Module 4: Principles of Asset Allocation

Mean-variance optimization (MVO)

is the most common approach to asset allocation. It assumes investors are risk averse, so they prefer more return for the same level of risk. Given an opportunity set of investable assets, their expected returns and variances, as well as the pairwise correlations between them, MVO identifies the portfolio allocations that maximize return for every level of risk. If the MVO analysis includes all investable risky assets, the result is the familiar “efficient frontier”

Lambda

Lambda is unique to each individual and is based on the investor’s willingness and capacity to take on risk. A risk-neutral investor will have a lambda of 0, although in practice it is typically assumed to be between 1 and 10 with an average level of 4.

MVO Constraints

The most common constraint in MVO is called the budget constraint or the unity constraint, which means the asset weights must add up to 100%. The next most common constraint used in MVO is the nonnegativity constraint, which means all weights in the portfolio are positive and between 0% and 100% (there are no short positions in the SAA).

Criticisms of MVO

1. GIGO: The quality of the output from the MVO (portfolio allocations) is highly sensitive to the quality of the inputs (i.e., expected returns, variances, and correlations). In other settings, this is often called the garbage in, garbage out (GIGO) problem. Although all three inputs are a source of estimation error in MVO, expected returns are particularly problematic, so we focus here on addressing the quality of the expected return inputs.

Criticisms of MVO

2. Concentrated asset class allocations: MVO often identifies efficient portfolios that are highly concentrated in a subset of asset classes, with zero allocation to others; in other words, lowest calculated standard deviation is not the same thing as practical diversification.

Criticisms of MVO

3. Skewness and kurtosis: MVO analysis, by definition, only looks at the first two moments of the return distribution: expected return and variance. It does not take into account skewness or kurtosis. But empirical evidence suggests quite strongly that asset returns are not normally distributed: there is significant skewness and kurtosis in actual returns

Criticisms of MVO

4. Risk diversification: MVO identifies an asset allocation diversified across asset classes but not necessarily the sources of risk. For example, equities and fixed-income securities are two different asset classes, but they are driven by some common risk factors, and diversifying across the two classes won’t necessarily diversify those risk factors.

Criticisms of MVO

5. Ignores liabilities: MVO also does not account for the fact that investors create portfolios as a source of cash to pay for something in the future: individual investors are looking to fund their consumption spending in retirement, for example, while pension funds are focused on funding the pension liability and repaying employees the retirement benefits promised to them. A more robust approach needs to account for the factors that affect these liabilities and the correlations between changes in value of the liabilities and returns on the asset portfolio.

Criticisms of MVO

6. Single-period framework: MVO is a single-period framework that does not take into account interim cash flows or the serial correlation of asset returns from one time period to the next. This means it ignores the potential costs and benefits of rebalancing a portfolio as capital market conditions change and asset allocations drift away from their optimal starting point.

Reverse optimization

Reverse optimization is just what it sounds like: instead of starting with expected returns (and the other inputs) and deriving optimal portfolio weights, start with what we assume to be “optimal” portfolio weights from the global market portfolio and derive the expected returns consistent with those weights. Then we use these return estimates (called implied returns) to do a traditional MVO and derive optimal portfolio weights for our particular investor.

Black-Litterman model

The Black-Litterman model is an extension of reverse optimization in which the implied returns (actually implied excess returns) from a reverse optimization are subsequently adjusted to reflect the investor’s unique views of future returns. For example, if reverse optimization:

Derives an expected return for emerging market equities of 6.5% and you believe this is too low, you could adjust the expected return by 75 basis points to 7.25%. You can then rerun the MVO using your adjusted return estimates.

Projects a return for U.K. large-cap equities of 8.2% and U.S. large-cap equities of 8.0% (a return differential of 20 basis points) and you believe that U.S. large-cap equities will outperform U.K. large-cap equities by 100 basis points, adjust the differential.

Resampled MVO

Resampling can also be used to address the GIGO and highly concentrated issues:

Resampling starts with the basic MVO using the best estimates of expected returns, sigma, and correlations to generate the efficient frontier and associated asset allocations for each point on the frontier.

Then, Monte Carlo simulation is used to generate thousands of random variations for the inputs around the initial estimates, resulting in efficient frontier and associated asset allocations for each point on the frontier.

The resampled efficient frontier is an average of all the simulated efficient frontiers, and the asset allocation for any single point on the resampled efficient frontier is an average of the possible portfolios for that point on the frontier.

Discuss the use of Monte Carlo simulation and scenario analysis to evaluate the robustness of an asset allocation

Monte Carlo simulation (MCS) can be used to:

• Address the limitations of MVO as a single-period model and the related issues of rebalancing and taxes in a multiperiod framework. In a single-period model, taxes are easy to incorporate into the analysis, and rebalancing the portfolio is irrelevant. However, in a multiperiod framework, rebalancing to move toward a strategic allocation target will involve buying and selling investments that trigger taxable capital gains and losses. Also, investors will save (add) money into and spend money out of their portfolio, resulting in interim cash flows. It is relatively straightforward to do this at each future point in an MCS.

• Guide individual investors to identify their risk tolerance level. MCS can be useful in illustrating the range and likelihood of possible outcomes given various assumptions. Clients planning for retirement can visually see how often and when they are likely to run out of money.

Describe and evaluate characteristics of liabilities that are relevant to asset allocation.

The following characteristics of liabilities are relevant to the asset allocation decision:

• Fixed versus contingent. Fixed liabilities have cash flows whose amount and timing are specified in advance, such as a fixed-rate corporate bond. Contingent liabilities have cash flows that depend on uncertain future events, such as the pension liability associated with a defined pension plan.

• Legal versus quasi-legal. Legal liabilities are obligations defined in a legal agreement. Quasi-legal liabilities are not legal obligations but are cash outflows expected to occur in the future and are essential to the mission of the institution. University endowments can be considered to have quasi-legal liabilities.

• Duration and convexity. These measure the change in value of a liability for a given change in interest rates. In the CFA curriculum, we typically talk about duration and convexity in relation to fixed-income securities, but the concept can be applied to any liability.

• Liability value versus size of sponsoring organization. A large liability in relation to the size of the sponsoring organization will necessarily be accounted for in the asset allocation decision; a small liability can usually be ignored as its effect on the optimal asset allocation is minimal.

• Factors that affect future cash flows. These factors include inflation, interest rates, risk premiums, and other economic conditions. DB pension obligations are influenced by the choice of the discount rate, for example.

• Timing considerations. These include longevity risk.

• Regulations affecting the determination of the liability’s value. These are typically found in the insurance industry.

Discuss approaches to liability-relative asset allocation.

Three approaches to liability-relative asset allocation:

• Surplus optimization. This is an extension of MVO in which we determine an efficient frontier based on the surplus with its volatility as our measure of risk, stated either in money or percentage terms.

• Two-portfolio approach. In this approach, we separate the asset portfolio into two subportfolios: a hedging portfolio and a return-seeking portfolio.

• Integrated asset-liability approach. This approach integrates both the assets and the liabilities in a joint optimization method.

Recommend and justify an asset allocation using a goals-based approach.

he goals-based approach to asset allocation is useful for individual investors who typically have a number of (sometimes conflicting) objectives with different time horizons and levels of urgency, which we will measure as specified required probabilities of success.

Discuss factors affecting rebalancing policy.

The higher the transaction costs, the wider the optimal corridor, as the benefits of rebalancing have to “pay” for higher costs to rebalance. The higher the investor’s risk tolerance, the wider the corridor, because the investor is less concerned about deviations from the optimal allocation. The higher the correlation of the asset class with the rest of the portfolio, the wider the corridor, because a portfolio tends to move with the asset class, and the allocations tend to stray more slowly from the target. Finally, the higher the volatility of the asset classes, the narrower the optimal corridor. This is because higher volatility increases the likelihood that the actual allocation will diverge over time from the target allocation.