Mean-Variance Optimization Vs. Naive Diversification In Portfolio Allocation 0 comments

Mar 4, 2013 12:33 PM

Mean-Variance Optimization vs. Naive Diversification in Portfolio Allocation

In response to the blog post "Understanding the Perils of Mean-Variance Optimization" written by Newfound Research on Seeking Alpha (posted Feb 26th, 2013), we wanted to test firsthand the utility of the MVO technique applied to asset allocation. We can start with a simplified unconstrained MVO model.

We take a base case scenario of a "naive" diversification (50:50) by investing equal dollar amounts between two assets - the S&P 500 futures (ticker: SP) and 30 Year Treasury bond futures (ticker: US) contracts - and back-test over a long period (1990-2012). The allocation was rebalanced on a weekly basis in order to keep us as tightly constrained as possible to the 50:50 allocation.

Having developed a base case portfolio (naive 50:50 allocation model), we can then build out an unconstrained MVO model. The unconstrained, or actively re-balanced, portfolio also invests in the same two assets (SP and US futures contracts), but with allocation weights determined by the change in ratios of the average return to variance or volatility. If for example:

SP: (Avg daily return = 0.04%; Variance = 0.02%)

US: (Avg daily return = 0.03%; Variance = 0.01%)

Then the conversion factor for US = 0.02% / 0.01% = 2

Using the conversion factor of 2, we come up with the weights:

We then run a back-test on these two simple re-balancing techniques (50:50 and MVO) in order to compare the two methods. The results can be seen below:

(click to enlarge)

(click to enlarge)

The results show an improved historical performance in the case of the MVO technique but the question remains: are they statistically significant (at 5% significance level)? To answer that question we performed a bootstrap test:

1. Take the equity curve of MVO model and subtracted the equity curve of 50:50 model to extract the excess returns attributable to the MVO process.

2. Normalize the new equity curve (excess returns) by subtracting the average return of the series from each daily return - zero centering

3. For each resample, select n instances of adjusted returns, at random (with replacement), and calculate their mean daily return (bootstrapped mean).

4. Perform 1000 of re-samples to generate a large number of bootstrapped means.

5. Form the sampling distribution of the means generated in the step above.

6. Derive the p-value of the initial back-test mean return (non zero-centered) based on the sampling distribution

The bootstrap results showed that the Z score was 0.478 which translates into a p value of 0.316 which falls short of being statistically significant at a 5% level.

In conclusion, the above exercise showed that using a MVO technique to asset allocation versus a naive 50:50 diversification did not add value at least for the use of those two instruments.

We will follow up with the results of the next logical step: how does a constrained MVO.

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## Mean-Variance Optimization Vs. Naive Diversification In Portfolio Allocation 0 comments

Mean-Variance Optimization vs. Naive Diversification in Portfolio AllocationIn response to the blog post "Understanding the Perils of Mean-Variance Optimization" written by Newfound Research on Seeking Alpha (posted Feb 26th, 2013), we wanted to test firsthand the utility of the MVO technique applied to asset allocation. We can start with a simplified unconstrained MVO model.

We take a base case scenario of a "naive" diversification (50:50) by investing equal dollar amounts between two assets - the S&P 500 futures (ticker: SP) and 30 Year Treasury bond futures (ticker: US) contracts - and back-test over a long period (1990-2012). The allocation was rebalanced on a weekly basis in order to keep us as tightly constrained as possible to the 50:50 allocation.

Having developed a base case portfolio (naive 50:50 allocation model), we can then build out an unconstrained MVO model. The unconstrained, or actively re-balanced, portfolio also invests in the same two assets (SP and US futures contracts), but with allocation weights determined by the change in ratios of the average return to variance or volatility. If for example:

SP: (Avg daily return = 0.04%; Variance = 0.02%)

US: (Avg daily return = 0.03%; Variance = 0.01%)

Then the conversion factor for US = 0.02% / 0.01% = 2

Using the conversion factor of 2, we come up with the weights:

W(SP) = 0.04 / (0.04 + 0.03 * 2) = 0.4

And

W(US) = (0.03 * 2) / (0.04 + 0.03 * 2) = 0.6

We then run a back-test on these two simple re-balancing techniques (50:50 and MVO) in order to compare the two methods. The results can be seen below:

(click to enlarge)(click to enlarge)The results show an improved historical performance in the case of the MVO technique but the question remains: are they statistically significant (at 5% significance level)? To answer that question we performed a bootstrap test:

1. Take the equity curve of MVO model and subtracted the equity curve of 50:50 model to extract the excess returns attributable to the MVO process.

2. Normalize the new equity curve (excess returns) by subtracting the average return of the series from each daily return - zero centering

3. For each resample, select n instances of adjusted returns, at random (with replacement), and calculate their mean daily return (bootstrapped mean).

4. Perform 1000 of re-samples to generate a large number of bootstrapped means.

5. Form the sampling distribution of the means generated in the step above.

6. Derive the p-value of the initial back-test mean return (non zero-centered) based on the sampling distribution

The bootstrap results showed that the Z score was 0.478 which translates into a

pvalue of 0.316 which fallsshortof being statistically significant at a 5% level.In conclusion, the above exercise showed that using a MVO technique to asset allocation versus a naive 50:50 diversification did

notadd value at least for the use of those two instruments.We will follow up with the results of the next logical step: how does a

constrainedMVO.Disclosure:I am long TLT, SPY.Additional disclosure:I actively trade a systematic macro strategy. My current positions include the following equities: SPY, TLT.Instablogsare blogs which are instantly set up and networked within the Seeking Alpha community. Instablog posts are not selected, edited or screened by Seeking Alpha editors, in contrast to contributors' articles.## Share this Instablog with a colleague

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