Kµ = log(0.5)
log(0.99) + 1 = 69.97 Kσ = log(0.5)
log(0.95) + 1 = 14.51
Hence, for the estimation of mean returns the most recent 70 days approximately get the same weight as all previous observations together. In contrast, for variances and covariances the half-life period is only 15 days.