portfolio optimization and setting the strategic asset...

Topic Three:

Portfolio Optimization and Setting the Strategic Asset Allocation

Overview of the Portfolio Optimization Process The preceding analysis demonstrates that it is possible for investors to

reduce their risk exposure simply by holding in their portfolios a sufficiently large number of assets (or asset classes). This is the notion of naïve diversification, but as we have seen there is a limit to how much risk this process can remove.

Efficient diversification is the process of selecting portfolio holdings so as to: (i) minimize portfolio risk while (ii) achieving expected return objectives and, possibly, satisfying other constraints (e.g., no short sales allowed). Thus, efficient diversification is ultimately a constrained optimization problem. We will return to this topic in the next session.

Notice that simply minimizing portfolio risk without a specific return objective in mind (i.e., an unconstrained optimization problem) is seldom interesting to an investor. After all, in an efficient market, any riskless portfolio should just earn the risk-free rate, which the investor could obtain more cost-effectively with a T-bill purchase.

Portfolio optimization is an effective tool for establishing the strategic asset allocation policy for a investment portfolio. It is most likely to be usefully employed at the asset class level rather than at the individual security level.

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The Portfolio Optimization Process As established by Nobel laureate Harry Markowitz in the 1950s, the

efficient diversification approach to establishing an optimal set of portfolio investment weights (i.e., {wi}) can be seen as the solution to the following non-linear, constrained optimization problem:

Select {wi} so as to minimize: subject to: (i) E(Rp) > R* (ii) Σ wi = 1

The first constraint is the investor’s return goal (i.e., R*). The second

constraint simply states that the total investment across all 'n' asset classes must equal 100%. (Notice that this constraint allows any of the wi to be negative; that is, short selling is permissible.)

Other constraints that are often added to this problem include: (i) All wi > 0 (i.e., no short selling), or (ii) All wi < P, where P is a fixed percentage

]w2w ... w[2w ] w ... [w n,1nn1nn1-n2,121212n

2n

21

21

2p −−+++++= ρσσρσσσσσ

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Solving the Portfolio Optimization Problem

In general, there are two approaches to solving for the optimal set of investment weights (i.e., {wi}) depending on the inputs the user chooses to specify: 1. Underlying Risk and Return Parameters: Asset class expected

returns, standard deviations, correlations) a. Analytical (i.e., closed-form) solution: “True” solution but

sometimes difficult to implement and relatively inflexible at handling multiple portfolio constraints

b. Optimal search: Flexible design and easiest to implement, but does not always achieve true solution

2. Observed Portfolio Returns: Underlying asset class risk and

return parameters estimated implicitly

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Example of Mean-Variance Optimization: Analytical Solution (Three Asset Classes, Short Sales Allowed)

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Example of Mean-Variance Optimization: Optimal Search Procedure (Three Asset Classes, Short Sales Allowed)

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Example of Mean-Variance Optimization: Optimal Search Procedure (Three Asset Classes, No Short Sales)

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Measuring the Cost of Constraint: Incremental Portfolio Risk

Main Idea: Any constraint on the optimization process imposes a cost to the investor in terms of incremental portfolio volatility, but only if that constraint is binding (i.e., keeps you from investing in an otherwise optimal manner).

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Efficient Frontier Area Map: No Short Selling Allowed

A frontier area map illustrates how the optimal allocation of assets changes as the investor’s expected return goal changes in the portfolio formation process

Return Goal (%)

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Mean-Variance Efficient Frontier With and Without Short-Selling

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Optimal Portfolio Formation With Historical Returns: Examples

Suppose we have monthly return data for the three recent years on the following six asset classes:

- Chilean Stocks (IPSA Index) - Chilean Bonds (LVAG Index) - Chilean Cash (LVAM Index) - U.S. Stocks (S&P 500 Index) - U.S. Bonds (SBBIG Index) - Multi-Strategy Hedge Funds (CSFB/Tremont Index)

Assume also that the non-CLP denominated asset

classes can be perfectly and costlessly hedged in full if the investor so desires

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Optimal Portfolio Formation With Historical Returns: Examples (cont.)

Consider the formation of optimal strategic asset allocations under a wide variety of conditions:

- With standard definition of the optimization problem: - Mean-Variance Optimization

- With and without hedging non-CLP exposure - With and Without 50% Constraint on non-CLP Assets

Each of these optimization examples will:

- Use the set of historical returns directly rather than the underlying set of asset class risk and return parameters

- Be based on historical return data from two different three-year period chosen to be before and after the global financial market meltdown of 2008-2009:

- August 2004 – July 2007 & February 2012 – January 2015 - Restrict against short selling (except those short sales

embedded in the hedge fund asset class)

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Aug 2004-Jul 2007: Mean-Variance Optimization: Non-CLP Assets 100% Unhedged

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Aug 2004-Jul 2007: Unconstrained Efficient Frontier: 100% Unhedged

E(R) σp Relative σp vs. Unhdgd Wcs Wcb Wcc Wuss Wusb Whf

6.00% 1.01% 1.000 7.87% 0.00% 92.13% 0.00% 0.00% 0.00% 7.00% 1.53% 1.000 12.20% 0.00% 87.80% 0.00% 0.00% 0.00% 8.00% 2.07% 1.000 16.53% 0.00% 83.47% 0.00% 0.00% 0.00% 9.00% 2.60% 1.000 20.87% 0.00% 79.13% 0.00% 0.00% 0.00%

10.00% 3.14% 1.000 25.20% 0.00% 74.80% 0.00% 0.00% 0.00% 11.00% 3.68% 1.000 29.53% 0.00% 70.47% 0.00% 0.00% 0.00% 12.00% 4.22% 1.000 33.87% 0.00% 66.13% 0.00% 0.00% 0.00% 13.00% 4.76% 1.000 38.20% 0.00% 61.80% 0.00% 0.00% 0.00% 14.00% 5.30% 1.000 42.53% 0.00% 57.47% 0.00% 0.00% 0.00% 15.00% 5.84% 1.000 46.87% 0.00% 53.13% 0.00% 0.00% 0.00%

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One Consequence of the Unhedged M-V Efficient Frontier

Notice that because of the strengthening CLP/USD exchange rate over the August 2004 – July 2007 period, the optimal allocation for any expected return goal did not include any exposure to non-CLP asset classes

This “unhedged foreign investment” efficient frontier is equivalent to the efficient frontier that would have resulted from a “domestic investment only” constraint.

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Aug 2004-Jul 2007: Mean-Variance Optimization: Non-CLP Assets 100% Hedged

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Aug 2004-Jul 2007: Unconstrained M-V Efficient Frontier: 100% Hedged


6.00% 0.81% 0.806 3.14% 0.00% 74.93% 0.00% 7.77% 14.16% 7.00% 1.22% 0.794 4.74% 0.00% 62.02% 0.00% 10.94% 22.30% 8.00% 1.63% 0.790 6.34% 0.00% 49.12% 0.00% 14.10% 30.44% 9.00% 2.05% 0.788 7.95% 0.00% 36.21% 0.00% 17.26% 38.57%

10.00% 2.47% 0.787 9.55% 0.00% 23.31% 0.00% 20.43% 46.71% 11.00% 2.89% 0.786 11.16% 0.00% 10.41% 0.00% 23.59% 54.85% 12.00% 3.32% 0.786 13.01% 0.00% 0.00% 0.00% 24.80% 62.19% 13.00% 3.76% 0.790 15.90% 0.00% 0.00% 0.00% 17.85% 66.25% 14.00% 4.23% 0.797 18.79% 0.00% 0.00% 0.00% 10.90% 70.31% 15.00% 4.71% 0.806 21.69% 0.00% 0.00% 0.00% 3.95% 74.36%

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Feb 2012-Jan 2015: Mean-Variance Optimization: Non-CLP Assets 100% Unhedged

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Feb 2012-Jan 2015: Unconstrained Efficient Frontier: 100% Unhedged


3.00% 0.22% 1.000 0.48% 1.56% 97.38% 0.00% 0.58% 0.00% 4.00% 0.22% 1.000 0.48% 1.56% 97.38% 0.00% 0.58% 0.00% 5.00% 0.23% 1.000 0.00% 1.63% 97.43% 0.16% 0.68% 0.10% 6.00% 0.64% 1.000 0.00% 8.59% 86.03% 5.38% 0.00% 0.00% 7.00% 1.13% 1.000 0.00% 14.50% 75.74% 9.76% 0.00% 0.00% 8.00% 1.64% 1.000 0.00% 20.42% 65.45% 14.13% 0.00% 0.00% 9.00% 2.14% 1.000 0.00% 26.33% 55.17% 18.50% 0.00% 0.00%

10.00% 2.66% 1.000 0.00% 32.24% 44.88% 22.87% 0.00% 0.00% 11.00% 3.17% 1.000 0.00% 38.16% 34.60% 27.25% 0.00% 0.00% 12.00% 3.68% 1.000 0.00% 44.07% 24.31% 31.62% 0.00% 0.00%

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Feb 2012-Jan 2015: Mean-Variance Optimization: Non-CLP Assets 100% Hedged

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Feb 2012-Jan 2015: Unconstrained M-V Efficient Frontier: 100% Hedged


3.00% 0.22% 1.010 0.39% 1.62% 96.39% 0.00% 1.60% 0.00% 4.00% 0.22% 1.010 0.39% 1.62% 96.39% 0.00% 1.60% 0.00% 5.00% 0.27% 1.146 0.00% 4.91% 93.92% 1.17% 0.00% 0.00% 6.00% 0.79% 1.242 0.00% 19.44% 73.99% 6.57% 0.00% 0.00% 7.00% 1.40% 1.241 0.00% 33.97% 54.05% 11.98% 0.00% 0.00% 8.00% 2.02% 1.237 0.00% 48.50% 34.12% 17.38% 0.00% 0.00% 9.00% 2.65% 1.234 0.00% 63.03% 14.19% 22.78% 0.00% 0.00%

10.00% 3.30% 1.243 0.00% 68.92% 0.00% 31.08% 0.00% 0.00% 11.00% 4.34% 1.370 0.00% 53.47% 0.00% 46.53% 0.00% 0.00% 12.00% 5.65% 1.536 0.00% 38.02% 0.00% 61.98% 0.00% 0.00%

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Another Consequence of the Hedged Frontier

Notice that because of the weakening CLP/USD exchange rate over the February 2012 – January 2015 period, the optimal allocation for the CLP-hedged solution was uniformly riskier than the comparable unhedged allocation for the same return goal

Notice also that the set of possible return goals is lower

than in the earlier period due primarily to the effect that Quantitative Easing following the global economic crisis had on lowering interest rates

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A Related Question About Foreign Diversification

What allocation to foreign assets in a domestic investment portfolio leads to a reduction in the overall level of risk?

Van Harlow, formerly of Fidelity Investments, performed the following analysis:

- Consider a benchmark portfolio containing a 100% allocation to U.S. equities

- Diversify the benchmark portfolio by adding a foreign equity allocation in successive 5% increments

- Calculate standard deviations for benchmark and diversified portfolios using monthly return data over rolling three-year holding periods during 1970-2005

- For each foreign allocation proportion, calculate the percentage of rolling three-year holding periods that resulted in a risk level for the diversified portfolio that was higher than the domestic benchmark

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Portfolio Risk Reduction and Diversifying Into Foreign AssetsUnited States, 1970-2005

0%

5%

10%

15%

20%

25%

30%

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65%

Foreign Stock Allocation

Fre

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f H

ighe

r R

isk

(vs

Dom

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c O

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Rol

ling

3 Y

ear

Per

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197

0-20

05

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Foreign Diversification Potential (cont.) Ennis Knupp Associates (EKA) have provided an alternative way of quantifying the

diversification benefits of adding global stocks to a U.S. stock portfolio:

EKA concludes that global diversification adds an important element of risk control within an investment program; the optimal allocation from a statistical standpoint is approximately 30%-40% of total equities, although they generally favor a slightly lower allocation due to cost considerations.

Impact of Diversification on Volatility1971 - 2001

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

0 10 20 30 40 50 60 70 80 90 100Percentage in Foreign Stocks

Volati

lity

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More on Mean-Variance Optimization: The Cost of Foreign Investment Restrictions

Start with the following base case: - Six asset classes: Three Chilean, Three Foreign (Including

Hedge Funds) - No Short Sales - 100% Hedged Foreign Investments - No Constraint on Total Foreign Investment

Consider the addition of a foreign investment constraint:

- 50% Limit on Foreign Asset Classes

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Additional Constraints: 50% Foreign Investment: August 2004 – July 2007

E(R) σp Relative σp v. Uncnstrd Wcs Wcb Wcc Wuss Wusb Whf

6.00% 0.81% 1.000 3.14% 0.00% 74.93% 0.00% 7.77% 14.17% 7.00% 1.22% 1.000 4.74% 0.00% 62.02% 0.00% 10.94% 22.30% 8.00% 1.63% 1.000 6.34% 0.00% 49.12% 0.00% 14.10% 30.44% 9.00% 2.06% 1.002 8.62% 0.00% 41.38% 0.00% 13.52% 36.48%

10.00% 2.50% 1.012 11.54% 0.00% 38.46% 0.00% 9.45% 40.55% 11.00% 2.97% 1.025 14.45% 0.00% 35.55% 0.00% 5.37% 44.63% 12.00% 3.44% 1.036 17.36% 0.00% 32.64% 0.00% 1.29% 48.71% 13.00% 3.92% 1.042 21.25% 0.00% 28.75% 0.00% 0.00% 50.00% 14.00% 4.41% 1.044 25.58% 0.00% 24.42% 0.00% 0.00% 50.00% 15.00% 4.92% 1.045 29.91% 0.00% 20.09% 0.00% 0.00% 50.00%

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Additional Constraints: 50% Foreign Investment: February 2012 – January 2015

E(R) σp Relative σp v. Uncnstrd Wcs Wcb Wcc Wuss Wusb Whf

3.00% 0.22% 1.000 0.39% 1.62% 96.39% 0.00% 1.60% 0.00% 4.00% 0.22% 1.000 0.39% 1.62% 96.39% 0.00% 1.60% 0.00% 5.00% 0.27% 1.000 0.00% 4.91% 93.92% 1.17% 0.00% 0.00% 6.00% 0.79% 1.000 0.00% 19.44% 73.99% 6.57% 0.00% 0.00% 7.00% 1.40% 1.000 0.00% 33.97% 54.05% 11.98% 0.00% 0.00% 8.00% 2.02% 1.000 0.00% 48.50% 34.12% 17.38% 0.00% 0.00% 9.00% 2.65% 1.000 0.00% 63.03% 14.19% 22.78% 0.00% 0.00%

10.00% 3.30% 1.000 0.00% 68.92% 0.00% 31.08% 0.00% 0.00% 11.00% 4.34% 1.000 0.00% 53.47% 0.00% 46.53% 0.00% 0.00% 12.00% N/A N/A N/A N/A N/A N/A N/A N/A

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Optimal Portfolio Formation With Historical Returns: Summary

Forming optimal portfolios using historical asset class returns is a fairly simple and convenient way to assess the quality of the organization’s strategic asset allocation policy. - However, a major caveat when using historical optimization is the implicit

assumption that the future risk-return relationships within the asset classes—as well as the interactions between them—will remain the same as in the historical data.

Using mean-variance analysis over this particular sample period,

there was a significant difference in the level of risk generated by unhedged versus hedged portfolios for all levels of the specified return goal.

When a constraint on foreign investing was imposed, the optimal solution changes if the constrained becomes binding

- In the 2004-2007 period, the restriction did reduce the amount of foreign investment and raised the risk level of the optimal allocation by up to 5% for most return goals

- In the 2012-2015, the restriction was only binding for the highest return goal, but made it impossible to achieve that goal with any combination of assets that also satisfied that restriction

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Notion of Downside Risk Measures: The most commonly employed measure of market risk is the variance

(or standard deviation) of the potential returns associated with a particular investment.

Return variance is a two-sided risk measure in that it does not discriminate between the possibility of receiving a return that is higher than expected versus the possibility of receiving a return that is lower than expected

- This distinction does not matter when return distributions are symmetric. - Asymmetric return distributions commonly occur when portfolios contain

either explicit or implicit derivative positions (e.g., using a put option to provide portfolio insurance)

Investors clearly care most about receiving less return than they were promised. That is, they focus on the downside risk - Variance is a misleading risk measure when the downside and upside

outcomes in a portfolio are not symmetric (e.g., put option protection).

Two popular measures of downside risk: - Semi-Variance - Lower Partial Moment

Potential Returns Considered Risky: An Illustration

Variance:

𝜎2 = � 𝑝𝑖 [𝑅𝑖 − 𝐸 𝑅 ]2∞

𝑅𝑖=−∞

Semi-Variance:

𝑆𝑆𝑆𝑆 − 𝜎2 =

� 𝑝𝑖 [𝑅𝑖 − 𝐸 𝑅 ]2𝐸(𝑅)

𝑅𝑖=−∞

30

E(R)

E(R)

Illustrating Risky Returns (cont.)

Lower-Partial Moment:

𝐿𝐿𝐿(𝑛) =

� 𝑝𝑖 [𝜏 − 𝑅𝑖 ]𝑛𝜏

𝑅𝑖=−∞

Note that: - n = 0: Probability of Loss - n = 1: Target Shortfall - n = 2: Target Semi-Variance

31

E(R) τ

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Example of Downside Risk Measures: To see how these alternative risk statistics compare to the variance consider the following probability distributions for two investment portfolios: Potential Prob. of Return for Prob. of Return for Return Portfolio #1 Portfolio #2 -15% 5% 0% -10 8 0 -5 12 25 0 16 35 5 18 10 10 16 7 15 12 9 20 8 5 25 5 3 30 0 3 35 0 3 Notice that the expected return for both of these portfolios is 5%: E(R)1 = (.05)(-0.15) + (.08)(-0.10) + ...+ (.05)(0.25) = 0.05 and E(R)2 = (.25)(-0.05) + (.35)(0.00) + ...+ (.03)(0.35) = 0.05

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Portfolio Return Probability Distributions

0

5

10

15

20

25

30

35

40

-15 -10 -5 0 5 10 15 20 25 30 35

Ret

urn

Prob

abili

ty (%

)

Potential Return (%)

Portfolio #1 Return Distribution

0

5

10

15

20

25

30

35

40

-15 -10 -5 0 5 10 15 20 25 30 35

Ret

urn

Prob

abili

ty (%

)

Potential Return (%)

Portfolio #2 Return Distribution

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Example of Downside Risk Measures (cont.): Clearly, however, these portfolios would be viewed differently by different investors. These nuances are best captured by measures of return dispersion (i.e., risk).

1. Variance As seen earlier, this is the traditional measure of risk, calculated the sum of the weighted squared differences of the potential returns from the expected outcome of a probability distribution. For these two portfolios the calculations are: (Var)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + ... + (.05)[0.25 - 0.05]2 = 0.0108 and (Var)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 + ... + (.03)[0.35 - 0.05]2 = 0.0114 Taking the square roots of these values leaves:

SD1 = 10.39% and SD2 = 10.65%

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Example of Downside Risk Measures (cont.): 2. Semi-Variance

The semi-variance adjusts the variance by considering only those potential outcomes that fall below the expected returns. For our two portfolios we have: (SemiVar)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + (.12)[-0.05 - 0.05]2 + (.16)[0.00 - 0.05]2 = 0.0054 and

(SemiVar)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 = 0.0034 Also, the semi-standard deviations can be derived as the square roots of these values:

(SemiSD)1 = 7.35% and (SemiSD)2 = 5.81% Notice here that although Portfolio #2 has a higher standard deviation than Portfolio #1, it's semi-standard deviation is smaller.

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Example of Downside Risk Measures (cont.): 3. Lower Partial Moments

For these two portfolios, we will consider two cases (n = 1 and n = 2), both having a threshold level of 0% (i.e., τ = 0):

(i) LPM1 (LPM1)1 = (.05)[0.00 - (-0.15)] + (.08)[0.00 - (-0.10)] + (.12)[0.00 - (-0.05)] = 0.0215 and

(LPM1)2 = (.25)[0.00 - (-0.05)] = 0.0125

(ii) LPM2 (LPM2)1 = (.05)[0.00 - (-0.15)]2 + (.08)[0.00 - (-0.10)]2 + (.12)[0.00 - (-0.05)]2 = 0.0022 and

(LPM2)2 = (.25)[0.00 - (-0.05)]2 = 0.0006

For comparative purposes, it is also useful to take the square root of the LPM2 values. These are:

(SqRt LPM2)1 = 4.72% and (SqRt LPM2)1 = 2.50%

Notice again that Portfolio #2 is seen as being less risky when the lower partial moment risk measures are used.

Portfolio Optimization Case Study #1: University of Texas Investment Management Company

Background: The University of Texas Investment Management Company (UTIMCO) is a private company whose client is the public endowment fund holding the assets of the University of Texas and Texas A&M University Systems. It currently has about USD 26 billion under management.

Investment Problem: The Board of Directors of UTIMCO faces a multi-dimensional investment problem that involves both short- and intermediate-term funding needs for the various campuses in the UT and A&M systems as well as long-term growth goals. Although UT is a public university, the UTIMCO staff feels that it must produce investment returns that are comparable to the endowments of Harvard and Yale Universities.

Portfolio Optimization Application (Fall 2005 & Fall 2014): Mean-downside risk optimization approach across multiple asset classes, including U.S. equity, non-U.S. equity, fixed-income, private equity, hedge funds, and real estate.

Miscellaneous Issues: - The downside risk threshold is the funding rate that is projected by the System’s

Board of Regents, which consists of politically appointed members. - Cambridge Associates was the primary economic consultant to the UTIMCO

Board at the time

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UTIMCO: Initial Asset Allocation and Issues to Address

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Benchmark for Developed International and Emerging Markets

Target and Upper Limit Identical in Hedge Funds

Target and Upper Limit Identical in Private Equity

Target and Lower Limit Identical in Fixed Income

Remove REITS From US Equity Category

Remove TIPS From Fixed Income Category

Reinstate Inflation Hedge Category

Liquidity Policy is Inconsistent With Asset Allocation Policy

Asset Category Policy Target Policy Range Benchmarks

U S Equities (Includes REITs) 25.0 15 to 45 Combination benchmark: 80% Russell 3000 Index plus 20% Wilshire Associates Real Estate Securities Index

Traditional US Equities 20.0 15 to 45 Russell 3000 Index REITs 5.0 0 to 10 Dow Jones Wilshire Real Estate Securities IndexGlobal ex US Equities MSCI All Country World Index ex USNon-US Developed Equity 10.0 5 to 15Emerging Markets Equity 7.0 0 to 10 Total Equity 42.0 20 to 60Equity Hedge Funds 10.0 5 to 15 90 day T-Bills + 4%Absolute Return Hedge Funds 15.0 10 to 20 90 day T-Bills + 3% Total Hedge Funds 25.0 15 to 25Venture Capital 6.0 0 to 10Private Equity 9.0 5 to 15 Total Private Capital 15.0 5 to 15 Venture Economics Periodic IRR indexCommodities 3.0 0 to 10 GSCI minus 1%

Fixed Income (Includes TIPS) 15.0 10 to 30 Combination benchmark: 66.7% Lehman Brothers Aggregate Bond Index plus 33.3% Lehman Brothers TIPS Index

Traditional Fixed Income 10.0 10 to 30 Lehman Brothers Aggregate Bond Index TIPS 5.0 0 to 10 Lehman Brothers US TIPS IndexCash 0.0 0 to 5 90 day T-Bills

Percent of Portfolio (%)

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UTIMCO: Developing the Downside Risk Threshold

May, 2005 39

Policy Payout Rate 4.75

+ Expense Rate 0.35 Only expenses that are not netted from returns are included here

+ Inflation Rate 3.00 Have used CPI, but HEPI is more appropriate. HEPI is about 1% higher.

+ Safety Margin 0.00 A Safety Margin could be useful in avoiding the Purchasing Power Wall and other calamities

= MAR 8.10 %

Calculating the Minimum Acceptable Return (MAR)

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UTIMCO: Existing and Recommended Constraints

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Risk & Return Assumptions Summary: PVA Assumptions: Constraints:

Data Item Consultant Average Historical UTIMCO

2003UTIMCO

2005

▪ 75th Pct PVA ▪ 25th Pct PVA

V/A Spread

Capture Ratio

▪ Exp PVA ▪ Std Dev

UTIMCO 2003 with

PVA

UTIMCO 2005 with

PVA

2003 Minimum %

2003 Maximum %

2005 Minimum %

2005 Maximum %

US Equity 20% 100% 20% 100% Nominal Returns 8.85% 11.53% 8.50% 8.50% 2.50% 35% 0.88% 9.13% 9.38% Real Returns 6.37% 6.86% 5.50% 5.50% -2.50% 6.13% 6.38% Std Deviation 16.44% 15.82% 17.00% 17.00% 5.00% 3.71% 17.40% 17.40%Non-US Developed Equity 10% 100% 10% 100% Nominal Returns 8.85% 11.86% 8.50% 8.50% 3.00% 35% 1.05% 9.25% 9.55% Real Returns 6.38% 7.19% 5.50% 5.50% -3.00% 6.25% 6.55% Std Deviation 17.48% 16.77% 19.00% 19.00% 6.00% 4.45% 19.51% 19.51%Emerging Markets Equity 0% 10% 0% 15% Nominal Returns 10.34% 15.04% 11.00% 10.50% 10.00% 25% 2.50% 12.50% 13.00% Real Returns 7.86% 10.36% 8.00% 7.00% -10.00% 9.50% 10.00% Std Deviation 24.80% 23.25% 26.00% 26.00% 20.00% 14.83% 29.93% 29.93%Absolute Return Hedge Funds 0% 20% 0% 25% Nominal Returns 6.91% 10.79% 7.00% 7.00% 4.00% 25% 1.00% 8.00% 8.00% Real Returns 4.42% 6.12% 4.00% 4.00% -4.00% 5.00% 5.00% Std Deviation 6.49% 6.15% 7.50% 7.50% 8.00% 5.93% 9.56% 9.56%Equity Hedge Funds 0% 20% 0% 20% Nominal Returns 8.46% 10.48% 8.00% 8.00% 5.00% 25% 1.25% 9.25% 9.25% Real Returns 5.97% 5.81% 5.00% 5.00% -5.00% 6.25% 6.25% Std Deviation 8.37% 8.16% 11.00% 10.00% 10.00% 7.41% 13.26% 12.45%Venture Capital 0% 10% 0% 10% Nominal Returns 14.24% 15.16% 14.00% 14.00% 15.00% 15% 2.25% 16.25% 16.25% Real Returns 11.57% 10.49% 11.00% 11.00% -15.00% 13.25% 13.25% Std Deviation 31.63% 18.78% 30.00% 30.00% 30.00% 22.24% 37.34% 37.34%

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UTIMCO: Existing and Recommended Constraints (cont.)

May, 2005 41

Risk & Return Assumptions Summary: PVA Assumptions: Constraints:

Data Item Consultant Average Historical UTIMCO

2003UTIMCO

2005

▪ 75th Pct PVA ▪ 25th Pct PVA

V/A Spread

Capture Ratio

▪ Exp PVA ▪ Std Dev

UTIMCO 2003 with

PVA

UTIMCO 2005 with

PVA

2003 Minimum %

2003 Maximum %

2005 Minimum %

2005 Maximum %

Private Equity 0% 10% 0% 15% Nominal Returns 11.85% 11.32% 11.50% 11.50% 10.00% 20% 2.00% 13.50% 13.50% Real Returns 9.38% 6.65% 8.50% 8.50% -10.00% 10.50% 10.50% Std Deviation 28.25% 9.04% 20.00% 24.00% 20.00% 14.83% 24.90% 28.21%REITS 0% 10% 0% 10% Nominal Returns 7.89% 14.54% 7.50% 7.50% 3.00% 25% 0.75% 8.25% 8.25% Real Returns 5.41% 9.87% 4.50% 4.50% -3.00% 5.25% 5.25% Std Deviation 13.64% 14.74% 15.00% 15.00% 6.00% 4.45% 15.65% 15.65%Commodities (Financial) 0% 10% 0% 10% Nominal Returns 6.40% 13.37% 5.00% 6.00% 3.00% 25% 0.75% 5.00% 6.75% Real Returns 3.70% 8.70% 2.00% 3.00% -3.00% 2.00% 3.75% Std Deviation 18.47% 18.43% 18.00% 18.00% 6.00% 4.45% 18.00% 18.54%TIPS 0% 10% 0% 15% Nominal Returns 4.94% 9.07% 5.50% 5.50% 1.00% 25% 0.25% 5.50% 5.75% Real Returns 2.40% 4.39% 2.50% 2.50% -1.00% 2.50% 2.75% Std Deviation 6.00% 3.69% 6.00% 6.00% 0.00% 1.48% 6.00% 6.18%US Fixed Income 10% 100% 10% 100% Nominal Returns 5.18% 8.80% 5.00% 5.75% 1.00% 25% 0.25% 5.25% 6.00% Real Returns 2.70% 4.13% 2.00% 2.75% -1.00% 2.25% 3.00% Std Deviation 5.34% 6.02% 6.00% 7.00% 2.00% 1.48% 6.18% 7.16%Cash 0% 0% -10% 0% Nominal Returns 3.33% 6.43% 4.00% 4.00% 0.00% 0% 0.00% 4.00% 4.00% Real Returns 0.86% 1.75% 1.00% 1.00% 0.00% 1.00% 1.00% Std Deviation 0.88% 0.91% 1.00% 1.00% 0.00% 0.00% 1.00% 1.00%Inflation Returns 2.48% 4.67% 3.00% 3.00% 3.00% 3.00% Std Deviation 1.25% 1.17% 2.00% 1.50%

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UTIMCO: Mean-Downside Risk Optimization Candidate Policy Portfolios Derived From 2005 Capital Market Assumptions

May, 2005 42

2005 EFFICIENT FRONTIERConstrained vs Unconstrained Frontiers

7.75%7.95%

8.15%

8.55%8.75%

8.95%9.05%

9.65%

9.05%8.95%

8.75%8.55%

8.35%8.15%

7.95%7.75%

8.35%

7.5%

8.0%

8.5%

9.0%

9.5%

10.0%

5.0% 6.0% 7.0% 8.0% 9.0% 10.0% 11.0%1 Yr Downside Risk

Expe

cted R

eturns

Unconstrained (no short sales)

GEF w Constraints

2003 Asset Allocation Policy

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UTIMCO: Selecting a Strategic Asset Allocation

The portfolio optimization process—regardless how the investment problem is framed—results in an optimal set of asset allocations that are efficient in the sense each optimal allocation minimizes risk for a given return goal

Once the efficient frontier is established, investors must next answer the following question: Which single allocation (or range of allocations) from the efficient frontier is appropriate for them?

Decisions Factors represent one approach to this problem. A decision factor is a measure or characteristic which may be used to relate specific goals to a particular decision.

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UTIMCO: Asset Allocation – May 2016

May, 2005 44 3 - 44

Portfolio Optimization Case Study #2: Chilean Pension System (Source: Fidelity Investments)

Background: System of private pension accounts since 1980. Beneficiaries select among several different investment managers (i.e., AFPs), which in turn over five different asset allocation alternatives. Constraints exist as to how much non-CLP investment can occur and what form the foreign investments must take.

Investment Problem: What are the optimal strategic asset allocations for the Chilean pension funds?

Portfolio Optimization Application (Fall 2004): Augmented mean-variance optimization using three Chilean asset classes (stocks, bonds, cash) and four foreign asset classes (U.S. stocks, U.S. bonds, Developed Non-U.S. stocks, Developed Non-U.S. bonds)

Miscellaneous Issue: Optimization process uses the “Resampled Frontier” approach to reduce estimation error problems

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Chile: Base Case Assumptions

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond US Stock US Bond

Risk Premium 7.19% 2.65% 0.00% 4.89% 1.66% 5.83% 1.41%

Real Cash Return 0.60% 0.60% 0.60% 0.60% 0.60% 0.60% 0.60%

Expected Real Return 7.79% 3.25% 0.60% 5.49% 1.66% 6.43% 2.01%

Volatility 25.02% 6.75% 1.50% 12.57% 3.33% 14.75% 5.05%

Base Case Assumptions:

-Expected real returns based on 1954 – 2003 risk premiums -Real returns for developed market stocks and bonds are GDP-weighted excluding US (equally-weighted returns for stocks and bonds are 5.73% and 1.39%, respectively)

- Chilean risk-premium volatility estimates exclude the period 1/72 – 12/75

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Chile: Base Case Assumptions (cont.)

Chile Stock

Chile Bond

Chile Cash

Developed Stock


Domestic Stocks 100.00% 21.85% 13.51% 35.28% -20.91% 38.78% -23.13%Domestic Bonds 100.00% 31.04% 2.71% -0.98% -1.39% 2.37%Domestic Cash 100.00% 1.79% 10.31% 6.27% 3.77%DM Stocks 100.00% 26.01% 71.23% 11.06%DM Bonds 100.00% 7.54% 73.19%US Stocks 100.00% 16.56%US Bonds 100.00%

- Correlation matrix is based on real returns from the period 1/93 – 6/03 using Chilean inflation and based in Chilean pesos - Real returns for developed market stocks and bonds are GDP-weighted excluding US

3 - 47

Chile: Notion of a Resampled Efficient Frontier

Created by Richard Michaud, resampling is a Monte Carlo technique for estimating the inputs of a mean-variance efficient frontier that results in well-diversified portfolios.

Concept of a Resampled Efficient Frontier: - Take a random sample of observation from a universe of asset class

returns (e.g., 30 of 60 months) and calculate the efficient frontier - Divide this efficient frontier into 20 regions by risk or expected return

and look at the median allocation in each of these regions - Repeat these steps for a new sampling of the asset class return

universe - Generate a large collection of efficient frontiers by repeated sampling of

the return universe (e.g., 500-1000 trials) - Average all of the “regional” allocations across the collection of

optimization trials – this is the resampled efficient frontier

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Chile: Notion of a Resampled Efficient Frontier (cont.)

Resampling provides a more realistic and reliable risk/return structure

Robust estimate of underlying distributions

While the weights on the actual frontier change erratically, the resampled weights are evenly distributed along the points on the efficient frontier

With the actual efficient frontier, a marginal change in risk or return can bring about a dramatic change in the optimal allocation. With the resampled frontier, the changes in weights are always smooth

Potential shortcomings of resampling:

Lack of theory (i.e., no reason why resampled portfolios will be optimal)

No framework for incorporating tactical views

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Chile: Traditional vs. Resampled Efficient Frontier

Asset Weights along Resampled Frontier

Scenario 1

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Points along Resampled Efficient Frontier

Asset 4

Asset 3

Asset 2

Asset 1

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Chile: Base Case Unconstrained Resampled Frontier (cont.)

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Chile: Base Case Unconstrained Resampled Frontier (cont.)

Unconstrained Frontier:

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Chile: Modifying the Unconstrained Optimization

Fund A Fund B Fund C Fund D Fund EChile Stock 60% 50% 30% 15% 0%Chile Bond 40% 40% 50% 70% 80%Chile Cash 40% 40% 50% 70% 80%

All Foreign Investments 30% 30% 30% 30% 30%Min Total Equity 40% 25% 15% 5% 0%Max Total Equity 80% 60% 40% 20% 0%

Constraint Set:

3 - 53

Chile: Modifying the Unconstrained Optimization (cont.)

Constrained Frontier for Fund A:

Point on EF

Chile Stock

Chile Bond

Chile Cash

Developed Stock


Expected Return Volatility

1 10.0% 20.0% 40.0% 24.2% 0.0% 5.8% 0.0% 3.37 5.65 2 10.6% 21.0% 38.4% 22.3% 0.2% 7.2% 0.3% 3.43 5.73 3 11.3% 22.4% 36.3% 20.7% 0.5% 8.2% 0.6% 3.51 5.85 4 12.1% 23.8% 34.1% 19.3% 0.7% 9.0% 0.9% 3.59 6.00 5 13.2% 24.9% 32.1% 18.0% 0.9% 9.6% 1.3% 3.68 6.18 6 14.4% 25.7% 30.2% 16.9% 1.1% 10.0% 1.7% 3.76 6.38 7 15.7% 26.3% 28.3% 15.9% 1.4% 10.3% 2.0% 3.85 6.62 8 17.2% 26.9% 26.4% 15.0% 1.6% 10.6% 2.3% 3.95 6.89 9 18.7% 27.2% 24.5% 14.2% 1.9% 10.9% 2.6% 4.05 7.19

10 20.3% 27.5% 22.7% 13.5% 2.1% 11.0% 2.8% 4.16 7.51 11 22.1% 27.7% 20.9% 13.0% 2.3% 11.1% 3.0% 4.27 7.86 12 23.9% 27.8% 19.3% 12.4% 2.4% 11.1% 3.2% 4.38 8.23 13 25.8% 27.8% 17.8% 11.9% 2.3% 11.2% 3.3% 4.49 8.63 14 27.8% 27.6% 16.4% 11.4% 2.3% 11.3% 3.2% 4.61 9.06 15 29.9% 27.2% 15.3% 10.9% 2.1% 11.4% 3.2% 4.74 9.53 16 32.3% 26.7% 14.1% 10.5% 2.0% 11.5% 3.1% 4.87 10.04 17 34.8% 25.9% 12.9% 10.1% 1.8% 11.5% 2.9% 5.01 10.61 18 37.6% 25.1% 11.6% 9.6% 1.7% 11.6% 2.8% 5.17 11.23 19 40.9% 24.2% 10.3% 8.8% 1.4% 11.6% 2.7% 5.34 11.97 20 45.6% 23.0% 8.0% 7.9% 1.2% 12.2% 2.2% 5.63 13.08

3 - 54

Chile: Comparing Optimal Allocations Across Constraints

Chile Stock

Chile Bond

Chile Cash

Developed Stock

Developed Bond

US Stock

US Bond

Expected Return Volatility

Unconstrained 42.8% 11.2% 0.0% 15.2% 1.6% 26.0% 3.2% 6.3% 13.9%Fund A 45.6% 23.0% 8.0% 7.9% 1.2% 12.2% 2.2% 5.6% 13.1%Fund B 35.5% 32.2% 10.1% 6.1% 1.7% 11.1% 3.3% 5.0% 10.6%Fund C 18.6% 40.3% 15.7% 6.0% 2.5% 10.7% 6.2% 4.0% 6.9%Fund D 6.5% 54.9% 14.5% 4.9% 5.4% 6.7% 7.1% 3.3% 4.9%Fund E 0.0% 63.8% 15.3% 0.0% 7.4% 0.0% 13.5% 2.6% 4.5%

Asset Allocations of Various Funds Using Point 20 on Unconstrained Frontier:

3 - 55

The Black-Litterman Optimization Process

Recall from our earlier discussion that the Black-Litterman (BL) model uses a quantitative technique known as reverse optimization to determine the implied returns for a series of asset classes that comprise the investment universe.

The main insight of the BL model is that if the global capital markets are in equilibrium, then the prevailing market capitalizations of these asset classes suggest the investment weights of an efficient portfolio with the highest Sharpe Ratio (i.e., risk premium per unit of risk) possible.

These investment weights can then be used, along with information about asset class standard deviations and correlations, to transform the user’s forecast of the global risk premium into asset class-specific risk premia (and expected returns) that are consistent with a capital market that is in equilibrium.

These equilibrium expected returns for the asset classes can then be used as inputs in a mean-variance portfolio optimization process or adjusted further given the user’s tactical views on asset class performance.

3 - 56

The Black-Litterman Process: An Example

Consider an investable universe consisting of the following five asset classes:

- US Bonds - Global Bonds-ex US - US Equity - Global Equity-ex US - Emerging Market Equity

As of May 2016, these asset classes had the following market capitalizations (in USD millions):

- US Bonds $19,982,690 (22.16%) - Global Bonds-ex US 24,680,102 (27.36%) - US Equity 20,131,204 (22.32%) - Global Equity-ex US 21,322,124 (23.64%) - Emerging Market Equity 4,078,236 ( 4.52%) Total: $90,194,356

3 - 57

Black-Litterman Example (cont.)

Consider also the following historical return standard deviations (January 1999 – April 2016): σusb = 3.51% σgb = 8.47% σuss = 15.14%

σgs = 17.61% σems = 23.00%

The historical correlation matrix, measured using all available pairwise historical return data: ρusb,gb = 0.5175 ρgb,gs = 0.4026 ρusb,uss = -0.0501 ρgb,ems = 0.2896 ρusb,gs = -0.0290 ρuss,gs = 0.8463 ρusb,ems = -0.0468 ρuss,ems = 0.7463 ρgb,uss = 0.1738 ρgs,ems = 0.8656

3 - 58


The remaining inputs that the user must specify are: (i) the global risk premium of the investment universe, and (ii) the risk-free rate. Using current market data we have:

- Global Risk Premium: 4.41% (10-yr Global Balanced) - Risk-Free Rate: 1.81% (10-yr US Treasury)

The heart of the BL process is to then calculate the

implied excess return for each asset class, using the following (stylized) formula:

[Risk Aversion Parameter] x [Covariance Matrix] x [Market Cap Weight Vector]

3 - 59


The risk aversion parameter is the rate at which more return is required as compensation for more risk. It is calculated as:

RAP = [Global Risk Premium] / [Market Portfolio Variance]

It can be shown in this example that the market portfolio variance is (9.25%)2 = 0.856%, so that:

RAP = (0.0441)/(0.00856) = 5.15

The covariance between two asset classes (Y and Z) is given by the formula:

Cov(Y,Z) = ρy,z x σy x σz

For instance, the covariance between US Equity and Global Equity-ex US is: (0.8463) x (15.14%) x (17.61%) = 0.023

3 - 60


The implied excess return (IER) for US Equity can then be computed as follows:

IERuss = (RAP) x {[Cov(uss,usb) x wusb] + [Cov(uss,gb) x wgb] +

… + [Cov(uss,ems) x wems]}

= (5.15) x {(0.000)(.2216) + … + (0.026)(.0452)} = 6.27% More formally, the solution for the entire asset class implied excess

return vector is given by:

0.30% 0.001 0.002 0.000 0.000 0.000 22.16% 2.31% 0.002 0.007 0.002 0.006 0.006 27.36% 6.27% = (5.15) x 0.000 0.002 0.023 0.022 0.026 x 22.32% 8.02% 0.000 0.006 0.023 0.031 0.035 23.64% 9.24% 0.000 0.006 0.026 0.035 0.053 4.52%

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The total expected return for US Equity is then simply

the IER plus the risk-free rate: 1.81% + 6.27% = 8.08%

The excess and total expected returns for the five asset

classes in this example are:

Excess Total - US Bonds: 0.30% 2.11% - Global Bonds: 2.31% 4.12% - US Equity: 6.27% 8.08% - Global Equity: 8.02% 9.83% - Emerging Equity: 9.24% 11.05%

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Black-Litterman Example: Excel Spreadsheet

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Black-Litterman Example: Proprietary Software (Zephyr Associates)

3 - 64

Mean-Variance Optimization with Black-Litterman Inputs

One of the criticisms that is sometimes made about the mean-variance optimization process that we have just seen is that the inputs (e.g., asset class expected returns, standard deviations, and correlations) must be estimated, which can effect the quality of the resulting strategic allocations.

Typically, these inputs are estimated from historical return data. However, it has been observed that inputs estimated with historical data—the expected returns, in particular—lead to “extreme” portfolio allocations that do not appear to be realistic.

Black-Litterman expected returns are often preferred in practice for the use in mean-variance optimizations because the equilibrium-consistent forecasts lead to “smoother”, more realistic allocations.

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BL Mean-Variance Optimization Example

3 - 66

Recall the implied expected returns and other inputs from the earlier example:

BL Mean-Variance Optimization Example (cont.)

3 - 67

These inputs can then be used in a standard mean-variance optimizer:


3 - 68

This leads to the following optimal allocations (i.e., efficient frontier):


3 - 69

3 - 70

Risk Parity and Asset Allocation: A Brief Overview

A common approach to allocating financial capital across a universe of asset classes is to think directly about how to deploy the available investment dollars in an “optimal” manner - An example of this is the celebrated mean-variance optimization approach of Markowitz, whereby the optimal asset class allocations are derived by minimizing portfolio risk subject to a specific return goal

This “efficient portfolio” approach to strategic asset allocation has been extremely useful in practice—albeit to varying degrees—over the past 60 years. However, it does have myriad shortcomings:

- The process requires estimates of several asset class investment characteristics: expected returns, standard deviations, correlations

- It is prone to producing “corner solutions” (i.e., extreme over- or under-allocations) when using historical data over abnormal past periods

- Some of the input variables (e.g., asset class correlations) are known to be quite unstable over time, which can lead to fragile solutions

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Risk Parity and Asset Allocation: Overview (cont.)

In some ways, a more important criticism of mean-variance optimization is that, by focusing of the allocation of capital rather than the way in which risk is allocated, the approach can lead to inefficient concentrations of assets for most return goals

- For instance, an investor with a higher return goal will invariably put more capital into higher risk assets that promise higher payoffs which leads to fund solutions that, while falling on the Efficient Frontier, may be dominated by other potential portfolios

- A consequence of this approach is that higher volatility assets tend to have an disproportionate impact on the risk of the total portfolio that may be out of sync with how the dollars are allocated in the fund

The basic idea of a risk parity approach to asset allocation is for the

investor to commit capital in the portfolio so as to equalize the risk contribution of each asset class

- Stated more plainly: Risk Parity = Equal Risk Contribution to the Total Portfolio by each asset class in the investable universe

- Notice that risk parity begins with the idea that it is the risk allocation that matters, not the dollar allocation (which then implies the level of portfolio risk)

3 - 72


The Risk Parity approach to asset allocation has an investment implication that is important to understand:

- The Risk Parity portfolio (i.e., the tangency portfolio proxy) will typically have an expected return that is less than the investor’s goal

- This means that a 100% allocation to the Risk Parity portfolio is not appropriate for an investor with a higher return hurdle

There are two ways that an investor desiring a higher expected return goal than that offered by the Risk Parity portfolio can react:

- The Markowitz solution would be to adjust the contents of the Risk Parity portfolio to increase the allocation to higher risk assets.

- The Risk Parity solution would be to use leverage to buy more of the same Risk Parity portfolio which maximizes the expected “reward-to-risk” ratio

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So, the ultimate distinction between the two investment approaches is how the investor increases the risk necessary to generate the higher expected return goal

- Markowitz: Alter the holdings of the Risk Parity portfolio to increase the allocation to riskier assets (and decrease the allocation to less-risky assets), using the same level of investment capital (i.e., no borrowing): #2 to #1

- Risk Parity: Borrow money to buy more of the original Risk Parity portfolio. Thus, the composition of the portfolio does not change and the additional risk comes from using financial leverage: #2 to #3 (or #4)

4

Asset Allocation with Risk Parity: An Example Suppose we have the following information about the

volatilities (i.e., standard deviations) and correlation coefficients for a four-asset class portfolio:

Also, although not necessary for the Risk Parity allocation computations, let us also consider the expected returns for these four asset classes as well as the risk-free rate:

E(R)1 = 9.1%, E(R)2 = 8.0%, E(R)3 = 5.1%, E(R)4 = 5.4%, RF = 2.7%

3 - 74

σ1 = 16.7% ρ12 = 0.71 ρ23 = 0.58

σ2 = 12.8 ρ13 = 0.42 ρ24 = 0.30

σ3 = 6.9 ρ14 = 0.22 ρ34 = 0.10

σ4 = 7.0

Risk Parity Example: Total Risk Contribution Approach

The best approach to constructing a Risk Parity portfolio insures that each asset class actually contributes equally to portfolio risk, but it comes at the expense of considerably more computational complexity

Specifically, the Equal Total Contribution to Risk approach

requires the investor to specify asset class correlations and to solve the following non-linear optimization problem:

- :

Select {Wi} so as to Minimize σp subject to: (i) ∑𝑊𝑖 = 1 (ii) [%TCR]1 = [%TCR]2 = … = [%TCR]N (iii) All Wi > 0

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Risk Parity Example: Total Risk Contribution Approach (cont.)

Since there are four asset classes in this example, the relevant constraint in the problem to be solved is:

[%TCR]1 = [%TCR]2 = [%TCR]3 = [%TCR]4 = 0.25 Solving the optimization problem in this example

generates the following Risk Parity asset allocation percentages:

W1 = 12.70% W2 = 15.00% W3 = 33.76% W4 = 38.55%

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The expected return and volatility level for this second approach to a Risk Parity Portfolio are:

E(R)p = (.1270)(.091)+(.1500)(.080)+(.3376)(.051)+(.3855)(.054) = 6.16% and:

σp = {[(.1270)2(.167)2+(.1500)2(.128)2+(.3376)2(.069)2+(.3855)2(.070)2] + [2(.1270)(.1500)(.167)(.128)(0.71)+2(.1270)(.3376)(.167)(.069)(0.42) +2(.1270)(.3855)(.167)(.070)(0.22) +2(.1500)(.3376)(.128)(.069)(0.58)

+2(.1500)(.3855)(.128)(.070)(0.30)+2(.3376)(.3855)(.069)(.070)(0.10)]}1/2 = 6.55%

The ex ante Sharpe Ratio for this Risk Parity Portfolio is:

𝑆𝑝 = (6.16 − 2.70)

6.55= 𝟎.𝟓𝟓𝟓

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3 - 78

0%

20%

40%

60%

80%

100%

1 2 3 4

Dollar Weight Risk Contribution

Implementing a Risk Parity Asset Allocation Scheme

3 - 79

Mean-Variance Efficient Portfolios

E(Rp) σp Sharpe Ratio 3.00% 2.92% 0.103 3.50% 3.41% 0.235 4.00% 3.90% 0.333 4.50% 4.39% 0.410 5.00% 4.87% 0.472 5.50% 5.37% 0.522 6.00% 6.17% 0.535 6.16% 6.49% 0.533 6.30% 6.79% 0.530 6.50% 7.26% 0.523 7.00% 8.54% 0.503 7.50% 10.00% 0.480 8.00% 11.67% 0.454 8.50% 13.47% 0.431 9.00% 16.03% 0.393

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

0.00% 5.00% 10.00% 15.00% 20.00%

Port

folio

Exp

ecte

d R

etur

n

Portfolio Volatility

Markowitz Efficient Frontier

Implementing a Risk Parity Asset Allocation Scheme (cont.)

Suppose the investor has a 7.50% return goal. How would she achieve that goal using the risk parity and mean-variance optimization approaches?

Notice two things about this mean-variance optimal allocation associated with a 7.50% return goal:

- It makes no allocation at all one of the four asset classes (i.e., Asset Class 3) - It produces an inferior ex ante Sharpe Ratio to that of the best available Risk

Parity portfolio (i.e., 0.480 vs. 0.528)

Comparing the allocation schemes for the 7.50% mean-variance portfolio and the best Risk Parity portfolio:

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Allocation: Mean-Variance: Risk Parity: W1 31.08% 12.70%

W2 36.54 15.00

W3 0.00 33.76

W4 32.38 38.55

Sharpe Ratio: 0.480 0.528


The Risk Parity portfolio has a very different asset allocation than the Mean-Variance efficient portfolio associated with the return goal and has a better reward-to-risk ratio

- However, the Risk Parity allocation only generates an expected return of 6.16%, which falls short of the investor’s 7.50% required return

- So, the investor will have to borrow additional funds to buy more of the Risk Parity portfolio to achieve the higher return goal

Assuming for simplicity that the investor can either borrow or

lend money at the risk-free rate (RF = 2.70% in this example), the amount of gross leverage required—call it W*—can be calculated be solving the following equation, which is based on the notion that the expected return from combining two investment portfolios is just a weighted average of the separate expected returns:

7.50% = W* ∙ (6.16%) + (1 – W*) ∙ (2.70%) 3 - 81


Solving for W* in this case leaves:

𝑊∗ = (7.50 −2.70)(6.16 −2.70)

= 1.3873

so that (1 – W*) = -0.3873 (i.e., a short position in RF). - This means that the investor will have to borrow 38.73 cents per every dollar of their initial capital in order to buy enough of the risk parity portfolio to generate the 7.50% required return

Intuitively, the return the investor expects to receive

comes from the net amount of two activities: - Investing 1.3873 at 6.16%: 1.3873 x 6.16% = 8.546% - Borrow 0.3873 at 2.70%: -0.3873 x 2.70% = -1.046%

Net Expected Return: 7.500%

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Additionally, assuming that the risk-free asset has no volatility or correlation with the Risk Parity portfolio—which is reasonable for the investor who is borrowing at this rate—the volatility of this position is:

σ* = (Gross Leverage) x (σrisk parity) = (1.3873) x (6.55%) = 9.09%

which is substantially less that the σ = 10.00% for the mean-variance efficient portfolio for this expected return goal Graphically, these concepts can be illustrated as follows:

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E(R)p

σp

7.50%

2.70%

Risk Parity Portfolio

1.3873 of Risk Parity Portfolio

Mean-Variance Efficient Frontier

Risk Parity Transformation Line

10.00% 9.09%

x

x x

portfolio optimization and setting the strategic asset...

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