Analyzing Private Equity within a Portfolio Context

A constrained optimization program is run to solve the equation at each level of risk aversion.16 In Exhibit 11 we present the base case of maximizing utility without using any class of private equity. We can see that a risk neutral investor allocates her entire portfolio to small-cap stocks. This is because the risk-neutral investor is not encumbered by concerns over risk; maximizing return is all that matters. An investor who is unconcerned with risk will not have her behavior affected by the volatilities or correlations of the various asset classes. This type of investor invests in the asset class that yields the highest expected returns: small-capitalization stocks.

However, as the investor's level of risk aversion increases, we see that she diversifies her portfolio between small-cap stocks and the less volatile investment-grade bonds. The reason is that these asset classes have less than perfect correlation with each other. By diversifying across a number of asset classes, the investor can reduce the volatility of her investment portfolio. This volatility dampening effect has greater utility as the level of risk aversion increases.

At At = 3, the high risk averse investor shifts almost three-fourths of her portfolio away from small-cap stocks and into investment-grade bonds. No allocation is made to cash at any level of risk aversion. This is due to the relatively low returns earned by this asset class over the time period studied.

The results for each category of private equity are presented in Exhibits 12 through 15. These exhibits demonstrate that, as an investor's risk aversion increases, so does her allocation to private equity.

16 To solve the utility maximization equation, we program an optimization as follows:

Maximize E(U) = XiWjEiR,) - ^^WjVPjPj subject to the constraints Yw>j = 1, and 0 < Wj < 1, where Ai = 0, 1, 2, 3 for different levels of relative risk aversion.

Risk Neutral Investor (A = 0)

Tbill

SBBIG

NASDAQ

S&P 500

0.000

0.000

1.000

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.048

0.048

0.135

0.261

Low Risk Aversion (A, = 1)

Tbill

SBBIG

NASDAQ

S&P 500

0.000

0.231

0.769

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.031

0.041

0.104

0.277

Moderate Risk Aversion (At = 2)

Tbill

SBBIG

NASDAQ

S&P 500

0.000

0.601

0.399

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.025

0.031

0.055

0.334

High Risk Aversion (At = 3)

Tbill

SBBIG

NASDAQ

S&P 500

0.000

0.724

0.276

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.023

0.028

0.040

0.373

In Exhibit 12, we see that a risk neutral investor will allocate all of her portfolio to venture capital. The reason is the superior returns earned by venture capital over this time period (it even outperforms small-cap stocks). The superior returns of venture capital are enough to outweigh its risk except for the high risk averse investor. We can see that such an investor diversifies her portfolio between venture capital and investment-grade bonds.

In Exhibit 13, with respect to LBOs, the low risk, moderate risk, and high risk aversion investor allocates approximately 64%, 84%, and 90%, respectively, of her portfolio to leveraged buyout investments. The allocation to LBOs is made to balance the risk of small-cap stocks. Small-cap stocks outperformed LBO firms over this time period, but they were also riskier. Therefore, as an investor's level of risk aversion increases, she makes a larger and larger allocation to LBO firms to balance the risk of small-cap stocks in her portfolio.

Risk Neutral Investor (A = 0)

Tbill

SBBIG

NASDAQ

S&P 500

Venture

O.OOO

O.OOO

O.OOO

O.OOO

1.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O7l

O.O7l

0.107

O.545

Low Risk Aversion (A; = 1)

Tbill

SBBIG

NASDAQ

S&P 500

Venture

O.OOO

O.OOO

O.OOO

O.OOO

1.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O6O

O.O7l

0.107

O.545

Moderate Risk Aversion (A; = 2)

Tbill

SBBIG

NASDAQ

S&P 5OO

Venture

O.OOO

O.OOO

O.OOO

O.OOO

1.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.048

O.O7l

0.107

O.545

High Risk Aversion (A; = 3)

Tbill

SBBIG

NASDAQ

S&P 5OO

Venture

O.OOO

O.269

O.OOO

O.OOO

O.73l

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O39

O.O57

O.O77

O.578

Exhibit 13: Maximizing Utility with Leveraged Buyouts

Risk Neutral Investor (A = 0)

Tbill

SBBIG

NASDAQ

S&P 5OO

LBO

O.OOO

O.OOO

1.000

O.OOO

O.OOO

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O48

O.O48

0.135

0.261

Low Risk Aversion (A; = 1)

Tbill

SBBIG

NASDAQ

S&P 5OO

LBO

O.OOO

O.OOO

O.354

O.OOO

O.646

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O36

O.O4O

O.O64

0.431

Moderate Risk Aversion (A; = 2)

Tbill

SBBIG

NASDAQ

S&P 5OO

LBO

O.OOO

O.OOO

0.159

O.OOO

0.841

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

O.O33

O.O38

O.O48

O.523

High Risk Aversion (A; = 3)

Tbill

SBBIG

NASDAQ

S&P 5OO

LBO

O.OOO

O.OOO

O.O94

O.OOO

O.9O6

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.031

O.O37

O.O45

O.547

Risk Neutral Investor (A = 0)

Tbill

SBBIG

NASDAQ

S&P 500

Mezzanine

0.000

0.000

1.000

0.000

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.048

0.048

0.135

0.261

Low Risk Aversion (A, = 1)

Tbill

SBBIG

NASDAQ

S&P 500

Mezzanine

0.000

0.000

0.569

0.000

0.431

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.033

0.039

0.077

0.347

Moderate Risk Aversion (A, = 2)

Tbill

SBBIG

NASDAQ

S&P 500

Mezzanine

0.000

0.000

0.320

0.000

0.680

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.030

0.034

0.047

0.464

High Risk Aversion (A, = 3)

Tbill

SBBIG

NASDAQ

S&P 500

Mezzanine

0.000

0.000

0.237

0.000

0.763

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.028

0.033

0.039

0.518

Similar results are presented in Exhibits 14 and 15 for mezzanine debt and distressed debt. A risk neutral investor will select small-cap stocks over either of these forms of debt due to the greater return earned by small-cap stocks. However, as the level of risk aversion increases, an investor will allocate a larger part of her portfolio to either mezzanine debt or distressed debt to balance the risks associated with small cap stocks. The allocation to mezzanine debt and distressed debt increases as an investor's level of risk aversion increases.

The important point of this analysis is that the Sharpe ratios and utility values for each category of private equity in Exhibits 12 through 15 dominate those in Exhibit 11 (which contains no component of private equity). Simply stated, the investor achieves a higher expected utility and a higher Sharpe ratio at each level of risk aversion, and with each component of private equity than she does without a component of private equity in her portfolio. 17

17 As a final point, the marginal utility of private equity is positive at each level of risk aversion where the marginal utility is determined by dE(U)/dwpe = E(Rpe) — 2AlGpe'LjwpjPpej. See Anson, "Maximizing Utility with Private Equity."

Risk Neutral Investor (A = 0)

Tbill

SBBIG

NASDAQ

S&P 500

Distressed

0.000

0.000

1.000

0.000

0.000

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.048

0.048

0.135

0.261

Low Risk Aversion (At = 1)

Tbill

SBBIG

NASDAQ

S&P 500

Distressed

0.000

0.000

0.282

0.000

0.718

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.037

0.041

0.061

0.461

Moderate Risk Aversion (At = 2)

Tbill

SBBIG

NASDAQ

S&P 500

Distressed

0.000

0.000

0.104

0.000

0.896

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.034

0.039

0.049

0.541

High Risk Aversion (A, = 3)

Tbill

SBBIG

NASDAQ

S&P 500

Distressed

0.000

0.000

0.045

0.000

0.955

Expected Utility

Expected Return

Standard Deviation

Sharpe Ratio

0.032

0.038

0.046

0.559

In summary, the four categories of private equity investing demonstrated positive portfolio benefits. In fact, private equity replaced investment-grade bonds (from Exhibit 11) as an effective portfolio diversifying agent. Each component of private equity improved the risk to reward performance of the investment portfolio.

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