**Efficient Portfolios and Multi-Factor Models****Summary:**Modern Portfolio Theory's concept of Efficient Portfolios can be useful for understanding and improving the risk/reward of portfolios and multi-factor trading systems. However, MPT's inputs are typically based on past market data, and so it should be used prudently as one of several factors influencing portfolio and trading model construction. This is especially true when optimizing Sharpe Ratios, where small changes in the data used in the optimization, such as the date range of the historical market data used, can result in Efficient Portfolios and trading systems that perform poorly in the future.

**Introduction**

Diversification has often been described as the only "free lunch" in investing, although it wasn't very helpful during the 2008 crisis. Market crises aside, it's still prudent to avoid having all your eggs in one basket - but in which baskets should they be? Modern Portfolio Theory (MPT) provides a simple framework for estimating optimal ways to diversify portfolios, but does so by making some simplifying, and at times, misleading assumptions:

- MPT assumes that market returns have a Normal Distribution, which may be a reasonable assumption during normal market conditions, but clearly isn't during periods like the 2008 crisis with its negatively skewed, fat-tailed risk.
- To construct portfolios that have optimal risk/reward in the future, MPT requires the future means, variances, and correlations of asset returns. Since this is unknowable, MPT is typically applied by using past market returns to estimate future means, variances, and correlations.
- Hence, MPT's solutions for optimal portfolios should be taken with a grain of salt, and it should be one of several factors influencing portfolio construction.

**Stepping Through MPT**

1. Big Picture: Correlation is the Key to MPT

The core concept of MPT is fairly simple: combine uncorrelated assets to smooth returns/reduce risk, and ideally, combine negatively correlated assets, as graphed to the right.
Of course, high-returning, negatively correlated assets as graphed to the right are rare in the real world, but adding uncorrelated assets to portfolios with even meager returns can boost portfolio risk/reward. Even adding assets with negative returns can boost portfolio risk/reward if their correlations with the portfolio is sufficiently negative, since their effect will be akin to owning insurance against market declines. |

Reward is typically defined as the portfolio's annual return, but defining Risk is less straightforward. For example, should negative asset returns be considered riskier than positive returns? MPT defines Risk as asset return volatility, including volatility due to positive asset returns, and is typically calculated by taking the standard deviation of the daily returns (multiplied by the square root of trading days per year to annualize it).

**2. Define "Risk" and "Reward"**Reward is typically defined as the portfolio's annual return, but defining Risk is less straightforward. For example, should negative asset returns be considered riskier than positive returns? MPT defines Risk as asset return volatility, including volatility due to positive asset returns, and is typically calculated by taking the standard deviation of the daily returns (multiplied by the square root of trading days per year to annualize it).

3. Assume Asset Returns have a Normal Distribution, and Analyze a Series of Simple, 2-Asset PortfoliosAssign an expected annual return and volatility to 2 theoretical assets, along with an assumed correlation for their returns. For simplicity, assume that the returns and volatilites have no randomness so their values never change. |

Then, calculate the expected return and standard deviation of combinations of these two assets. If you vary the asset weights from 0% of asset A and 100% of asset B, to 1% of A and 99% of B, etc., all the way to 100% of A and 0% of B, then graphing the resulting returns and StdDevs of the 100 simulated portfolios will visualize the

**Efficient Frontier**.

If two portfolios have the same amount of expected risk, then the more efficient portfolio is the one with the higher expected return. Therefore, only the portfolios above the bend of the curve lie on MPT's Efficient Frontier.

MPT generates a potentially huge number of efficient portfolios above the bend of the curve, so how does one determine which portfolio to select? Sometimes, constraints will dictate portfolio choice, such as the maximum amount of expected risk that can be assumed, or a minimum expected return required to support a spending policy while maintaining the purchasing power of the endowment's assets.

Alternatively, metrics such as the Sharpe Ratio can be optimized, which is explored in Part 2. The Sharpe Ratio is the ratio of return to risk, and its formula is:

Sharpe Ratio = (Expected Annual Return - Risk Free Interest Rate) / Annualized Standard Deviation of the Daily Returns

Expected Annual Return = 14.6%

Risk Free Rate (e.g. 3 month T-bills) = 4%

Excess Return (over Risk Free) = 14.6%-4% = 10.6%

Annualized StDev of Daily Returns = 11.9%

Sharpe Ratio = 10.6%/11.9% = 0.89

Sharpe Ratio = (Expected Annual Return - Risk Free Interest Rate) / Annualized Standard Deviation of the Daily Returns

**Example**:Expected Annual Return = 14.6%

Risk Free Rate (e.g. 3 month T-bills) = 4%

Excess Return (over Risk Free) = 14.6%-4% = 10.6%

Annualized StDev of Daily Returns = 11.9%

Sharpe Ratio = 10.6%/11.9% = 0.89

As noted above, correlation is the key to MPT. Below plots the risk/reward of the 2-Asset portfolios for differing correlations between assets A and B. Notice when correlation=0.8 there is almost no bend in the curve, and there is little diversification benefit as well. If correlation=1 then the curve would be a straight line, and there would be no diversification benefit since any of the portfolios could be replicated with one of the assets - they could be replicated with Asset B and an allocation to cash earning the risk-free rate, or by levering Asset A above 100% and borrowing at the risk-free rate.

Of course, markets in the real world have a lot of randomness, and the graph of risk/reward vs correlation of actual markets may look closer to the following graph. The Efficient Frontier is now fuzzier, but is still represented by the portfolios on the top perimeter of the plot.

MPT can also be applied to multi-factor trading models in a similar manner, with each factor's returns replacing the asset class returns used above. However, return distributions of factors, or "alphas", tend to be more chaotic and non-Normal than stock market returns, and factor returns are often infrequent, making it easier to be

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**fooled by randomness**In Part 2, MPT and Sharpe Ratio optimization is applied to some actual market data, which shows how care must be taken when applying it - especially when optimizing portfolio allocations.

Wade Vagle, CFA, CAIA

Get in touch at Wade@SchoolsThatLast.com