CFA 2018 level 3 schweser practice exam CFA 2018 level 3 question bank CFA 2018 CFA 2018 r17 principles of asset allocation IFT notes

DEVELOPING ASSET-ONLY ASSET ALLOCATIONS 2.1 Mean–Variance Optimization: Overview This section addresses LO.a: LO.a: describe the use of mean–variance optimization in asset allocation;

Principles of Asset Allocation

1 Introduction 3

2 Developing Asset-Only Asset Allocations 3

2.1 Mean–Variance Optimization: Overview 3

2.2 Monte Carlo Simulation 9

2.3 Criticisms of Mean–Variance Optimization (MVO) 7

2.4 Addressing the Criticisms of Mean–Variance Optimization 7

2.4.1 Reverse Optimization 10

2.4.2 Black-Litterman Model 10

2.4.3 Adding Constraints beyond the Budget Constraints 11

2.4.4 Resampled Mean–Variance Optimization 11

2.4.5 Other Non-Normal Optimization Approaches 13

2.5 Allocating to Less Liquid Asset Classes 13

2.6 Risk Budgeting 14

2.7 Factor-Based Asset Allocation 15

3 Developing Liability-Relative Asset Allocations 16

3.1 Characterizing the Liabilities 16

3.2 Approaches to Liability-Relative Asset Allocation 17

3.2.1 Surplus Optimization 17

3.2.2 Hedging/Return-Seeking Portfolio Approach 21

3.2.3 Integrated Asset–Liability Approach 22

3.2.4 Comparing the Approaches 22

3.3 Examining the Robustness of Asset Allocation Alternatives 22

3.4 Factor Modeling in Liability-Relative Approaches 23

4 Developing Goals-Based Asset Allocations 23

4.1 The Goals-Based Asset Allocation Process 24

4.2 Describing Client Goals 25

4.3 Constructing Sub-Portfolios 25

4.4 The Overall Portfolio 28

4.5 Revisiting the Module Process in Detail 29

4.6 Periodically Revisiting the Overall Asset Allocation 30

4.7 Issues Related to Goals-Based Asset Allocation 30

5 Heuristics And Other Approaches To Asset Allocation 31

6 Portfolio Rebalancing In Practice 32

Summary from the Curriculum 34

Examples from the Curriculum 39

Example 1 Mean–Variance-Efficient Portfolio Choice 1 39

Example 2 A Strategic Asset Allocation Based on Distinguishing a Nominal Risk-Free Asset 40

Example 3 Monte Carlo Simulation for a Retirement Portfolio with a Proposed Asset Allocation 41

Example 4 Problems in Mean–Variance Optimization 43

Example 5 Risk Budgeting in Asset Allocation 45

Example 6 Surplus Optimization 46

Example 7 The Hedging/Return-Seeking Portfolios Approach 46

Example 8 Liability-Relative Asset Allocation: Major Approaches 47

Example 9 Robustness and Risk Assessment in Liability-Relative Asset Allocation 47

Example 10 Understanding Client Goals 47

Example 11 Selecting a Module 48

Example 12 Tolerance Bands for an Asset Allocation Error! Bookmark not defined

This document should be read in conjunction with the corresponding reading in the 2018 Level III CFA® Program curriculum Some of the graphs, charts, tables, examples, and figures are copyright

2017, CFA Institute Reproduced and republished with permission from CFA Institute All rights reserved

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1 Introduction

A diversified, multi-asset class portfolio is typically created using the following two separate steps: 1) Asset allocation decision, which refers to the long-term allocation, based on estimates of risks and returns for each asset class

2) Implementation decision, which involves determining the specific investments (individual securities, pooled investment vehicles, and separate accounts) that will be used to implement the targeted allocations

These two steps can be carried out simultaneously but it is preferable to perform them separately because it is practically difficult to perform them simultaneously and unlike implementation decisions, the strategic asset allocation decisions/policies are revisited somewhat infrequently (e.g., annually or less frequently)

2 DEVELOPING ASSET-ONLY ASSET ALLOCATIONS

2.1 Mean–Variance Optimization: Overview

This section addresses LO.a:

LO.a: describe the use of mean–variance optimization in asset allocation;

Under Markowitz approach, when assets are not perfectly correlated they can be combined such that portfolio risk is less than the weighted average risk of assets In other word, the Markowitz approach focuses on asset’s impact on portfolio risk, not the risk of the asset itself

Mean-variance optimization (MVO) provides a framework for determining how much to allocate to each asset in order to maximize portfolio’s expected return at a given level of risk Mean–variance

optimization is based on three sets of inputs: returns, risks (standard deviations), and pair-wise

correlations for the assets in the opportunity set

Markowitz assumes that the investor’s goal is to maximize his utility function, where the utility is given

by following objective function:

𝑈𝑚= 𝐸 (𝑅𝑚) − 0.005𝜆𝜎𝑚2Where,

o Um = the investor’s utility for asset mix (allocation) m  Um can be interpreted as a equivalent return—that is, the utility value of the risky return offered by the asset mix, stated in terms of the risk-free return that the investor would value equally

certainty-o Rm = the return for asset mix m

o λ = the investor’s risk aversion coefficient  it is investor specific and depends on two things, i.e., willingness to take risk and ability to take risk

 The greater the λ, the more risk-averse the investor is It usually lies between 1 and 10

 When value of λ = 0, it means the investor is risk-neutral  implying that utility solely depends on the expected return

 When value of λ = 4, it represents a moderately risk-averse investor

 Risk aversion is inversely related to risk tolerance

o σ2

m= the expected variance of return for asset mix m

o In the above equation, we use 0.005 when E(Rm) and σm are expressed as percentages rather than as decimals If those quantities are expressed as decimals, then we use 0.5

According to the utility function, utility is enhanced by high expected returns and diminished by high risk In other words, a risk-averse investor “penalizes” the expected rate of return of a risky portfolio by

a certain percentage (or penalizes the expected profit by a dollar amount) to account for the risk

involved That is, the greater the risk the investor perceives, the larger the risk aversion co-efficient, the larger the penalization, and consequently, the more conservative asset allocation will be

Under Markowitz’s mean–variance optimization model, we create the minimum variance frontier, which

is the set of portfolios with the lowest risk for a given level of return From this set, the efficient frontier, the portfolio with the highest return for a given level of risk (or lowest expected risk for a given level of expected return), is identified Once the investor identifies the efficient frontier, the goal is to identify the portfolio with the risk that best fits with their preferences In other words, MVO is a risk budgeting tool which helps investors spend their risk budget wisely

Exhibit 2 given below shows a base case of the efficient frontier The horizontal axis has standard

deviation in % while vertical axis has expected return in %

 The efficient frontier line gives us the most efficient portfolio for any given level of risk That

portfolio will have a certain weightage of different asset classes that we will have in our opportunity set

 The efficient mix at the far left of the frontier with the lowest risk is referred to as the global

minimum variance portfolio, while the portfolio at the far right of the frontier is the maximum

expected return portfolio

 When there are no constraints, such as a budget constraint that requires sum of weights to sum to 1 (or 100% in percentage terms) and non-negativity constraint that allows only positive weights or allocations (i.e., no negative or short positions), the maximum expected return portfolio will consist

of a 100% allocation to the single asset with the highest expected return (but not necessarily the highest level of risk)

We can observe that at “Global Minimum Variance Portfolio”, the slope of the efficient frontier is

greatest This slope represents the rate at which expected return increases per unit increase in risk As

we move from left to right, the slope decreases and it is lowest at maximum return portfolio This implies that as we move left to right along the efficient frontier, the risk-aversion of investor decreases The “kinks” in the line representing the slope (times 10) of the efficient frontier represent corner

portfolios in which an asset either enters or leaves the efficient mix As shown in Exhibit 2, a lower risk aversion coefficient leads to a riskier (higher) point on the efficient frontier, while a higher risk aversion coefficient leads to a more conservative (lower) point on the efficient frontier The asset mix with the highest Sharpe ratio lies at the vertical line (at volatility of 10.88%); it intersects the Sharpe ratio line at a value of 3.7 (an unscaled value of 0.37) This portfolio is also represented by the intersection of the slope line and the Sharpe ratio line

Risk Aversion

Risk preference (or willingness to take risk) refers to the degree of investment risk one is comfortable taking An investor’s willingness to bear risk is based primarily on the investor’s attitudes and beliefs about investments (various asset types)

Risk capacity (or ability to take risk) refers to “investor’s ability to tolerate portfolio losses and the potential decrease in future consumption associated with those losses” An investor's ability to bear risk depends on net worth, income, the size of an emergency fund in relation to consumption needs, and the rate at which the individual saves out of gross income It is an objective measure

Time Horizon

While setting up a strategic asset allocation, investment advisors prefer to use annual capital market assumptions even if the investment horizon of investors is longer (e.g 10 years) This is because annual capital market assumptions reflect the expectations associated with the evaluation horizon (e.g., one year or three years)

This section addresses LO.b:

LO.b: recommend and justify an asset allocation using mean–variance optimization;

Exhibit 3 given below reflects an efficient frontier asset allocation area graph The asset allocation of the minimum variance portfolio is the vertical cross section at the far left, with nearly 100% cash; while the vertical cross section at the far right, with 45% in emerging markets and 55% in global REITs, is the optimal asset allocation of the maximum variance portfolio (standard deviation of 20.5%) The vertical line as illustrated in the exhibit below reflects the asset mix with the highest Sharpe ratio

The chart shows that we can select the actual asset allocation given different level of risk For example, the vertical line shows that at ~10.2 level of risk, we will have a small amount of cash (reflected as red line portion – approx 5%); similarly, the yellow line (approx 30%) reflects allocation to Global ex UK REITS, and so on

When the investor has a numerical return objective, he can select asset allocation by identifying the efficient portfolios expected to meet that return objective For example, if the return objective is 7%, the asset allocation with a 7% expected return can be selected

Determining allocation to cash: There are two different approaches to determining an allocation to cash

and cash equivalents, such as government bills

 Approach 1: Cash and cash equivalents can be included among other assets for which efficient frontier is constructed

 Approach 2: Separate cash from risky assets and define efficient frontier based only on ‘risky’ assets In this approach, we take cash on a point on return or vertical axis and construct a line that goes from that point from the vertical axis to a tangency point of efficient frontier This portfolio is called the optimal risky portfolio or the tangency portfolio

Two-fund separation theorem: The second approach can also be referred to the two-fund separation

theorem which assumes that all investors hold some combination of the risk-free asset and an optimal portfolio of risky assets

Refer to Example 1 from the curriculum

Refer to Example 2 from the curriculum

LO.a: critique the use of mean–variance optimization in asset allocation;

2.3 Criticisms of Mean–Variance Optimization (MVO)

1) Input sensitivity: In a mean-variance approach, the output (asset allocation) is highly sensitive to small changes in the inputs (capital market assumptions)

2) Estimation error: Due to the sensitivity of output to changes in inputs, mean-variance approach suffers from estimation error i.e inputs in a forward-looking context are forecasts and therefore they contain errors

3) Lack of diversification: The asset allocations suggested by MVO tend to be highly concentrated in a subset of the available asset classes Further, the asset allocations suggested by MVO may appear diversified across assets but the sources of risk may be concentrated

4) Assumes normal distribution of asset class returns and focuses only on mean and variance of

returns: Mean-variance optimization only focuses on mean and variance of returns whereas, in reality, investors are concerned about more than the mean and variance of returns, e.g., VAR, drawdown risk etc Hence, a mean-variance approach is not appropriate to use for non-normally distributed asset class returns

5) Static approach: The mean-variance framework is a single period model and is not useful for periods objectives As a result, it does not take into account trading/rebalancing costs and taxes 6) Strategic asset allocations suggested by mean-variance analysis are unintuitive and require

multi-professional judgment before adoption

7) Most portfolios have an objective to meet liability obligations or to meet consumption needs but MVO allocations do not take into account the factors that may affect value of the liability or the consumption needs

8) In MVO, the composition of efficient portfolios is typically more sensitive to expected return

estimates than that of volatilities and correlations In addition, it is difficult to accurately estimate expected returns

2.4 Addressing the Criticisms of Mean–Variance Optimization

The criticisms in MVO can be addressed by the following ways:

1) We can improve the quality of inputs by using reverse optimization (discussed in section 2.4.1) 2) We can limit the impact of estimation error by using a constrained optimization, whereby, we set a maximum or minimum allocation for a single asset or group of assets These constraints help us in establishing a more diversified asset allocation

3) We can limit the impact of incorrect expected return on asset allocation by conducting sensitivity tests to understand the effect on asset allocation to changes in expected returns

This section addresses LO.c:

LO.c: interpret and critique an asset allocation in relation to an investor’s economic balance sheet;

Traditionally, an asset allocation is done without considering the human capital or other assets which are not well-defined financial assets But as discussed in the previous reading, during asset allocation,

we should select asset allocation for an investor in relation to his/her economic balance sheet (i.e including extended portfolio assets)

Example: To understand asset allocation in relation to an investor’s economic balance sheet, we will

discuss a hypothetical example of Emma Beel who is a 45-year- old tenured university professor in London

Beel has GBP 1,500,000 in liquid financial assets, largely due to a best-selling book Her employment as a tenured university professor is viewed as very secure and produces cash flows that resemble those of a very large, inflation-adjusted, long-duration bond portfolio The net present value of her human capital

is estimated at GBP 500,000 Beel inherited her grandmother’s home on the edge of the city, valued at GBP 750,000 Capital market assumptions are given in Exhibit 1 below

 Based on the results of a risk tolerance questionnaire, Beel has moderate risk tolerance (hence, the

risk aversion coefficient is 4.0)

Refer to exhibit 6 below which summarizes Beel’s assets (financial and human)

Refer to Exhibit 7 below

 Since Beel’s house and human capital are non-tradable assets, we allocate a fixed 27.27% to UK residential real estate and 18.18% to human capital in the optimization opportunity set

 It is important to note that because of the constrained allocations to human capital and UK

residential real estate, the remaining allocations associated with Beel’s liquid financial assets do not include UK equities, UK fixed income, or global REITs as these assets are highly correlated with either UK residential real estate or UK human capital

This section addresses LO.g:

LO.g: discuss the use of Monte Carlo simulation and scenario analysis to evaluate the robustness of

an asset allocation;

2.2 Monte Carlo Simulation

Monte Carlo simulation generates a number of strategic asset allocations using random scenarios for investment returns, inflation, investor’s time horizon, and other relevant variables and provides

information about the range of possible investment outcomes as well their probability of occurrence from a given asset allocation

Strengths of Monte Carlo Simulation:

 Unlike traditional MVO, Monte Carlo can handle multiple periods

 Monte Carlo provides a more realistic picture of future outcomes when terminal wealth is path dependent (i.e when deposits and withdrawals are made from the fund)

 Monte Carlo simulation allows us to evaluate the robustness of an asset allocation by evaluating the impact of changes in the distribution of the portfolio’s expected value through time, potential maximum drawdowns, trading/rebalancing costs and taxes on the probability of meeting various portfolio goals

Refer to Example 3 from the curriculum

2.4.1 Reverse Optimization

As the name implies, reverse optimization works in the opposite direction In reverse optimization, we take a set of optimal asset allocation weights, covariances and the risk aversion coefficient as inputs into the model and solve for expected returns These reverse-optimized returns are also referred to as implied or imputed returns It is a forward-looking optimization approach In estimating the inputs, the optimal asset allocation weights can either be market-capitalization weights, weights of existing policy portfolio, the average asset allocation policy of a peer group, or a fundamental weighting scheme 2.4.2 Black–Litterman Model

The Black-Litterman approach determines asset allocation by first determining the equilibrium returns (which reflect average investors’ expectations) using a reverse optimization method and then adjusting them for investor’s views regarding the performance of various assets by taking into account strength of those views

Steps of Black-Litterman Approach:

1) The equilibrium returns are calculated using the equilibrium capital market weights and covariance matrix for all asset classes

2) The expected returns are formulated by back-solving the equilibrium returns calculated in step 1 3) The investor’s expectations regarding the performance of various asset classes are expressed along with the strength of those views The strength of views or confidence level, measured as variance, determines the weight assigned to each view, i.e the greater the variance, the smaller the precision

of one’s view and the lower the one’s confidence

In the Black–Litterman model, investors’ views can be incorporated in two ways, i.e

i An absolute return forecast is associated with a given asset class; or

ii Return differential of an asset (or group of assets) is expressed relative to another asset (or group of assets)

4) The expected returns calculated in step 2 are adjusted for the investor’s views (determined in step 3) using a Bayesian procedure

 When an investor has no views on the asset classes, equilibrium weights of an asset are the same as optimal weights of an asset class; as a result, market implied returns are used The default strategic asset allocation for average investors is the weights of assets classes in a well-diversified global portfolio (e.g MSCI World index)

 However, in case of absence of views, an investor with below-average risk tolerance can

combine global market portfolio with the risk-free asset to reduce the overall risk; whereas, an

investor with above-average risk tolerance can invest in the global market portfolio and use

leverage by borrowing at the risk-free rate to obtain a higher expected return

5) Efficient frontier and portfolios are created by using MVO

Strengths of Black-Litterman Approach:

 The Black-Litterman model enables investors to combine their unique views regarding the

performance of various assets with market equilibrium returns

 The Black-Litterman model tends to develop more diversified portfolios because it considers weights

of a well-diversified global portfolio and combines investor’s views with equilibrium returns which helps dampen the effect of any extreme investor’s views on the optimization process

 Unlike unconstrained MVO, this model does not result in unintuitive portfolios (e.g portfolios with large short positions in asset classes)

 The Black-Litterman model ensures greater consistency across the estimates as it anchors the estimates to equilibrium returns

2.4.3 Adding Constraints beyond the Budget Constraints

This section addresses LO.f:

LO.f: describe how client needs and preferences regarding investment risks can be incorporated into asset allocation;

Standard constraints used in mean-variance optimization are

 Budget (unity) constraint: sum of asset class weights = 1

 Non-negativity constraint

Other possible constraints:

1 Specify a set allocation to a specific asset For example, 20% to real estate or 30% to human capital

2 Specify an asset allocation range for an asset For example, 5% to 20% allocation to the emerging market

3 Specify an upper limit, due to liquidity considerations For example, we can set an upper limit to an alternative asset class to address liquidity considerations

4 Specify the relative allocation of two or more assets For example, allocation to emerging equities must be less than the allocation to developing equities

5 Hold one or more assets representing the systematic characteristics of the liability short

Rationale for adding other (real-world) constraints into the optimization problem: This may help us in

overcoming some of the potential shortcomings of mean–variance optimization as discussed above However, such constraints should not be added just to control the output of a mean–variance

optimization Further, a large number of constraints does not guarantee an optimal asset allocation 2.4.4 Resampled Mean–Variance Optimization

Resampled MVO is an optimization process that addresses the issues of input uncertainty, estimation

error, and diversification associated with traditional MVO process Resampled MVO is basically a

combination of Markowitz’s MVO framework and Monte Carlo simulation

In Resampled MVO, a sensitivity analysis is conducted on a large scale using a Monte Carlo simulation that generates a simulated set of capital market assumptions using the sample values of asset classes’ means, variances, and covariances

 These simulated set of capital market assumptions are then used as an input in the traditional MVO to create a set of resampled efficient portfolios These intermediate frontiers are referred

to as simulated frontiers

 This process is repeated many times and the asset allocations from each of the simulated

portfolios are integrated into one frontier called "Resampled Efficient Frontier" The simulated

efficient portfolios on the Resampled efficient frontier are ranked in terms of return from highest to lowest

Strengths of Resampled MVO Approach:

 Resampled MVO leads to more diversified asset allocation

 Resampled MVO recognizes that forward looking estimates are inherently subject to error

Asset allocations from MVO (Area graph A) Asset allocations from Resampled MVO (area graph B)

It is clearly evident in the asset allocations graphs above that in area graph B (using resampled MVO), the changes in different asset classes are much smoother than that of area graph A (using MVO)

Criticisms of Resampled MVO are as follows:

 Some resampled efficient frontiers are concave in shape, implying declining expected return with a decline in expected risk

 The asset allocations in riskier asset may be over-diversified

 The resampled efficient frontier cannot completely eliminate estimation error as it also suffers from estimation errors issue

 The resampled efficient frontier approach lacks a theoretical foundation

2.4.5 Other Non-Normal Optimization Approaches

As discussed earlier, MVO focuses on mean and variance and assumes normally distributed asset

returns However, asset returns are not normally distributed and investors may also be concerned about skewness and kurtosis (or left tail risk)

Key non-normal optimization approaches include the following:

 Mean-semivariance optimization

 Mean-conditional value-at-risk optimization

 Mean-variance skewness optimization

 Mean-variance skewness kurtosis optimization

Long-Term versus Short-Term Inputs

From time horizon perspective, there are two types of inputs, such as

a) Unconditional inputs: Unconditional inputs refer to long-term (10-plus- year) capital market

assumptions that ignore current market conditions, such as valuation levels, the business cycle, and interest rates

b) Conditional inputs: Conditional inputs refer to shorter-term capital market assumptions that

explicitly take into consideration the current market conditions

Refer to Example 4 from the curriculum

This section addresses LO.d:

LO.d: discuss asset class liquidity considerations in asset allocation;

2.5 Allocating to Less Liquid Asset Classes

Less liquid asset classes include direct real estate, infrastructure and private equity Illiquid assets can provide a return premium and risk reduction characteristics that can enhance an investor’s overall wealth objective over the long term

It is quite challenging to determine an appropriate allocation to these assets because of the following limitations:

Limitations of illiquid asset classes:

 It is difficult to accurately measure the risk and return characteristics of these asset classes and to make capital market assumptions for these less liquid asset classes

 The illiquid indexes do not reflect their true volatility due to the illiquid nature of the constituents that make up these asset classes

 unlike traditional asset classes, no low-cost passive investment vehicles exist that can be used to used to track their performance

 Illiquid asset classes cannot be readily diversified to eliminate idiosyncratic risk

 The risk and return characteristics of actual investment vehicles (i.e direct real estate funds,

infrastructure funds, and private equity funds) are significantly different from that of the asset classes themselves In other words, the risk and return characteristics of investment vehicles

represent a company-specific (idiosyncratic) risk

Practical options of investing in less liquid assets:

 Exclude less liquid asset classes; then consider real estate funds, infrastructure funds, and private equity funds

 Include less liquid asset classes in the asset allocation decision and model the specific risk

characteristics associated with the implementation vehicles

 Include less liquid asset classes in the asset allocation decision and model the inputs to represent the highly diversified characteristics associated with the true asset classes For example, we can use listed real estate indexes, listed infrastructure, and public equity indexes to estimate the risk of the less liquid asset classes and their correlation with the other asset classes in the opportunity set

Cash, the Risk-Free Asset, and Liquidity Needs

 From a single-period perspective, 30-day or 90-day Government bills can be treated as a risk-free asset class with zero standard deviation and zero correlations with other asset classes

 From a multi-period perspective, T-bills are treated as a risky asset class with positive standard

deviation and non-zero correlations with other asset classes

 We can either use cash or another asset as a proxy for risk-free asset in the optimization process

where its optimal allocation is determined by optimization model or we can exclude the risk-free asset from the optimization and determine the optimal allocation to cash-like assets by setting aside

an amount equal to the present value of the short-term liquidity needs

This section addresses LO.e:

LO.e: explain absolute and relative risk budgets and their use in determining and implementing an asset allocation;

2.6 Risk Budgeting

Risk budgeting involves identifying objective estimates for total portfolio risk (i.e target risk level) and allocate the risk to a portfolio’s constituents efficiently in an attempt to keep the portfolio within the acceptable risk bounds The goal of risk budgeting is to maximize return per unit of risk

The marginal contribution to total risk (MCTR) is the rate at which risk changes with a small change in

the current weights of a specific portfolio holding MCTR helps us in determining optimal positions for the portfolio

For asset class i, MCTR is calculated as follows:

MCTRi = (Beta of asset class i with respect to portfolio) × (Portfolio return volatility or standard

deviation)

 The beta takes account both the asset’s own volatility and the asset’s correlations with other portfolio assets

The absolute contribution to total risk (ACTR) for an asset class measures its total contribution to

portfolio return volatility

For asset class i, ACTR can be calculated as follows:

ACTRi = (Weighti) × (MCTRi) Percentage of total risk that each position contributes = Each asset class ACTR / Total portfolio risk

From a risk-budgeting perspective, “an asset allocation is optimal when the ratio of excess return to MCTR is the same for all assets and matches the Sharpe ratio of the tangency portfolio”

Where,

Ratio of excess return to MCTR = (Expected return – Risk-free rate)/MCTR Refer to Example 5 from the curriculum

This section addresses LO.h:

LO.h: describe the use of investment factors in constructing and analyzing an asset allocation;

2.7 Factor-Based Asset Allocation

As discussed in Reading 16, in factor-based approaches, asset allocations are based on risk factors and the desired exposure to each factor and thus, asset classes are described with respect to their

sensitivities to each of the risk factors Factors are the underlying exposures that explain and influence

an investment’s risk For example, the underlying factor affecting the risk of a broad market-cap

weighted stock portfolio is the market factor, also called equity risk Typical factors used in asset

allocation include market or equity exposure, size, valuation, momentum, liquidity, duration (term), credit, and volatility

Let’s consider a size factor; to have a positive exposure to size, we go long on small-cap stock and short

on large-cap stock In other words, Size factor return = Small-cap stock return – Large-cap stock return This can be viewed as a zero (dollar) investment, or self-financing investment, in which we can sell short the underperforming attribute and used those proceeds to take a long position in the better-performing attribute

 Considering the above size-factor equation, we can observe that if large-cap stocks

underperform small-cap stocks, the realized size return would be positive

Other examples include the following:

Factor/Asset Class Factor Definition

Duration Long Treasury bonds – Treasury bills

In general, if an analyst is better at making risk factors assumptions, then it is preferred to factors-based asset allocation In contrast, if an analyst has more expertise in making capital market assumptions, then

it is better to use asset allocations based on asset classes It is important to note that when the

opportunity sets have similar exposures, then the asset allocation based on risk factors would be similar

to asset allocation based on asset classes Refer to Exhibit 22 below

3 DEVELOPING LIABILITY-RELATIVE ASSET ALLOCATIONS

The focus of liability-relative asset allocation is to establish an asset allocation that is able to fund future cash flow liabilities The liability-relative asset allocation is used by large institutional investors with legal liabilities—for example, banks, insurance companies, and pension plans

This section addresses LO.j:

LO.j: describe and evaluate characteristics of liabilities that are relevant to asset allocation;

3.1 Characterizing the Liabilities

It is important to understand the characteristics of the liabilities that we need to fund as it helps us in

determining the composition of the liability-matching portfolio and its basis risk with respect to the

liabilities These characteristics affect the choice of appropriate discount rate(s) to estimate the present value of the liabilities and thus the degree to which assets are adequate in relation to those liabilities The characteristics of liabilities that can affect asset allocation decisions are as follows:

Sr No Characteristics

1

Fixed versus contingent cash flows

 Fixed or non-contingent liabilities: The amounts and timing of payments are fixed in advance

by the terms of a contract For example, corporate bond with a fixed coupon rate

 Contingent liabilities: The amounts and timing of payments depend upon future, uncertain events For example, liabilities of a defined benefit (DB) pension plan depend on the employees’ retirement dates, longevity, and cash payout rules

2

Legal versus quasi-liabilities

 Legal liabilities represent some contractual obligation to make

 Quasi-liabilities represent expected cash payments to be made in future which are essential to the mission of the entity we are dealing with For example, liabilities of an endowment of a university are quasi-liabilities

3 Duration and convexity of liability cash flows

4 Value of liabilities as compared with the size of the sponsoring organization

5 Factors driving future liability cash flows

6 Timing considerations, such as longevity risk

7 Regulations affecting liability cash flow calculations

Let us discuss a hypothetical example of a DB pension plan Assume that Market value of plan assets is US$2.500 billion, present value of liabilities is US$2.261 billion, given a 4% discount rate for high-quality corporate bonds

The following steps of the valuation exercise for a DB pension plan occur on a fixed annual date:

1 Calculate the market value of assets

2 Project liability cash flows (via actuarial principles and rules)

3 Determine an appropriate discount rate for liability cash flows

4 Compute the present value of liabilities, the surplus value, and the funding ratio

Surplus = Market value (assets) – Present value (liabilities) = 2.500 billion – 2.261 billion = 0.239 billion Funding ratio = US$2.5 billion/US$2.261 billion = 1.1057  indicating that plan is about 10.6%

overfunded

It is important to note that the surplus value and the funding ratio are highly dependent upon the discount rate assumption If the discount rate is assumed to be 2%, the surplus drops to −US$0.539 billion and the funding ratio equals 0.8226 This implies that if liabilities are fixed in nature, the discount rate should be the risk-free rate with reference to the duration of the liability cash flows, for example, a 10-year zero-coupon bond yield for a liability with a (modified) duration of 10

*Note: Funding ratio = Market value (assets)/Present value (liabilities)

 When funding ratio equals 1 (or the surplus is 0, the plan is said to be fully funded

 When funding ratio is greater (less) than 1, the plan is said to be over-funded (under-funded) This section addresses LO.k and l:

LO.k: discuss approaches to liability-relative asset allocation;

LO.l: recommend and justify a liability-relative asset allocation;

3.2 Approaches to Liability-Relative Asset Allocation

There are following three approaches to Liability-Relative Asset Allocation

1) Surplus optimization

2) Hedging/return-seeking portfolios approach

3) Integrated asset–liability approach

3.2.1 Surplus Optimization

Surplus optimization is a special case of traditional mean-variance optimization In surplus optimization,

as the name implies, the focus is on maximizing the surplus return per unit of surplus risk (surplus volatility) over any given time horizon Surplus optimization approach is simple and is most appropriate for risk-averse investors The assumptions used in surplus optimization are similar to those of the

traditional Markowitz model (i.e expected returns and a covariance matrix)

The objective function for surplus optimization is given as follows:

𝑈𝑚𝐿𝑅= 𝐸 (𝑅𝑠,𝑚) − 0.005𝜆𝜎2 (𝑅𝑠,𝑚) Where,

o 𝑈𝑚𝐿𝑅 is the surplus objective function’s expected value for a particular asset mix m;

o E(Rs,m) is the expected surplus return for asset mix m

Expected surplus return = (Δ asset value – Δ liability value) / Initial asset value

 Δ liability value (or liability return) measures time value of money for liabilities plus any expected changes in the discount rate and future cash flows over the planning horizon

o λ (lambda) indicates the investor’s risk aversion The more risk averse the investor, the greater the penalty for surplus return volatility

Interpretation of utility function: We calculate a utility for a particular asset mix based on its expected surplus return, variance of surplus return and the investor’s risk aversion For a given level of risk, we identify an asset mix which provides the highest utility

Steps of Surplus Optimization approach:

1 Select asset categories and determine the planning horizon The asset categories can be US Equity,

US Bonds, Private equity, etc

2 Estimate expected returns and volatilities for the asset categories and estimate liability returns This involves making capital market expectations

3 Determine constraints For example, policy and legal limits on the amount of capital invested in individual assets or asset categories

4 Estimate the correlation matrix (asset categories and liabilities) and the volatilities

5 Compute the surplus efficient frontier and compare it with the asset-only efficient frontier

6 Select a recommended portfolio mix

Let us discuss an example of LOWTECH who has been following an asset-only approach but has now decided to adopt a liability-relative approach With respect to surplus optimization, the trustees want to maintain surplus return volatility at a level that tightly controls the risk that the plan will become

underfunded, and they would like to keep volatility of surplus below US$0.25 billion (10%) Exhibit 25 shows LOWTECH’s asset categories and current allocation for a one-year planning horizon The current allocation for other asset categories, such as cash, is zero In this example, only two constraints, i.e., budget and non-negativity, are used

Exhibit 26 shows the plan sponsor’s capital market assumptions over a three- to five-year horizon

Exhibit 28 shows a surplus efficient frontier that results from the optimization program based on the inputs Surplus risk (i.e., volatility of surplus) in money terms (US$ billions) is on the x-axis, and expected surplus in money terms (US$ billions) is on the y-axis

From Exhibit 28, we can observe that like the asset-only efficient frontier, the surplus efficient frontier has a concave shape and the current mix is sub-optimal as it lies below the surplus efficient frontier Hence, we can choose either a higher expected surplus with the same surplus risk or lower surplus risk for the same expected surplus by using optimization model

Exhibit 29 shows the asset class weights for surplus efficient portfolios

Since the investment committee wants to minimize the volatility of the surplus, it is appropriate to choose an asset allocation toward the left-hand side of the surplus efficient frontier For instance, a surplus efficient portfolio with about 60% bonds and the remainder in other assets (illustrated in Exhibit 29) will drop surplus volatility by about 50%

Exhibit 30 shows the weights for portfolios on the usual asset-only efficient frontier based on the same capital market assumptions reflected in Exhibit 29

We can observe that the asset mixes are different for the two approaches

1 The most conservative mix for the surplus efficient frontier (in Exhibit 29) consists mostly of the US corporate bond index (the hedging asset) because it results in the lowest volatility of surplus over the one-year horizon Bonds have an inverse correlation with changes in the present value of the frozen liability cash flows (because the liabilities reflect negative cash flows)

2 The most conservative mix for the asset-only efficient frontier (in Exhibit 30) consists mainly of cash However, it is pertinent to note that as the degree of risk aversion decreases, the asset mixes of both asset-only and surplus optimization approaches become similar

Refer to Example 6 from the curriculum

3.2.2 Hedging/Return-Seeking Portfolio Approach

As the name implies, asset allocation in hedging/return-seeking approach is determined in two parts:

 Part 1: Asset allocation for liability hedging portfolio or to fund liabilities

 The goal of the liability hedging portfolio is to minimize the volatility of the plan’s

assets relative to changes in the value of the plan’s unique liability

 The liability hedging portfolio comprises of assets whose returns depend on the same factor(s) that affect returns of the liabilities For example, if payments are dependent upon future inflation, the hedging portfolio would include index-linked (inflation-linked) Treasury bonds

 Possible techniques which can be used to hedge against liabilities include cash flow matching, duration matching, and immunization

 Part 2: Asset allocation for return-seeking portfolio This involves investing in risky assets that can generate a return above and beyond the liability benchmark

 The goal of the growth portfolio is to generate excess returns to strengthen the plan’s funded status over the long term and to help pay for future benefit accruals, reducing the size of needed contributions to the plan The growth portfolio is typically composed

of public and private equity, hedge funds, commodities, and high-yield credit (among other assets)

 The return-seeking portfolio can be managed independently of the hedging portfolio (for example, using mean–variance optimization or another method)

Taking the previous example of LOWTECH, if it has a small surplus then under this approach, the pension plan needs to invest most of its capital in the hedging portfolio The hedging portfolio (i.e US$2.261 billion, given 4% discount rate) can be invested in long-bond indexed investment The remaining

US$0.239 billion can be invested in risky assets with higher expected return, such as stocks, real estate, and hedge funds Note that if the discount rate were 2% rather than 4%, the pension plan would be underfunded In such a case, the pension plan sponsor either have to increase the funding ratio

(increase contributions) in order to ensure liabilities payments or use a variant of the two-portfolio approach

The variants of the two-portfolio approach include the following:

 Partial hedge, whereby lower capital is allocated to the hedging portfolio to generate higher

expected returns This approach is useful when the sponsor is able to increase contributions if the funding ratio does not increase in the future to 1 or above

 Dynamic hedge, whereby a higher capital is allocated to the hedging portfolio as the funding ratio increases The approach is often referred to as the liability glide path

These variants are less conservative compared with the basic approach as they do not perfectly hedge the liabilities The basic approach is most appropriate for conservative investors, such as life insurance companies, and for overfunded/fully funded institutional investors that can fully hedge their liabilities

Limitations of Two-Portfolio Approach: The basic two-portfolio approach cannot be directly applied

under following circumstances

1) When the funding ratio is less than 1, the investor cannot create a fully hedged portfolio unless there is a sufficiently large contribution from the sponsor

2) When a true hedging portfolio is not available and the investor can partially hedge the portfolio For example, when losses are linked with weather-related causes, such as hurricanes or earthquakes When hedges are imperfect, investors are exposed to basis risk In such cases, investors can use an insurance contract to hedge basis risk

Refer to Example 7 from the curriculum

3.2.3 Integrated Asset–Liability Approach

In this approach, the asset and liability decisions are integrated and jointly optimized This approach involves selecting liabilities and linking the asset performance with changes in the liability values These include asset–liability management (ALM) for banks and some other investors and dynamic financial analysis (DFA) for insurance companies

Advantages of Integrated asset-liability approach:

 It helps in improving the institution’s overall surplus

 It does not assume linear correlation and take into account the transaction costs, turnover

constraints, and other real-world constraints in the optimization model

 It is the most comprehensive approach

3.2.4 Comparing the Approaches

Surplus Optimization Hedging/Return-Seeking

Portfolios

Integrated Asset-Liability

Portfolios

 Assumes linear correlation  Assumes linear or non-linear

correlation

 Assumes linear or non-linear correlation

 Can be used for all levels of

risk and any funded ratio

 Prefer to use for conservative level of risk and positive funded ratio (for basic approach)

 Can be used for all levels of risk and any funded ratio

 Based on single period

Refer to Example 8 from the curriculum

3.3 Examining the Robustness of Asset Allocation Alternatives

We can evaluate the robustness of an asset allocation by using the following:

 Performing “What-if” sensitivity analysis;

 Scenario analysis based on historical time periods

 Multi-stage simulation analysis: This gives us a probability of both good and bad outcomes In this analysis, we can begin with a set of scenarios for the underlying driving economic factors Each scenario gives us a set of probabilistic outcomes for the institutional investor’s asset portfolio and the cash flows for its liabilities at each stage of the planning horizon

Exhibit 32 shows the decision structure for the simulation of an insurance company over several periods, including modeling of the company’s business strategy and the required capital rules

Refer to Example 9 from the curriculum

3.4 Factor Modeling in Liability-Relative Approaches

The liability-relative approaches can be implemented in a factor-based model, whereby assets and liabilities are linked with the underlying driving factors Two primary factors are future economic

conditions and inflation The main idea is that in factor modeling, the asset categories should be

positively correlated with underlying driving risk factors

This section addresses LO.m:

LO.m: recommend and justify an asset allocation using a goals-based approach;

4 DEVELOPING GOALS-BASED ASSET ALLOCATIONS

In goals-based approaches, asset allocations focus on addressing an investor’s goals, given investor’s own time horizon and the required probability of success Under goals-based approach, an investor’s overall portfolio is divided into sub-portfolios, reflecting various goals with their own specific asset allocation This approach is also known as Goals-based investing (GBI) Goals-based asset allocation is most appropriate for clients having multiple goals, time horizons, and urgency levels

In goals-based investing approach, it is important to consider both taxable and tax-exempt investments and for determining the appropriate funding cost for the goal (or “discount rate” for future cash flows),

we consider probability- and horizon-adjusted expectations (called “minimum expectations” or

minimum return expected to be earned over the given time horizon with a given minimum required probability of success) rather than expected average returns

There is a difference between liabilities for an institutional investor and goals for an individual investor Exhibit 33 given below summarizes the difference between individuals and institutions goals

Risk measure Volatility (return or surplus) Probability of missing goal

Return determination Mathematical expectations Minimum expectations

Forecasting techniques

for goals

Can be easily forecasted through averaging using rule of large numbers

Cannot be forecasted precisely using rule of large numbers and averaging

4.1 The Goals-Based Asset Allocation Process

Exhibit 35 illustrates the major elements of the goals-based asset allocation process

It is evident in the above chart that there are two fundamental parts to this process The first part focuses on the creation of portfolio modules, while the second part focuses on identifying client goals and matching each of these goals to the appropriate sub-portfolio of a suitable asset size Under goals-based investing, given the time horizon and the required probability of success, we need to adjust the expected return to formulate the relevant discount rate which is used to determine the level of capital needed to meet that goal This capital is invested in the optimized portfolio asset allocation, called a declining-balance portfolio Under this portfolio, the portfolio balance tends to decline over time until the end of the horizon That is, the investor gets income and some of the principal is liquidated every year

Typically, advisors prefer to use pre-optimized modules, provided that the investment constraints imposed by the client are compatible with those used in the creation of the module set Incompatible constraints include constraints related to geographical or credit emphases or de-emphases, base

currency, the use of alternative strategies, or the acceptability of illiquid investments

4.2 Describing Client Goals

As an investment advisor, it is important to realize that individual investors have a range of goals and these goals are not always well-formulated Hence, in describing goals, the first step is to identify

between “cash goals” and “labeled goals” Goals for which anticipated cash flows are available are known as cash goals whereas goals having no precise details about funds needed are called “labeled goals” Examples of labeled goals include “to have minimal risk”, “to preserve capital”, “to preserve purchasing power”, “to achieve long-term growth” etc