Reading 17 Asset Allocation

Question 1

An ALM Example: A Defined-Benefit Pension Plan

Answer 1

For a DB pension plan, net worth is called pension surplus. The funding ratio, calculated by dividing the value of pension assets by the present value of pension liabilities, measures the relative size of pension assets compared with pension liabilities. Some countries state requirements pertaining to pension plan contributions in terms of the funding ratio.

Question 2

Conditional Return Correlations

Answer 2

Global correlations tend to increase in times of increased volatility. Correlations appear to depend on, i.e., are conditional on, global volatility.

Question 3

Asset Allocation and Human Capital

Answer 3

Individual investors, strategic asset allocation must also consider human capital.

Several intuitive theoretical conclusions:

1. Investors with safe labor income (thus safe human capital) will invest more of their financial portfolio into equities. A tenured professor is an example of a person with safe labor income; an at-will employee in a downsizing company is an example of a person with risky labor income.
2. Investors with labor income that is highly positively correlated with stock markets should tend to choose an asset allocation with less exposure to stocks. A stockbroker with commission income is an example of a person who has that type of labor income.
3. The ability to adjust labor supply (high labor flexibility) tends to increase an investor’s optimal allocation to equities.

Investors with a higher degree of labor flexibility should take more risk in their investment portfolios. The intuition is that labor flexibility can function as a kind of insurance against adverse investment outcomes; for example, working longer hours or retiring later can help offset investment losses. Younger investors typically have such flexibility.

Risky human capital may have two components: a component correlated with stock market returns and a component uncorrelated with stock market returns. The two types affect the asset allocation decision differently.

When the investor’s labor income is risky but not correlated with the stock market, the investor’s optimal strategic asset allocation over time follows by and large the same pattern as the case where the investor’s human capital is risk free—so long as the risk of human capital (i.e., income variance over time) is small. This effect occurs because the investor’s human capital does not add to his or her stock market exposure. When the risk of uncorrelated human capital is substantial, however, the investor’s optimal allocation to stocks is less than it would be otherwise, all else equal.

By contrast, when a large part of an investor’s human capital is correlated with the stock market, the appropriate strategic asset allocation involves a much higher allocation to bonds at young ages.

In summary, to effectively incorporate human capital in developing the appropriate asset allocation, an individual’s investment advisor must determine 1) whether the investor’s human capital is risk-free or risky and 2) whether the human capital’s risk is highly correlated with the stock market. Advisors should keep in mind the following themes:

• Investors should invest financial capital assets in such a way as to diversify and balance out their human capital.
• A young investor with relatively safe human capital assets and/or greater flexibility of labor supply has an appropriate strategic asset allocation with a higher weight on risky assets such as stocks than an older investor. The allocation to stocks should decrease as the investor ages. When the investor’s human capital is risky but uncorrelated with the stock market, the optimal allocation to stocks may be less but still decreases with age.
• An investor with human capital that has high correlation with stock market returns should reduce the allocation to risky assets for financial assets and increase the allocation to financial assets that are less correlated with the stock market.

Question 4

Weaknesses of the Fed model?

Answer 4

The Fed model has the following weaknesses:

• Compares a real variable to a nominal variable. The S&P500 earnings yield will not automatically adjust to incorporate changes in inflation and could be considered real. The yield on a treasury is adjusted to incorporate changes in inflation ans is thus considered nominal.
• Ignores the equity risk premium. Assuming the yield on treasuries is the same as the earning yield on the S&P ignores the inherent risk of equities.
• Ignores earnings growth. Growth expectations affect earnings, but treasury yields have no growth components. By assuming the yield on a treasury should be the same as corporate earnings yield, the model implicitly assumes zero growth in earnings.

Question 5

The Resampled Efficient Frontier

Answer 5

1. The Michaud approach to asset allocation is based on a simulation exercise using MVO and a data set of historical returns. Using the sample values of asset classes’ means, variances, and covariances as the assumed true population parameters, the simulation generates sets of simulated returns and, for each such set (simulation trial), MVO produces the portfolio weights of a specified number of mean–variance efficient portfolios (which may be called simulated efficient portfolios). Information in the simulated efficient portfolios resulting from the simulation trials is integrated into one frontier called the resampled efficient frontier.
2. Resampled efficient portfolio for a given return rank as the portfolio defined by the average weights on each asset class for simulated efficient portfolios with that return rank.
3. The set of resampled efficient portfolios represents the resampled efficient frontier.

Question 6

Strategic Asset Allocation Implementation Choices

Answer 6

A passive position can be implemented through:

1. a tracking portfolio of cash market securities—whether self-managed, a separately managed account, an exchange-traded fund, or a mutual fund—designed to replicate the returns to a broad investable index representing that asset class;
2. a derivatives-based portfolio consisting of a cash position plus a long position in a swap in which the returns to an index representing that asset class is received; or
3. a derivatives-based portfolio consisting of a cash position plus a long position in index futures for the asset class.

Active investing can be implemented through:

1. a portfolio of cash market securities that reflects the investor’s perceived special insights and skill and that also makes no attempt to track any asset-class index’s performance, or
2. a derivatives-based position (such as cash plus a long swap) to provide commodity-like exposure to the asset class plus a market-neutral long–short position to reflect active investment ideas.

Semiactive investing can be implemented through (among other methods):

1. a tracking portfolio of cash market securities that permits some under- or overweighting of securities relative to the asset-class index but with controlled tracking risk, or
2. a derivatives-based position in the asset-class plus controlled active risk in the cash position (such as actively managing its duration).

Question 7

There are several explanations why growth does not translate into higher equity returns in emerging markets…

Answer 7

There are several explanations why growth does not translate into higher equity returns in emerging markets:

• Current and future economic growth is already reflected in stock prices.
• The growth is concentrated in private companies, government-owned companies, and subsidiaries of foreign companies.
• The benefits of growth are captured by labor.
• Weak corporate governance results in management squandering shareholder wealth.

Question 8

Investment Characteristics of Emerging Markets

Answer 8

Investing in emerging markets entails issues and risks that are not present or as pronounced in the developed world. The most prominent concerns are those of investability, non-normality and dilution in returns, the growth illusion, corporate governance, contagion, currency issues, institutional investor and analyst performance, and changes from market integration.

1. A foreign investor cannot participate in emerging markets if they are not investable. Perhaps the most obvious limitation to investability is liquidity, where thin trading results in a non-executed transaction or one where the price impact diminishes the return.
2. For emerging markets however, extreme returns are more frequent than under a normal distribution, which results in a fat-tailed (leptokurtic) distribution. Both large positive (positive skewness) and large negative (negative skewness) returns have been found in emerging markets.
3. Corporate governance is a concern in emerging markets for foreign minority investors.
4. Although most studies find that these markets provide return and diversification benefits over the longer term, the benefits are negated over the short-term if investors exit markets suddenly because of herding behavior or margin calls from the initial crisis. Using evidence from the 2008 financial crisis, it also appears that developed country crises can spread to emerging markets. In sum, contagion is an issue that should be of concern to investors in emerging markets.
5. Traditionally, emerging market stock returns and currency changes were positively correlated as the emerging market investor experienced losses on both the stock and currency position during crises. However, more recent evidence indicates that stock prices generally increase when emerging currencies decline, reducing the risk to a foreign investor.
6. It appears that some investors are able to exploit market inefficiencies in emerging markets.
7. As prices rise in a newly integrated market, the expected return for the market should decline as asset pricing will now depend on the covariance.
8. In summary, market integration generally increases stock prices, decreases volatility, may decrease diversification benefits, improves market microstructure, increases informational efficiency, lowers the cost of capital, and increases economic growth.

Question 9

Strategic versus Tactical Asset Allocation

Answer 9

1. Strategic asset allocation sets an investor’s desired long-term exposures to systematic risk. “Long term” has different interpretations for different investors, but five years is a reasonable minimum reference point. Tactical asset allocation involves making short-term adjustments to asset-class weights based on short-term predictions of relative performance among asset classes. TAA can subsume a range of approaches, from occasional and ad hoc adjustments to frequent and model-based adjustments.
2. TAA creates active risk (variability of active returns—i.e., portfolio returns minus benchmark returns). In exchange for active risk, the manager using TAA hopes to earn positive active returns that sufficiently reward the investor after deducting expenses.
3. Strategic asset allocations are reviewed periodically or when an investor’s needs and circumstances change significantly. The policy portfolio should be revised only to account for changes in the investor’s long-term capital market forecasts, not to reflect short-term forecasts.

Question 10

Qualitative and quantitative return objectives of investors

Answer 10

Investors have both qualitative and quantitative investment objectives.

1. Qualitative return objectives describe the investor’s fundamental goals, such as to achieve returns that will:

• provide an adequate retirement income (for an individual currently in the workforce);
• maintain a fund’s real purchasing power after distributions (for many endowments and foundations);
• adequately fund liabilities (for investors such as pension plans and insurance companies); or
• exceed the rate of inflation in the long term (from the prospectus of an inflation-protected bond fund).

1. Because strategic asset allocation involves meeting an investor’s long-term needs, precise statements of numerical return objectives must take account of the effects of compounding.
2. Careful specification of the numerical return objective should reflect the costs of earning investment returns and inflation as well as their compound effects through time.

Question 11

Alternative Investments

Answer 11

One concern for many investors, however, is the availability of resources to directly or indirectly research investment in these groups.

Question 12

Risks of International Assets

Answer 12

1. Currency risk is a distinctive issue for international investing because exchange rate fluctuations affect both the magnitude and volatility of return.
2. Currency risk may not be a large concern, however, for several reasons. First, the correlation between the asset return in local-market currency terms and change in the currency value will be less than one, providing some offset to exchange rate volatility considered in isolation.
3. A second reason why currency risk may not be a large concern is that in a global portfolio, the average correlation between currencies will be less than one, providing further risk reduction.
4. Finally, in terms of standard deviations, currency volatility is approximately half that of stock market volatility. Compared with bond volatility, however, currency volatility is typically twice as large. As a result, exchange rate risk is often considered to be a more critical consideration for bond portfolios compared with common share portfolios.
5. Political risk is a potential concern for the global investor and can come in many forms. In its most general form, political risk results when a country does not have responsible fiscal and monetary policies necessary for economic growth and/or when the country does not possess the appropriate legal and regulatory structure necessary for the growth of financial markets.
6. Many researchers believe developed world investors underinvest in nondomestic markets, a phenomenon called home country bias. This bias could prevent investors from allocating assets optimally. One explanation for home country bias may be investors’ relative lack of familiarity with nondomestic markets. Investors may be less comfortable with foreign markets because of differences in language, information availability, culture, or business practices. Other reasons for home country bias include high transactions costs, low foreign asset liquidity, or foreign political risk. In addition, some institutions (e.g., insurance companies) may have domestic liabilities that should not be matched with foreign assets.

Question 13

Strategic asset allocation reflects what systematic risk exposure?

Answer 13

Strategic asset allocation reflects the investor`s desired systematic risk exposure.

Question 14

Experience-Based Approaches to Asset allocation

Answer 14

1. A 60/40 stock/bond asset allocation is appropriate or at least a starting point for an average investor’s asset allocation. From periods predating modern portfolio theory to the present, this asset allocation has been suggested as a neutral (neither highly aggressive nor conservative) asset allocation for an average investor. The equities allocation is viewed as supplying a long-term growth foundation, the fixed-income allocation as supplying risk-reduction benefits. If the stock and bond allocations are themselves diversified, an overall diversified portfolio should result.
2. The allocation to bonds should increase with increasing risk aversion.
3. Investors with longer time horizons should increase their allocation to stocks. One idea behind this rule of thumb is that stocks are less risky to hold in the long run than the short run, based on past data.
4. A rule of thumb for the percentage allocation to equities is 100 minus the age of the investor. This rule of thumb implies that young investors should adopt more aggressive asset allocations than older investors. For example, it would recommend a 70/30 stock bond allocation for a 30-year-old investor.

Question 15

ALM approach and AO approach, definition

Answer 15

1. The asset/liability management (ALM) approach involves explicitly modeling liabilities and adopting the optimal asset allocation in relationship to funding liabilities.
2. In contrast to ALM, an asset-only (AO) approach to strategic asset allocation does not explicitly involve modeling liabilities. In an AO approach, any impact of the investor’s liabilities on policy portfolio selection is indirect (e.g., through the level of the return requirement). Compared with ALM, an AO approach affords much less precision in controlling risk related to the funding of liabilities.
3. One example of an AO approach to strategic asset allocation is the Black–Litterman model. This model takes a global market-value-weighted asset allocation (the “market equilibrium portfolio”) as the default strategic asset allocation for investors.

Question 16

The ALM approach tends to be favored when…

Answer 16

In general, the ALM approach tends to be favored when:

• the investor has below-average risk tolerance;
• the penalties for not meeting the liabilities or quasi-liabilities are very high;
• the market value of liabilities or quasi-liabilities are interest rate sensitive;
• risk taken in the investment portfolio limits the investor’s ability to profitably take risk in other activities;
• legal and regulatory requirements and incentives favor holding fixed-income securities; and
• tax incentives favor holding fixed-income securities.

Question 17

Opportunities in International Assets

Answer 17

1. A possible explanation for the reduced effectiveness of international stock diversification in mitigating portfolio risk during times of stress is that stock markets are becoming more integrated (linked) internationally
2. Although country factors have become less important and industry factors more important, diversification across borders should still be a consideration for most investors for several reasons:

• First, it has been demonstrated that the relative importance of the factors varies by industry and country. For example, equity returns in some industries are more related to a country factor, whereas in other industries the industry factor dominates. The relative importance of factors also varies by country.
• Second, smaller economies may have very limited industry representation such that the residents must look outside their border for greater industry representation. For example, an agrarian economy may not have a tech industry.
• Third, the stocks within the domestic industry may not provide the greatest opportunity. The financial markets in some countries may be less efficient, providing an opportunity for active managers.
• Fourth, the finding that currency factors are important for returns indicates that diversifying across countries can provide risk reduction.

3. International diversification can more generally benefit the investor in the following ways:

• First, foreign markets may offer better valuations than domestic markets.
• Second, although global markets tend to crash together in the short term, over the long term a global equity portfolio provides better protection against adverse events than a local portfolio, especially when equally weighted.
• Third, world bond markets’ correlations are usually low, often lower than that between equity markets. Diversifying with nondomestic bonds offers opportunities for a better risk–return tradeoff, especially for lower risk portfolios.
• Fourth, although particular markets may have outperformed in the past and certain economies may be poised for future growth, the past often does not forecast the future and the valuations of strong economies will reflect their prospects. No one market is always going to be the best investment and it is very difficult to predict which markets will outperform.

4. The implication for asset allocation from the above discussion is that investors should diversify across countries and industries. Indexing provides an efficient means of diversification by providing global exposure to diverse economies, industries, currencies, and political regimes.

Question 18

Costs of International Assets

Answer 18

1. The costs of trading in foreign securities can be higher than that for domestic securities. If liquidity is lower abroad, trading costs such as commissions and the bid–ask spread are likely higher.
2. Additionally, withholding taxes on income are frequently assessed by foreign governments.
3. In addition to the higher trading costs of nondomestic investment, investors may face the problem of low free-float. Free-float refers to the proportion of stock that is publicly traded, which will be low when the government, other companies, founding families, and/or management have large equity holdings. An equity market may have large capitalizations, but a foreign investor may not be able to invest much in it. The major capitalization-weighted indices in the world have been adjusted for free-float.

Question 19

SAA for Defined-Benefit Plans

Answer 19

Plan sponsors typically face a range of constraints motivated by regulatory and liquidity concerns.

From an ALM perspective, the following are desirable characteristics for an asset allocation:

• The risk of funding shortfalls is acceptable.
• The anticipated volatility of the pension surplus is acceptable. Low pension surplus volatility is generally associated with asset allocations whose duration approximately matches the duration of pension liabilities, because pension liabilities behave similarly to bonds as concerns interest rate sensitivity.
• The anticipated volatility of the contributions is acceptable.

In either an ALM or AO approach, if pension liabilities are fixed in nominal terms, inflation is not a concern. Otherwise, the advisor needs to consider how much inflation protection the asset allocation is expected to afford.

Question 20

Steps in the BL Model

Answer 20

1. Define equilibrium market weights and covariance matrix for all asset classes.

Purpose: Inputs for calculating equilibrium expected returns

2. Back-solve equilibrium expected returns

Purpose: Form the neutral starting point for formulating expected returns

3. Express views and confidence for each view

Purpose: Reflect the investor’s expectations for various asset classes; the confidence level assigned to each view determines the weight placed on it

4. Calculate the view-adjusted market equilibrium returns

Purpose: Form the expected return that reflects both market equilibrium and views

5. Run mean–variance optimization

Purpose: Obtain efficient frontier and portfolios

The first step in the BL model is to calculate the equilibrium returns, because the model uses those returns as a neutral starting point. Because we cannot observe equilibrium returns, we must estimate them based on the capital market weights of asset classes and the asset-class covariance matrix.

Incorporating equilibrium returns has two major advantages.

• First, combining the investor’s views with equilibrium returns helps dampen the effect of any extreme views the investor holds that could otherwise dominate the optimization. The result is generally better-diversified portfolios than those produced from a MVO based only on the investor’s views, regardless of the source of those views.
• Second, anchoring the estimates to equilibrium returns ensures greater consistency across the estimates.

The BL model largely mitigates the problem of estimation error-maximization by spreading any such errors throughout the entire set of expected returns.

Question 21

Asset/Liability Modeling with Simulation

Answer 21

Managers often use Monte Carlo simulation together with surplus optimization (or sometimes standard mean–variance optimization) to provide more detailed insight on the performance of asset allocations under consideration. Simulation is particularly important for investors with long time horizons, because the MVO or surplus optimization is essentially a one-period model.

A simple asset allocation approach that blends surplus optimization with Monte Carlo simulation follows these steps:

• Determine the surplus efficient frontier and select a limited set of efficient portfolios, ranging from the MSV portfolio to higher-surplus-risk portfolios, to examine further.
• Conduct a Monte Carlo simulation for each proposed asset allocation and evaluate which allocations, if any, satisfy the investor’s return and risk objectives.
• Choose the most appropriate allocation that satisfies those objectives.

The first step in the three-step ALM employs the model presented in Sharpe (1990). The objective function is to maximize the risk-adjusted future value of the surplus (or net worth). Formally, in a mean–variance context, doing so amounts to maximizing the difference between the expected change in future surplus and a risk penalty. The risk penalty is a function of the variance of changes in surplus value and the investor’s risk tolerance (or risk aversion).

U^ALM_m=E(SR_m)−0.005R_Aσ²(SR_m)

where:

U^ALM_m = the surplus objective function’s expected value for a particular asset mix m, for a particular investor with the specified risk aversion

E(SR_m) = expected surplus return for asset mix m, with surplus return defined as (change in asset value – change in liability value)/(initial asset value)

σ²(SR_m) = variance of the surplus return for the asset mix m

R_A = risk-aversion level

Question 22

SAA for Foundations and Endowments

Answer 22

Fiduciaries of endowments and foundations should focus on developing and adhering to appropriate long-term investment and asset allocation policies. Low-cost, easy-to-monitor, passive investment strategies are often their primary approach to implementing a strategic asset allocation.

Because of limited resources to fund the costs and complexities of due diligence, small endowments have a constrained investment opportunity set compared with large endowments.

Question 23

The Unconstrained MVF

Answer 23

1. The simplest optimization places no constraints on asset-class weights except that the weights sum to 1. We call this form an unconstrained optimization, yielding the unconstrained minimum-variance frontier.
2. The Black (1972) two-fund theorem states that the asset weights of any minimum-variance portfolio are a linear combination of the asset weights of any other two minimum-variance portfolios.

Question 24

Shortfall risk, downside risk, Roy’s safety-first criterion

Answer 24

Another way for an investor to quantify risk is in terms of shortfall risk, the risk that a portfolio’s value will fall below some minimum acceptable level during a stated time horizon.

Shortfall risk is one example of the larger concept of downside risk (risk relating to losses or worse than expected outcomes only). Downside risk concepts include not only shortfall risk but concepts such as semivariance and target semivariance that also may be applied in asset allocation and are discussed in statistical textbooks (as well as defined in the glossary).

The oldest shortfall risk criterion is Roy’s safety-first criterion. Roy’s safety-first criterion states that the optimal portfolio minimizes the probability over a stated time horizon that portfolio return, R_P, will fall below some threshold level R_L that the investor insists on meeting or exceeding. The safety-first optimal portfolio maximizes the safety-first ratio (SFRatio):

SFRatio=(E(R_P)−R_L)/σ_P

Another shortfall risk approach. An investor could also specify a given maximum probability of not meeting a return threshold. That probability can be translated into a standard deviation test, if we assume a normal distribution of portfolio returns. For example, suppose that a 2.5 percent probability of failing to meet a return threshold is acceptable. Given a normal distribution of returns, the probability of a return that is more than two standard deviations below the expected return is approximately 2.5 percent. Therefore, if we subtract two standard deviations from a portfolio’s expected return and the resulting number is above the client’s return threshold, the resulting portfolio passes that shortfall risk test. If the resulting number falls below the client’s threshold, the portfolio does not pass that shortfall risk test. Shortfall probability levels of 5 percent and 10 percent translate into 1.65 and 1.28 standard deviations below the mean, respectively, under a normality assumption.

Question 25

Strategic asset allocation vs. horse race system

Answer 25

1. Strategic asset allocation is superior to the *horse race system* as a method for controlling the systematic risk exposures. Using strategic asset allocation, it is possible to maintain maximum control over the risk exposures 2. In contrast, the horse race system creates incentives for the investment managers to take on a higher level of risk than is appropriate. The managers have the incentive to greatly overweight the highest-expected-return asset class in order to finish first in the race, particularly if they are lagging other managers. The resulting portfolio will tend to be less diversified and have higher risk than the policy portfolio.

Question 26

Strategic asset allocation and Tactical asset allocation, definition

Answer 26

1. **Strategic asset allocation** is an integrative element of the planning step in portfolio management. In strategic asset allocation, an investor’s return objectives, risk tolerance, and investment constraints are integrated with long-run capital market expectations to establish exposures to IPS-permissible asset classes. The aim is to satisfy the investor’s investment objectives and constraints. Thus strategic asset allocation can be viewed as a process with certain well-defined steps. Performing those steps produces a set of portfolio weights for asset classes; we call this set of weights the strategic asset allocation (or the **policy portfolio**). Thus “strategic asset allocation” may refer to either a process or its end result. 2. **Tactical asset allocation (TAA)**, which involves making short-term adjustments to asset-class weights based on short-term expected relative performance among asset classes.

Question 27

SAA for Insurance Companies

Answer 27

An insurer’s strategic asset allocation must complement and coordinate with the insurer’s operating policy. Investment portfolio policy thus seeks to achieve the most appropriate mix of assets 1) to counterbalance the risks inherent in the mix of insurance products involved and 2) to achieve the stated return objectives. The insurer must consider numerous factors in arriving at the appropriate mix, the most important of which are asset/liability management concerns, regulatory influences, time horizons, and tax considerations. Insurers are taxable enterprises, in contrast to defined-benefit pension plans, endowments, and most foundations. Therefore, insurers focus on after-tax return and risk. Like defined-benefit plans, however, insurers face contractual liabilities to insureds. As a result, an ALM approach to strategic asset allocation is generally chosen. ALM considerations include yield, duration, convexity, key rate sensitivity, value at risk, and the effects of asset risk on capital requirements given the spread of risk-based capital regulation. **Portfolio segmentation** is the creation of subportfolios within the general account portfolio, according to the product mix for each individual company. In this approach, the insurer groups liabilities according to the product line of business or segment. (Some insurers segment by individual product line; others group similar lines according to such characteristics as duration.) An asset type’s appropriateness is measured on at least three bases: expected return, interest rate risk (duration), and credit risk characteristics. Segmentation offers the following advantages: * provides a focus for meeting return objectives by product line; * provides a simple way to allocate investment income by line of business; * provides more-accurate measurement of profitability by line of business. For example, the insurer can judge whether its returns cover the returns it offers on products with investment features such as annuities and guaranteed investment contracts (GICs); * aids in managing interest rate risk and/or duration mismatch by product line; and * assists regulators and senior management in assessing the suitability of investments. Fixed-income investments constitute the majority holding of most life and non-life insurers. Casualty insurance companies traditionally maintain a bond portfolio to offset insurance reserves, with capital and surplus funds invested largely in common stocks. In addition to credit quality, insurers must also consider bonds’ taxability. In the United States, bonds issued by states and municipalities are generally exempt from taxation at the federal level (state bonds are also exempt from state taxes, and municipal bonds from municipal and state taxes, in general). For tax reasons, non-life insurers have often been major purchasers of such tax-exempt bonds. For a life insurance company, the selection of bond maturities is substantially dictated by its need to manage the interest rate risk exposure arising from asset/liability duration mismatch. Consequently, life insurers typically structure the bond portfolio’s maturity schedule in line with the estimated liability cash outflows, at least in the short and intermediate term. Insurers hold equity investments for several reasons. Life insurers market a variety of products such as variable annuities and variable life insurance policies that may be linked to equity investments. Insurers then hold equity investments in the separate account(s) associated with those products. Another important function of the investment operation is to provide growth of surplus to support the expansion in insurance volume; common stocks, equity investments in real estate, and venture capital have been the investment alternatives most widely used to achieve surplus growth. Insurers (particularly life insurers) generally maintain limited liquidity reserves; most life insurers depend on their fixed-income portfolio’s maturity schedule and their ability to control interest rate risk to assure that surrenders and/or policy loans can be funded with little or no loss of principal income. Casualty insurers, especially those with relatively short duration liabilities, tend to have higher liquidity requirements than other insurers.

Question 28

Cash Equivalents and Capital Market Theory

Answer 28

1. The term “risk-free rate” suggests a single-period perspective; a reported positive standard deviation for cash equivalents suggests a *multiperiod perspective*. 2. When we assume a nominally risk-free asset and take a single-period perspective, mean–variance theory points to choosing the asset allocation represented by the perceived tangency portfolio *if the investor can borrow or lend at the risk-free rate*. (Borrowing in this context means using margin to buy risky assets, resulting in a negative weight on the risk-free asset.) 3. **Capital allocation line**, which describes the combinations of expected return and standard deviation of return available to an investor from combining his or her optimal portfolio of risky assets with the risk-free asset. 4. *Leveraging the tangency portfolio* may be practically irrelevant for many investors. 5. The tangency portfolio is the efficient portfolio with the *highest Sharpe ratio*.

Question 29

Asset/Liability Management

Answer 29

1. Using an ALM approach, asset allocation must consider the risk characteristics of the liabilities in addition to those of the assets, because the focus is on funding the liabilities. 2. The efficient frontier is more precisely the “asset-only” efficient frontier, because it fails to consider liabilities. **Net worth** (the difference between the market value of assets and liabilities), also called **surplus**, summarizes the interaction of assets and liabilities in a single variable. The ALM perspective focuses on the **surplus efficient frontier**. Mean–variance surplus optimization extends traditional MVO to incorporate the investor’s liabilities. 3. The leftmost point on the surplus efficient frontier is the *minimum surplus variance (MSV) portfolio*, the efficient portfolio with the least risk from an ALM perspective. The MSV portfolio might correspond to a cash flow matching strategy or an immunization strategy. 4. The rightmost point on the surplus efficient frontier represents the *highest-expected-surplus portfolio*. Similar to traditional MVO, the highest-expected-surplus portfolio typically consists of 100 percent in the highest-expected-return asset class. In fact, at high levels of risk, the asset allocations on the surplus efficient frontier often resemble high-risk asset-only efficient portfolios.

Question 30

Dynamic approach and Static approach, definitions

Answer 30

1. A **dynamic approach** recognizes that an investor’s asset allocation and actual asset returns and liabilities in a given period affect the optimal decision that will be available next period. The asset allocation is further linked to the optimal investment decisions available at all future time periods. 2. In contrast, a **static approach **does not consider links between optimal decisions at different time periods, somewhat analogous to a driver who tries to make the best decision as she arrives at each new street without looking further ahead. This advantage of dynamic over static asset allocation applies both to AO and ALM perspectives. With the ready availability of computing power, institutional investors that adopt an ALM approach to strategic asset allocation frequently choose a dynamic rather than a static approach. A dynamic approach, however, is more complex and costly to model and implement. Nonetheless, investors with significant future liabilities often find a dynamic approach to be worth the cost.

Question 31

Cash flow matching and Immunization, definitions

Answer 31

1. A **cash flow matching** approach structures investments in bonds to match (offset) future liabilities or quasi-liabilities. When feasible, cash flow matching minimizes risk relative to funding liabilities. 2. An **immunization** approach structures investments in bonds to match (offset) the weighted-average duration of liabilities.Because duration is a first-order approximation of interest rate risk, immunization involves more risk than does cash flow matching with respect to funding liabilities.

Question 32

SAA for Banks

Answer 32

Bank’s securities portfolio plays an important role in 1) managing the balance sheet’s overall interest rate risk, 2) managing liquidity (assuring adequate cash is available to meet liabilities), 3) producing income, and 4) managing credit risk.

Question 33

According to the modern portfolio theory, which risk is awarded?

Answer 33

According to the modern portfolio theory, *only systematic risk* is rewarded. Total risk (may be measured by standard deviation) is comprised of systematic and unsystematic.

Question 34

Other Considerations in Asset Allocation for Individual Investors

Answer 34

**Mortality risk** is the risk of loss of human capital if an investor dies prematurely. Of course, it is the investor’s family that bears the effects of mortality risk. Life insurance has long been used to hedge this risk. **Longevity risk** is the risk that the investor will outlive his or her assets in retirement. Longevity risk is also related to income needs and so logically should be directly related to asset allocation, unlike mortality risk. Longevity risk cannot be completely managed through asset allocation. One reaction to this risk might be to bear greater investment risk in an effort to earn higher long-term returns. If the investor can tolerate additional risk, this approach may be appropriate. Such a strategy reduces longevity risk only in expectation, however; the higher mean return may not be realized and the investor may still outlive his resources. Transferring longevity risk in whole or in part to an insurer through an annuity contract may be rational. Insurers can profitably accept longevity risks by 1) spreading the risks among a large group of annuitants and 2) making careful and conservative assumptions about the rate of return earned on their assets. A **life annuity** type of instrument should be considered for many retirement plans. A life annuity guarantees a monthly income to the annuitant for the rest of his life. Life annuities may have one of three forms: In a **fixed annuity**, the periodic income payments are constant in amount; in a **variable annuity**, the payments vary depending on an underlying investment portfolio’s returns; and an **equity-indexed annuity** provides a guarantee of a minimum fixed payment plus some participation in stock market gains, if any.

Question 35

Major Steps in Asset Allocation

Answer 35

_C1_ Capital market conditions ⇒ _C2_ Prediction procedure ⇒ _C3_ Expected returns, risks, correlations _I1_ Investor`s assets, liabilities, net worth and risk attitudes ⇒ _I2_ Investor`s risk tolerance function ⇒ _I3_ Investor`s risk tolerance _M1_ Optimizer ⇒ _M2_ Investor`s asset mix ⇒ _M3_ Returns * We can portray the relationship between the investor’s circumstances (box I1) and risk tolerance (box I3) with a **risk tolerance function**. * From period to period, any (or all) of the items in boxes C1, C3, I1, I3, M2, and M3 may change. However, the items in boxes C2, I2, and M1 should remain fixed, because they contain decision rules (procedures).

Question 36

Criteria for Specifying Asset Classes

Answer 36

Five criteria that will help in effectively specifying asset classes: 1. *Assets within an asset class should be relatively homogeneous*. Assets within an asset class should have similar attributes. 2. *Asset classes should be mutually exclusive*. Overlapping asset classes will reduce the effectiveness of strategic asset allocation in controlling risk and also introduce problems in developing asset-class return expectations. 3. *Asset classes should be diversifying*. For risk-control purposes, an included asset class should not have extremely high expected correlations with other asset classes or with a linear combination of the other asset classes. 4. *The asset classes as a group should make up a preponderance of world investable wealth*. From the perspective of portfolio theory, selecting an asset allocation from a group of asset classes satisfying this criterion should tend to increase expected return for a given level of risk. Furthermore, including more markets expands the opportunities for applying active investment strategies, assuming the decision to invest actively has been made. 5. *The asset class should have the capacity to absorb a significant fraction of the investor’s portfolio without seriously affecting the portfolio’s liquidity*. Practically, most investors will want to be able to reset or rebalance to a strategic asset allocation without moving asset-class prices or incurring high transaction costs.

Question 37

The first step in the portfolio construction?

Answer 37

The steps in the asset allocation process are: 1. Determine the investror's risk, return, and constraints. 2. Formulate long-term capital market expectations 3. Determine the mix of assets (allocation) that best meets objectives of the IPS 4. Monitor the portfolio 5. Adjust the portfolio as necessary for strategic or tactical asset allocation. Strategic asset allocation is the first step in the portfolio *construction* process which is step 3 above.

Question 38

Advantages of Resamped Efficient Frontier, Black-Litterman approach and Monte-Carlo simulation compared to the standard MVO process

Answer 38

Advantages of using a _resampled efficient frontier_: * The resampled efficient frontier approach generates portfolios that are *more stable* through time than those derived using standard mean-variance optimization (MVO). A portfolio, that is more stable would reduce turnover and transaction costs. * The resampled efficient frontier approach generates portfolios that are *more diversified* than those derived using standard MVO. A more diversified portfolio should be less volatile. Advantages of using the _Black-Litterman approach_: * The BL approach *incorporates the investor's views* while standard MVO does not. * The BL approach generates portfolios that are *more diversifeid* than those derived standard MVO.A more diversified portfolio should be less volatile. Advantages of using _a Monte-Carlo simulation_: * Monte-Carlo simulation *allow for portfolio rebalancing under changing tax rates* and in multi-period situations. MVO does not consider these factors. * Monte-Carlo simulation *can compute path-dependent terminal wealth*.

Question 39

Characteristic Liability Concerns of Various Investors

Answer 39

**Type of Investor:** Individual **Type of Liability (Quasi-Liability)**: Taxes, mortgage payments (living expenses, wealth accumulation targets) **Penalty for Not Meeting**: Varies **Asset Allocation Approach in Practice**: AO most common, ALM \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor: **Pension plans (defined benefit) **Type of Liability (Quasi-Liability):** Pension benefits **Penalty for Not Meeting: **High, legal and regulatory **Asset Allocation Approach in Practice: **ALM, AO \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor: **Pension plans (defined contribution) **Type of Liability (Quasi-Liability):** Retirement needs **Penalty for Not Meeting:** Varies **Asset Allocation Approach in Practice: **Integrated with individual’s asset allocation approach \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor**: Foundations and endowments **Type of Liability (Quasi-Liability):** Spending commitments, capital project commitments **Penalty for Not Meeting**: High **Asset Allocation Approach in Practice:** AO, ALM \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor: **Life insurance companies **Type of Liability (Quasi-Liability):** Death proceeds, annuity payments, return guarantees on investment products **Penalty for Not Meeting**: Very high, legal and regulatory **Asset Allocation Approach in Practice:** ALM \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor:** Non-life insurance companies **Type of Liability (Quasi-Liability):** Property and liability claims **Penalty for Not Meeting: **Very high, legal and regulatory **Asset Allocation Approach in Practice: **ALM \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ **Type of Investor:** Banks **Type of Liability (Quasi-Liability):** Deposits **Penalty for Not Meeting:** Very high, legal and regulatory **Asset Allocation Approach in Practice:** ALM

Question 40

Tactical Asset Allocation

Answer 40

**Tactical asset allocation (TAA)** involves deliberately underweighting or overweighting asset classes relative to their target weights in the policy portfolio in an attempt to add value. TAA is active management at the asset-class level. Thus in a top-down perspective, TAA would follow the strategic asset allocation decision and stand one level above decisions about how to manage money within an asset class. TAA can be conducted independently of the within-class investment decisions by using derivative securities, a cost-efficient means for changing asset-class exposures. In that case, TAA can be described as an overlay strategy. TAA is based on short-term expectations and perceived disequilibria. TAA is frequently based on the following three principles: 1. *Market prices tell explicitly what returns are available*. Cash yields reveal the immediate nominal return accorded short-term investors. The yield to maturity of T-bills is the nominal reward for holding them to maturity. Thus, at least for this and similar pure discount instruments, investors have objective knowledge of prospective returns. 2. *Relative expected returns reflect relative risk perceptions*. When investors perceive more risk, they demand payment for assuming it. If expected equity returns are particularly high compared with bond expected returns, the market is clearly according a substantial risk premium to stocks. It does so when investors in general are uneasy with the outlook for stocks. 3. *Markets are rational and mean reverting*. If the TAA manager can identify departures from equilibrium in the relative pricing of asset classes, the manager may try to exploit them with knowledge that departures from equilibrium compress a proverbial spring that drives the system back towards balance. The tactical asset allocator should be aware that if a rule for trading leads to superior performance, investors on the losing side of the trades may eventually stop playing; market prices will then adjust to reflect changes in supply and demand, and a trading rule may cease to work. Furthermore, the tactical asset allocator should be aware that deviations from fair value based on historical analysis could persist if the economic environment has changed. Factors such as: * changes in assets’ underlying risk attributes; * changes in central bank policy; * changes in expected inflation; and * position in the business cycle Relative to the strategic asset allocation, however, TAA is a *source of tracking risk*. To manage that risk, in practice, TAA managers often are limited to making adjustments within given *bands or tactical ranges* around target asset-class weights. TAA must overcome a transaction-costs barrier to be advantageous. The potential benefits of any tactical adjustment must be examined on an *after-costs basis*.

Question 41

Traditional asset classes include...

Answer 41

Traditional asset classes include the following: 1. *Domestic common equity*. Market capitalization sometimes has been used as a criterion to distinguish among *large-cap, mid-cap, and small-cap domestic common equity* as asset classes. 2. *Domestic fixed income*. Maturity sometimes has been used to distinguish among *intermediate-term and long-term domestic bonds* as asset classes. Recently, inflation protection has been used to distinguish between *nominal bonds and inflation-protected bonds* as asset classes. 3. *Non-domestic (international) common equity*. Developed market status sometimes has been used to distinguish between *developed market equity and emerging market equity*. 4. *Non-domestic fixed income*. Developed market status sometimes has been used to distinguish between *developed market fixed income and emerging market fixed income*. 5. *Real estate*. The term *alternative investments* is now frequently used to refer to all risky asset classes excluding the four listed above. 6. *Cash and cash equivalents.*

Question 42

The Efficient Frontier

Answer 42

1. According to mean–variance theory, in determining a strategic asset allocation an investor should choose from among the efficient portfolios consistent with that investor’s risk tolerance. Efficient portfolios make efficient use of risk; they offer the maximum expected return for their level of variance or standard deviation of return. 2. Efficient portfolios plot graphically on the **efficient frontier**, which is part of the **minimum-variance frontier** (MVF). Each portfolio on the minimum-variance frontier represents the portfolio with the smallest variance of return for its level of expected return. The graph of a minimum-variance frontier has a turning point (its leftmost point) that represents the **global minimum-variance (GMV) portfolio**. The GMV portfolio has the smallest variance of all minimum-variance portfolios. The portion of the minimum-variance frontier beginning with and continuing above the GMV portfolio is the efficient frontier.

Question 43

The criticism of relying on pairwise correlations for risk-control purposes

Answer 43

1. The criticism of relying on pairwise correlations is that an asset class may be highly correlated with some linear combination of other asset classes even when the pairwise correlations are not high. 2. For each current asset class, find the linear combination of the other asset classes that minimizes *tracking risk* with the proposed asset class. (**Tracking risk** is defined as the square root of the average squared differences between the asset class’s returns and the combination’s returns.)

Question 44

Strategic Asset Allocation for Individual Investors

Answer 44

What characteristics of individual investors distinguish them from other investors in ways that may affect the strategic asset allocation decision? Individual investors are *taxable* and must *focus on after-tax returns*. Tax status distinguishes individual investors from tax-exempt investors (such as endowments) and even other taxable investors such as banks, which are often subject to different tax schedules than individual investors. **Human capital**, the present value of expected future labor income, is not readily tradable. In addition to human capital, an individual has **financial capital**, which consists of more readily tradable assets such as stocks, bonds, and real estate.

Question 45

Asset allocation review

Answer 45

## Footnote **Asset allocation review** is a periodic review of the appropriateness of a portfolio`s asset allocation

Question 46

Calculation of return on Capital Allication Line (CAL) with a specified standard deviation

Answer 46

1. Pick the corner portfolio (P) with the highest Sharpe ratio - tangent portfolio 2. The CAL is the straight line drawn betwen the risk-free asset and the tangent portfolio 3. Calculate the weight of portfolio using the formula: σ_cal = x•σ_P + (1-x)•σ_rf, where σ_rf = 0 =\> x = σ_cal/σ_P 4. Calculate the expected return on the portfolio: R_cal=x•R_P + (1-x) R_rf

Question 47

Currency risk in foreign bond investing

Answer 47

* In foreign bonds, the unhedged bond volatility is *roughly half* of the currency volatility. * For foreign equity the relationship is reversed with unhedged equity returns twice as volatile as the currency return. * Relatively this suggests *currency volatility and hedging currency is more important for bonds* than for equity.

Question 48

Key economic role of strategic asset allocation due to observations

Answer 48

The strategic asset allocation specifies the investor’s desired exposures to *systematic risk*

Question 49

The Empirical Debate on the Importance of Asset Allocation

Answer 49

1. Asset allocation is important as *the fraction of the variation in returns over time* attributable to asset allocation, based on regression analysis. 2. An alternative perspective is asset allocation’s importance in explaining the *cross-sectional variation of returns*—that is, the proportion of the variation among funds’ performance explained by funds’ different asset allocations. In other words, to what degree do differences in asset allocation explain differences in rates of return over time for a group of investors? 3. Investors need to keep in mind their own specific risk and return objectives and establish a strategic asset allocation that is expected to satisfy both. Sidestepping strategic asset allocation finds no support in the empirical or normative literature.

Question 50

Numerical risk aversion

Answer 50

**Numerical risk aversion** can be measured in an interview or questionnaire in which the investor expresses preferences among sets of choices involving risky and certain returns. Risk aversion is the inverse of risk tolerance: A lower value of risk aversion means a higher tolerance for risk. To give approximate guidelines for the scale we will use, an R_A of 6 to 8 represents a high degree of risk aversion (i.e., a low risk tolerance), while an R_A of 1 to 2 represents a relatively low degree of risk aversion (i.e., a high risk tolerance). **U_m=E(R_m)−0.005R_Aσ²_m** where: U_m = the investor’s expected utility for asset mix m E(R_m) = expected return for mix m R_A = the investor’s risk aversion σ_²m = variance of return for mix m In equation, E(R_m) and σ_m are expressed as percentages rather than as decimals.

Question 51

Behavioral Influences on Asset Allocation

Answer 51

1. If the advisor establishes that a client is *loss averse,* one approach may be to incorporate an appropriate shortfall risk constraint or objective in the asset allocation decision. Managing assets with such a constraint or objective should reduce the chance that the client finds himself facing the prospect of a substantial loss. 2. If the investor displays *mental accounting* the investor will place his total wealth into separate accounts and buckets. Each bucket is associated with a different level of risk tolerance depending on a purpose the investor associates with it. 3. The *money’s source* may affect how an individual invests: An investor may be more likely to invest in a risky venture with cash that is drawn from a windfall gain rather than from salary. 4. A *multistrategy approach* has greater complexity than the standard finance approach of developing one strategic asset allocation for the client, because it involves many optimizations rather than just one. Furthermore, developing a set of asset allocations for stand-alone portfolios ignores the correlations between assets across portfolios; the resulting overall asset allocation may fail to use risk efficiently. 5. The *fear of regret* may play a role in actual asset allocation decisions in at least two ways. First, it may be a psychological factor promoting diversification. Second, regret avoidance may limit divergence from peers’ average asset allocation if the investor is sensitive to peer comparisons.

Question 52

Rebalancing to the Strategic Asset Allocation

Answer 52

What does “*rebalancing*” mean? We should distinguish between: 1) changes to the policy portfolio itself because of *changes in the investor’s investment objectives and constraints*, or because of *changes in his or her long-term capital market expectations* and 2) adjusting the actual portfolio to the strategic asset allocation because asset price changes have moved portfolio weights away from the target weights beyond tolerance limits. Although “rebalancing” is used sometimes to refer to the first type of adjustments, in industry practice rebalancing usually refers to 2) and thus we should know some basic facts about it in that sense. Rebalancing may be done on a *calendar basis* (such as quarterly) or on a *percentage-of-portfolio basis*. Percentage-of-portfolio rebalancing occurs when an asset-class weight first passes through a rebalancing threshold (also called a trigger point). The percentage-of-portfolio approach done in a disciplined fashion *provides a tighter control* over risk than calendar-basis rebalancing.

Question 53

The Black–Litterman Approach

Answer 53

Two versions of the Black–Litterman approach exist: * *Unconstrained Black–Litterman (UBL) model.* Taking the weights of asset classes in a global benchmark such as MSCI World as a neutral starting point, the asset weights are adjusted to reflect the investor’s views on the expected returns of asset classes according to a Bayesian procedure that considers the strength of the investor’s beliefs. We call this unconstrained Black–Litterman model, or UBL model, because the procedure does not allow non-negativity constraints on the asset-class weights. * *Black–Litterman (BL) model.* This approach reverse engineers the expected returns implicit in a diversified market portfolio (a process known as **reverse optimization**) and combines them with the investor’s own views on expected returns in a systematic way that takes into account the investor’s confidence in his or her views. These view-adjusted expected return forecasts are then used in a MVO with a constraint against short sales and possibly other constraints. In practice, the UBL model is *an improvement on simple MVO* because the absence of constraints against short sales in the UBL model usually does not result in unintuitive portfolios (e.g., portfolios with large short positions in asset classes), a common result in unconstrained MVO. Investors often formally want to recognize such constraints in optimization. As a result, the second version of the Black–Litterman approach, the *BL model*, is probably *more important in practice.* *BL model* also may be viewed as an asset allocation process with two desirable qualities: * The resulting asset allocation is well diversified. * The resulting asset allocation incorporates the investor’s views on asset-class returns, if any, as well as the strength of those views. A *practical goal of the BL model* is to create stable, mean–variance-efficient portfolios which overcome the problem of expected return sensitivity.

Question 54

Additive vs. compound formulation of a return objective

Answer 54

1. An *additive formulation* of a return objective can serve as a starting point. Because additive formulations provide an intuitive wording of a return objective, such formulations are common in actual investment policy statements. The differences between additive and multiplicative formulations can be essentially negligible for low levels of spending rates and inflation. Nevertheless, portfolio managers should prefer the multiplicative formulation for strategic asset allocation purposes; managers should also observe the distinction between compound and arithmetic mean rates of growth. 2. If an investor’s return requirement is based on the compound rate of return needed to achieve a goal, the corresponding arithmetic mean one-period return needed to achieve that goal will be *higher* than the return requirement stated as a compound rate of return 3. If the investor states an arithmetic mean annual return objective based on a compound growth rate calculation, the investor’s return objective should reflect an appropriate *upward adjustment* from the compound growth rate.

Question 55

Adding the asset class to the portfolio is optimal if....

Answer 55

*Adding the asset class* (denoted new) to the portfolio is optimal if the following condition is met: (E(R_new)−R_F)/σ_new\>([E(R_p)−R_F]/σ_p)Corr(R_new,R_p) This expression says that for the investor to gain by adding the asset class, that asset class’s Sharpe ratio must exceed the product of the existing portfolio’s Sharpe ratio and the correlation of the asset class’s return with the current portfolio’s return.

Question 56

The Importance of the Quality of Inputs

Answer 56

The most important inputs in mean–variance optimization are the *expected returns*. Unfortunately, mean returns are also the *most difficult input to estimate*.

Question 57

The Sign-Constrained MVF: The Case Most Relevant to Strategic Asset Allocation

Answer 57

1. The Black theorem is helpful background for the case of optimization that is most relevant to practice, MVO, including the constraints that the asset-class weights be non-negative and sum to 1. We call this approach a **sign-constrained optimization** because it excludes negative weights, and its result is the sign-constrained minimum-variance frontier. A negative weight would imply that the asset class is to be sold short. 2. Adjacent **corner portfolios** define a segment of the minimum-variance frontier within which 1) portfolios hold identical assets and 2) the rate of change of asset weights in moving from one portfolio to another is constant. As the minimum-variance frontier passes through a corner portfolio, an asset weight either changes from zero to positive or from positive to zero. The GMV portfolio, however, is included as a corner portfolio irrespective of its asset weights. 3. *In a sign-constrained optimization, the asset weights of any minimum-variance portfolio are a positive linear combination of the corresponding weights in the two adjacent corner portfolios that bracket it in terms of expected return (or standard deviation of return)*. The foregoing statement is the key observation about the structure of a sign-constrained optimization; we may call it the **corner portfolio theorem**. 4. The efficient frontier bows out toward the left, reflecting less-than-perfect positive correlation between corner portfolios. As a result, the linear approximation provides a quick approximation (and *upper limit*) for the standard deviation.