## Abstract

This thesis concerns risk and performance of private equity funds. Private equity funds are illiquid investments with mostly unobservable returns and with special institutional rules distinguishing them from other assets typically considered in finance. This thesis studies (i) how to risk-adjust the performance of private equity funds using data on cash flows instead of returns, and (ii) how to optimally allocate capital between private equity and publicly traded assets. It consists of three chapters which can be read independently.

The first chapter concerns risk adjustment of private equity cash flows. Recent liter-ature has developed methods to risk-adjust private equity cash flows using stochastic discount factors (SDFs). In this chapter, we find that those methods result in unrealistic time discounting, which can generate implausible performance estimates. We propose and evaluate a modified method which estimates a set of SDF parameters so that the subjective term structure of interest rates is determined by market data. Our method is based on a decomposition of private equity performance in a risk-neutral part and a risk adjustment, and it keeps the risk-neutral part constant as we add or remove risk factors from the SDF. We show that (i) our approach allows for economically meaning-ful measurement and comparison of risk across models, (ii) existing methods estimate implausible performance when time discounting is particularly degenerate, and (iii) our approach results in lower variation of performance across funds.

The second and third chapters study optimal portfolio allocation with private equity funds and publicly traded assets. In the second chapter, we study the portfolio problem of an investor (or limited partner, LP) that invests in stocks, bonds, and private equity funds. Stocks and bonds are liquid assets, while private equity is illiquid. The LP repeat-edly commits capital to private equity funds. This capital is only gradually contributed and eventually distributed back to the LP, requiring the LP to hold a liquidity buffer for its uncalled commitments. We solve the problem numerically for LPs with different risk aversion, and we find that optimal private equity allocation is not monotonically declining in risk aversion, despite private equity being riskier than stocks. We investigate the optimal dynamic investment strategy of two LPs at opposite ends of the risk aversion spectrum, and we find two qualitatively different strategies with intuitive heuristics. Fur-ther, we introduce a secondary market for private equity partnership interests to study optimal trading in this market and implications for the LP’s optimal investments.

The third chapter considers a portfolio problem with private equity and several liquid assets. This chapter focuses on average portfolio allocation over time, as opposed to dynamic strategies generating that allocation, and derives an approximate closed-form solution despite complex private equity dynamics. In this chapter, optimal portfolio al-location is well approximated by static mean-variance optimization with margin require-ments. Margin requirements are self-imposed by the investor, and because private equity needs capital commitment, the investor assigns greater margin requirement to private equity than liquid assets. Due to that greater margin requirement, the risky portfolio of constrained investors can optimally underweight private equity relative to the tangency portfolio, even when private equity has positive alpha and moderately high beta with respect to liquid assets.

The first chapter concerns risk adjustment of private equity cash flows. Recent liter-ature has developed methods to risk-adjust private equity cash flows using stochastic discount factors (SDFs). In this chapter, we find that those methods result in unrealistic time discounting, which can generate implausible performance estimates. We propose and evaluate a modified method which estimates a set of SDF parameters so that the subjective term structure of interest rates is determined by market data. Our method is based on a decomposition of private equity performance in a risk-neutral part and a risk adjustment, and it keeps the risk-neutral part constant as we add or remove risk factors from the SDF. We show that (i) our approach allows for economically meaning-ful measurement and comparison of risk across models, (ii) existing methods estimate implausible performance when time discounting is particularly degenerate, and (iii) our approach results in lower variation of performance across funds.

The second and third chapters study optimal portfolio allocation with private equity funds and publicly traded assets. In the second chapter, we study the portfolio problem of an investor (or limited partner, LP) that invests in stocks, bonds, and private equity funds. Stocks and bonds are liquid assets, while private equity is illiquid. The LP repeat-edly commits capital to private equity funds. This capital is only gradually contributed and eventually distributed back to the LP, requiring the LP to hold a liquidity buffer for its uncalled commitments. We solve the problem numerically for LPs with different risk aversion, and we find that optimal private equity allocation is not monotonically declining in risk aversion, despite private equity being riskier than stocks. We investigate the optimal dynamic investment strategy of two LPs at opposite ends of the risk aversion spectrum, and we find two qualitatively different strategies with intuitive heuristics. Fur-ther, we introduce a secondary market for private equity partnership interests to study optimal trading in this market and implications for the LP’s optimal investments.

The third chapter considers a portfolio problem with private equity and several liquid assets. This chapter focuses on average portfolio allocation over time, as opposed to dynamic strategies generating that allocation, and derives an approximate closed-form solution despite complex private equity dynamics. In this chapter, optimal portfolio al-location is well approximated by static mean-variance optimization with margin require-ments. Margin requirements are self-imposed by the investor, and because private equity needs capital commitment, the investor assigns greater margin requirement to private equity than liquid assets. Due to that greater margin requirement, the risky portfolio of constrained investors can optimally underweight private equity relative to the tangency portfolio, even when private equity has positive alpha and moderately high beta with respect to liquid assets.

Original language | English |
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Place of Publication | Frederiksberg |
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Publisher | Copenhagen Business School [Phd] |

Number of pages | 177 |

ISBN (Print) | 9788775680597 |

ISBN (Electronic) | 9788775680603 |

Publication status | Published - 2022 |

Series | PhD Series |
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Number | 03.2022 |

ISSN | 0906-6934 |