Capital Commitment

Key Question Being Explored in This Insight:

How does ex-ante commitment in private equity funds impact investor portfolios, and why do commitment risk premiums persist even with increased liquidity and diversification?

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View the page of: Ludovic Phalippou

Institution: ESSEC Business School

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How does ex-ante commitment in private equity funds impact investor portfolios, and why do commitment risk premiums persist even with increased liquidity and diversification?

Image courtesy of interviewee. December 11, 2024

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This video discusses the complexities and risks associated with investing in private equity, particularly focusing on two main types of risk: commitment timing risk and commitment quantity risk.

1. **Commitment Timing Risk**: This video risk arises from the uncertainty of when a general partner will call for committed capital. Investors (limited partners) do not know when their money will be needed, which can lead to liquidity issues.

2. **Commitment Quantity Risk**: This video risk involves the fluctuating value of the investor's overall portfolio by the time the capital is called. For instance, if an investor commits a portion of their wealth to private equity but their overall wealth decreases, the committed amount may represent a larger percentage of their now smaller portfolio, putting financial strain on them.

The speaker describes a model developed to analyze these risks, emphasizing that commitment quantity risk is particularly costly for investors since it can lead to substantial proportions of their portfolios being tied up in illiquid investments at inopportune times. They found that commitment quantity risk is not diversifiable through multiple fund investments due to synchronized financial market movements affecting commitment ratios.

This video also touches upon how private equity investments typically require upfront commitments, creating a disconnect between initial ownership percentages and actual portfolio distributions upon capital calls, which directly impacts investors’ liquidity.

In response to these challenges, the speaker suggests under-allocating to private equity and potentially using co-investment opportunities at the time of capital calls to manage risk effectively. This video strategy allows investors to retain flexibility and adjust their commitments based on the prevailing market conditions, making it a prudent approach for better managing liquidity needs connected with private equity investments.

Overall, the text reflects on the importance of modeling these risks and adapting investment strategies accordingly, while also noting a trend toward more frequent co-investment offers that help investors mitigate pressure from capital commitments.

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Here are five key points from the video:

1. **Capital Commitment Risks**: The video discusses the unique risks associated with capital commitment in private equity, specifically commitment timing risk (uncertainty about when capital will be called) and commitment quantity risk (the risk that the ratio of committed capital to total wealth fluctuates, impacting the investor's portfolio).

2. **Model Complexity**: The author highlights the complexity of modeling these risks, which took over a decade to develop. The model involves stochastic processes to simulate the timing of capital calls and the distribution of proceeds from private equity investments.

3. **Investor Behavior**: It explains that investors typically under-commit to private equity to manage the potential risks. When capital is called, investors may wish to increase their commitment, but they face commitment quantity risk if the market environment has changed.

4. **Co-Investment Opportunities**: The video also mentions the popularity of co-investment opportunities offered by general partners to limited partners. These allow investors to adjust their commitments at the time of capital calls, helping mitigate commitment quantity risk.

5. **Liquidity and Allocation**: The discussion emphasizes that while liquidity risk is often stressed, it is the commitment quantity risk that significantly affects investor behavior and portfolio allocation. Investors are likely to keep their private equity allocations reasonable (generally around 15-20% of the portfolio) to maintain liquidity and manage risks effectively.

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Here are some key quotes from the video, highlighting major concepts and insights regarding investment in private equity and the associated risks:

1. **Mean-Variance Framework**: "So when you go to any investment course, you are basically taught, a variation of mean-variance, which was created 50, 60 years ago, or more, 70."

2. **Commitment in Private Equity**: "What has happened in recent years is that for a number of assets, now you need to commit in advance, the capital."

3. **Capital Call Risks**: "There are two risks that are associated with capital commitment. The first risk is a risk that the limited partner does not know when the capital will be called."

4. **Commitment Quantity Risk**: "The second risk arises because at the time of capital commitment, the limited partner chooses the amount to commit. [...] We call this risk commitment, quantity risk."

5. **Liquidity Frictions**: "Commitment, quantity risk is very costly to the investor, whereas commitment, timing risk is not."

6. **Under-Commitment**: "In order to avoid reaching these high values of commitment to wealth ratios, what investors do is that they under allocate, they under commit to private equity."

7. **Desire for Adjustability**: "They would like to adjust their commitment at the time of capital call. [...] They'd be willing to pay 1.2% of their initial wealth in order to have this possibility."

8. **Diversification of Commitment Risks**: "We actually can't diversify commitment, quantity risk. There's no way to diversify commitment, quantity risk by increasing the amount of funds."

9. **Effect of Market Dynamics**: "Commitment, quantity risk matters. [...] If there's a drop in the stock market, the commitment wealth ratio will increase for all funds at the same time."

10. **Co-Investment Opportunities**: "Co-investment opportunities are offered by general partners to some limited partners, so their favorite ones, and they basically allow the limited partners to get rid of commitment, quantity risk."

11. **Insight on Liquid Allocation**: "If you are being reasonable, if you calibrate the model right, in any case, you don't want to have more than 15, 20% in private equity."

These quotes collectively illustrate the complexities and risks tied to capital commitment in private equity, emphasizing the importance of managing commitment quantity and timing risks, as well as the dynamics of portfolio allocation.

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So when you go to any investment course,

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you are basically taught, a vari, a variation

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of mean variance, which was created 50, 60 years ago,

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or more, 70.

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And, it's the same in practice.

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And what you do then is that you say, well, how do I decide

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how to allocate between different type of assets?

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And you say how much you expect in terms of return

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and that kind of risk,

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and then they tell you, okay,

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you should put 20% in real estate

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and 10% on, on gold and whatever.

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And the problem is that, that

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that has happened on in recent years is that for a number

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of, of of assets, now you need

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to commit in advance, the capital.

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And so if somebody says,

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you should put 10% in private equity, it's like,

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I cannot just put 10% in private equity.

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I have to commit, an amount of money.

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And then I don't know when they will call this money.

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It may represent 10% of my portfolio,

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but it depends what my portfolio will do in terms

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of evolution, right?

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So for example, if, if I have a hundred million right now

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and I'm committing 10 million,

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and they call it in two, three years time,

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if the stock market went very well

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and my portfolio is worth 200, then my 10 million committed,

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it is just 5% of my portfolio.

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Um, but if the market tanked

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and went to like 50 million, my portfolio tanked

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and went to 50 million, then my 10 million now is 20%

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of my portfolio and it's probably too much.

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Um, and so, so this creates a new type of risk.

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And, and, and there's a need for modeling this,

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this, this issue.

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We started more of a paper by thinking,

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let's just model the, the fact that you have

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to commit in advance and have to wait until it's deployed.

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But it was more with the idea that then it creates a sorts

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of like illiquidity and,

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and, a, a sort of debt

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that you have hanging on you like an

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unfunded commitment, we say.

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So, but in the end, it turned out that most

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of the action came from the fact that I like

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what I just described, but the rest

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of your portfolio value is gonna fluctuate a lot.

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And when the money is gonna be deployed,

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it may represent a too low or way too high amount of money.

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And this is in fact what really annoys you as an investor,

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not so much the fact that your money is not really

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accessible or the fact that, you have made the, you,

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you have this debt hanging on you.

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So most private equity funds are

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so-called blind pooled funds.

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The way it works, is

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that you have a company which acts as general partner.

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You can think about KKR. So those are the gps.

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They manage the funds.

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The capital is provided by the limited partners

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who are the invest the investors.

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So those are the LPs.

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So when the fund is raised,

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there's an initial fundraising period of about 18 months

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during which the limited partners will commit.

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So ante to providing cash to the GPS on demand

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over a pre-specified investment period of about five years.

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So if you're the LP,

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and I'm the gp, if you commit to providing, $50 million

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to the funds, then you basically trust me

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to find the right investment opportunities.

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You won't give me your money immediately.

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I will go looking for these opportunities.

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And then once I will have identified them, I will get back

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to you and ask for the money.

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So this where this can happen years

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after the fundraising period.

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So there's two risks that are associated

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with capital commitment.

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The first risk, is a risk

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that the limited partner does not know when the

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capital will be called.

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They don't know when the general partner will get back

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and ask for the capital to be transferred.

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This risk, we refer to it as commitment, timing, risk.

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The second risk arises

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because at the time of capital commitment,

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the limited partner chooses the amount to commit.

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So they commit a positive amount of money,

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but their wealth is going to fluctuate between the time

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of commitment and the time at which the capital is invested.

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So if you invested this $50 million,

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and if that was 10% of your wealth at the time

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of capital commitment, your wealth is going to fluctuate.

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If there's a stock market crash between capital commitment

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and capital call, imagine that the 10%

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of $500 million have dropped to $250 million.

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Uh, this means that the capital commitment is now 20%

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of your wealth at the time of capital core.

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We call this risk commitment, quantity risk.

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So these two risks are very specific

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to private equity funds.

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And then there are different liquidity fee,

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liquidity risk, which are less unique

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to private equity funds.

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So there's the, illiquidity of the secondary market.

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If you're the LP during the holding period, you might want

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to sell stakes in your private, equity funds.

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You can go and secondary market,

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in which case you sell these stakes at a discount.

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So there's also uncertainty on the time of distribution of,

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investment proceeds.

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So as a gp, I, I invested your money, this can last years,

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and then it's up to me to decide when to exit the investment

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and to return you the investment proceeds.

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So you have no control over that.

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So this, a liquidity risk was studied in a paper

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by Ang Pap Elau and Westerfield in 2014.

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And they have shown that it has effect significant effects

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on portfolio allocation and on welfare.

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What we focus on is capital commitment.

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So, risks that are focused on on commit commitment risk.

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So commitment, timing and commitment, quantity risk.

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But we also study the interaction between these risks

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and the liquidity of the second of, of the secondary market.

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And the other risks, the,

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The, the model was very complex, so it, it,

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it took us more than 10 years to solve it.

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And, and it, it, it, it, it was really, yeah, a bit crazy.

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Um, yeah, it's mathematically quite demanding.

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We, at, at one point, we even hired a,

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a PhD students in math here at Oxford to like,

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who specialize in like partial

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differential equation systems.

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And then he failed because I think what was,

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what was necessary is that not only you're good at math,

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but you understand the economics so that you,

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you can really like set up things correctly

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to to, to solve them.

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Um, so the model is quite complicated, in, in,

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in the details of it.

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But the, the overall idea is relatively simple.

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You, you have, when you commit capital, you start a,

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a pass, some process, we call it.

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It's, it's, it's, it's a process that has no memory.

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Every period your money may be called or not called.

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And, it has a certain expected duration

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because every period, let's say you have one chance in free

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to be called, and if a model would be at year frequency,

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it would mean you can expect to be called after three years.

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But if you could be called after one year,

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because the, the flip may, may, may be, you know,

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on the right side, as soon as the first year,

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so you have this person process that you start

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and you commit a certain amount of money to it,

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you wait until it kicks.

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When it ticks, it calls your money in,

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your money is deployed, the post on process starts again.

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When it ticks again, your money comes back to you and, and,

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and you don't have access to that money when, when,

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when invested and then you commit again and, and so on.

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And, and so this is quite demanding,

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but with just one process, it's quite okay.

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But when you start getting to two, three

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or more, then it gets like really hard.

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Um, and in practice, people have 20 or 30 funds.

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So if you are doing it on 2030 funds,

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it'll be like technically almost impossible to program.

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I mean, conceptually it's feasible, but, but,

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but managing to program more these

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PDs you'd get lost in them.

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Like it is nearly impossible to, to, to program.

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Um, and so we have a limit case

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where you have an infinite number of funds

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because things simplify a little bit.

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Um, but we, we, we solved it only

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for one, two, and three funds.

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Um, and so that, that, that's, that's a model,

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that we had to write.

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And the data, we used data that are now publicly available

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or relatively easily available with, with frequent.

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Um, so by now,

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practically fund cash flow detailed data

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are, are widely available.

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And so we had like 5,000 of them to calibrate the model on.

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So we could do some, some things

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that when we started the project like 10, 15 years ago,

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it, it would've been harder to,

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to do in terms of, yeah, calibration. So it's

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Really important to have a model which captures the

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different features of private equity.

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We have a dynamic, portfolio model.

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Uh, so it's a model, with which features,

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risk averse investor with infinite horizon

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and with access to liquid stocks, bonds,

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and private equity thoughts.

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So we model investments from the perspective of the lp.

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So at times zero, the LP commits a positive amount of money

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to the private equity fund.

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In our baseline model, he only has access to one fund.

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Um, but this amount, this amount

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of money is not transferred yes, yet, between the time

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of capital commitment

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and the time of capital call, the LP is going

207
00:09:06.985 --> 00:09:11.265
to pay a management fee to the GP in order to model the fact

208
00:09:11.265 --> 00:09:14.105
that capital caller is a random time, it's a time

209
00:09:14.105 --> 00:09:16.025
that is unknown to the limited partner.

210
00:09:16.405 --> 00:09:18.905
We use a stochastic process, a hitting time,

211
00:09:19.425 --> 00:09:22.185
a so-called process in order to model the time

212
00:09:22.185 --> 00:09:26.585
of capital call when this hitting process hits on,

213
00:09:27.205 --> 00:09:28.225
the capital is called.

214
00:09:28.605 --> 00:09:31.305
And the limited partner then has the option between

215
00:09:31.915 --> 00:09:33.745
defaulting on their commitment

216
00:09:34.165 --> 00:09:37.025
or transferring the money to the private equity fund.

217
00:09:37.295 --> 00:09:39.865
Default is however very costly for the investor

218
00:09:40.135 --> 00:09:41.225
because it means

219
00:09:41.225 --> 00:09:44.105
that the investor is gonna lose forever access

220
00:09:44.125 --> 00:09:45.945
to the private, private equity markets.

221
00:09:47.775 --> 00:09:50.355
So if the investor chooses not to default,

222
00:09:50.985 --> 00:09:52.795
then they transfer the amount,

223
00:09:52.795 --> 00:09:54.915
which is then gonna grow at the private

224
00:09:54.915 --> 00:09:56.195
equity fund returner.

225
00:09:56.455 --> 00:09:58.955
But this amount, during the whole holding period,

226
00:09:58.965 --> 00:10:00.035
which is unknown,

227
00:10:00.625 --> 00:10:02.915
this amount is gonna be stuck in private equity.

228
00:10:03.255 --> 00:10:05.275
So basically the investor is not gonna be able

229
00:10:05.275 --> 00:10:06.675
to consume out of this amount.

230
00:10:07.815 --> 00:10:10.595
The investor is only able to consume out

231
00:10:10.595 --> 00:10:11.915
of their liquid wells.

232
00:10:11.975 --> 00:10:14.595
So the ones that is invested in liquid stocks

233
00:10:14.695 --> 00:10:16.915
and bonds with the, for the amount

234
00:10:16.915 --> 00:10:18.515
that is invested in private equity,

235
00:10:18.985 --> 00:10:20.595
there's no way to consume out of that.

236
00:10:22.085 --> 00:10:23.625
We have a second question process,

237
00:10:24.025 --> 00:10:26.785
a set a second hitting processor to model the fact

238
00:10:26.815 --> 00:10:30.505
that the distribution of the investment proceeds,

239
00:10:30.605 --> 00:10:32.065
is also a random time.

240
00:10:33.085 --> 00:10:34.945
So when this personal process hits,

241
00:10:35.455 --> 00:10:38.265
then the investment proceeds are,

242
00:10:39.305 --> 00:10:40.945
provided back to the investor.

243
00:10:41.525 --> 00:10:43.585
The investor can commit to a new fund,

244
00:10:43.805 --> 00:10:45.385
and basically the process repeat.

245
00:10:46.755 --> 00:10:50.855
So the investor maximizes the utility of consumption

246
00:10:51.115 --> 00:10:52.775
and gets the consumption out

247
00:10:52.775 --> 00:10:55.695
of investments in those liquid stocks, bonds,

248
00:10:55.915 --> 00:10:57.175
and private equity funds.

249
00:10:57.595 --> 00:11:00.535
And in this setup, we can basically study the effect

250
00:11:00.535 --> 00:11:02.175
of the different liquidity frictions

251
00:11:02.395 --> 00:11:04.975
by shutting down each liquidity friction

252
00:11:04.975 --> 00:11:06.095
at a time, one at a time.

253
00:11:06.675 --> 00:11:10.975
So for example, we shut down, we shut down, commitment,

254
00:11:11.235 --> 00:11:14.455
timing, risk, yeah, by letting,

255
00:11:14.455 --> 00:11:18.055
the investor know when the request process is gonna hit.

256
00:11:18.055 --> 00:11:20.495
So by, by letting the investor know when the capital is

257
00:11:20.495 --> 00:11:24.655
going to be called, we shut down commitment, quantity risk

258
00:11:24.915 --> 00:11:27.775
by letting the investor adjust the amount that is allocated

259
00:11:27.795 --> 00:11:30.535
to private equity at the time of capital call.

260
00:11:31.595 --> 00:11:33.255
We can also study the interactions

261
00:11:33.255 --> 00:11:36.295
between these two fractions and a secondary market

262
00:11:36.315 --> 00:11:38.535
and liquidity by varying the discounts

263
00:11:38.535 --> 00:11:39.575
on the secondary market.

264
00:11:39.795 --> 00:11:41.775
So basically, if you put the discount to zero,

265
00:11:41.995 --> 00:11:44.895
you have a secondary market that is perfectly lake quota.

266
00:11:46.345 --> 00:11:48.685
So we have also some extensions of our model.

267
00:11:49.025 --> 00:11:50.245
We have an extension

268
00:11:50.425 --> 00:11:52.885
to a model in which two font can

269
00:11:52.885 --> 00:11:53.925
be be traded simultaneously.

270
00:11:54.505 --> 00:11:57.325
We have another extension in which an infinite amount

271
00:11:57.325 --> 00:11:59.325
of fonts can be traded simultaneously.

272
00:11:59.585 --> 00:12:01.445
And those extensions allow us

273
00:12:01.445 --> 00:12:04.765
to study whether commitment risk can be diversified

274
00:12:05.145 --> 00:12:07.005
by investing in more than one font.

275
00:12:07.105 --> 00:12:08.645
So we actually spent a fair amount

276
00:12:08.645 --> 00:12:10.725
of time on, calibration.

277
00:12:10.825 --> 00:12:12.605
Uh, so the calibration

278
00:12:12.605 --> 00:12:16.645
of the parameters driving the dynamics of liquid stocks

279
00:12:16.645 --> 00:12:17.965
and bonds is very standard.

280
00:12:18.115 --> 00:12:20.685
It's based on s and p 500 log returns,

281
00:12:20.985 --> 00:12:22.525
and on three months treasury bond,

282
00:12:22.945 --> 00:12:24.965
what's new is the calibration

283
00:12:24.965 --> 00:12:26.885
of the parameters driving the dynamics

284
00:12:26.885 --> 00:12:27.965
of private equity funds.

285
00:12:28.545 --> 00:12:30.845
So for that, we use frequent data.

286
00:12:31.235 --> 00:12:34.285
This is data on, fund cash flows.

287
00:12:34.785 --> 00:12:38.005
So we have, data as of 2020,

288
00:12:38.265 --> 00:12:41.965
and we consider all the United States, raised funds

289
00:12:42.475 --> 00:12:46.525
between 1991 and, to 2015.

290
00:12:47.745 --> 00:12:51.885
So these funds include venture capital, growth, equity,

291
00:12:52.025 --> 00:12:53.485
and leverage buyout funds.

292
00:12:53.705 --> 00:12:56.845
So we have a very large amount of, of wso.

293
00:12:57.585 --> 00:13:01.005
The first parameter that we calibrate is a parameter

294
00:13:01.035 --> 00:13:04.205
that drives the process that

295
00:13:04.845 --> 00:13:06.325
triggers capital core.

296
00:13:06.865 --> 00:13:09.605
So this parameter is intensity of capital core.

297
00:13:10.185 --> 00:13:13.205
In order to calibrate that parameter, what we do is

298
00:13:13.205 --> 00:13:16.485
that we calculate the empirical distribution

299
00:13:16.865 --> 00:13:18.285
of cash inflows.

300
00:13:18.425 --> 00:13:21.525
So the data distribution, those inflows,

301
00:13:21.595 --> 00:13:23.205
they contain management fees

302
00:13:23.265 --> 00:13:26.325
and code capital, we have them for each fund.

303
00:13:26.345 --> 00:13:27.645
So we have the whole distribution,

304
00:13:27.645 --> 00:13:30.685
and we match this distribution to the model implied

305
00:13:31.245 --> 00:13:35.285
distribution, and we obtain an intensity of capital call,

306
00:13:35.615 --> 00:13:38.845
which basically, means that the, the, the commitments,

307
00:13:38.985 --> 00:13:41.765
period is going to be on average three years.

308
00:13:43.435 --> 00:13:46.765
Then, to calibrate the intensity

309
00:13:46.865 --> 00:13:49.885
of capital distribution, we could be doing the same thing

310
00:13:50.075 --> 00:13:52.805
with, the distributions in the data.

311
00:13:53.385 --> 00:13:56.605
The problem with that is that we would need about 15 years

312
00:13:56.825 --> 00:13:59.205
of data for each funder in order

313
00:13:59.385 --> 00:14:01.805
to have the whole time series of distributions.

314
00:14:02.225 --> 00:14:04.365
So that would mean that we had, we would have

315
00:14:04.365 --> 00:14:07.285
to limit the exercise to funds that were raised

316
00:14:07.305 --> 00:14:11.445
before 2005 in order to work around that issue.

317
00:14:11.755 --> 00:14:14.845
What we do is that we compute the distribution

318
00:14:15.105 --> 00:14:18.045
of performance measures of funds, instead

319
00:14:18.045 --> 00:14:20.205
of computing the distribution of outlets,

320
00:14:20.865 --> 00:14:23.365
we choose a very standard measure of performance,

321
00:14:23.365 --> 00:14:25.245
which was a public market equivalent,

322
00:14:25.575 --> 00:14:28.925
which was introduced in 2005 by Kaplan insurer,

323
00:14:29.305 --> 00:14:33.245
and which basically calculates the ratio of funds

324
00:14:33.795 --> 00:14:35.485
inflows discounted with the s

325
00:14:35.485 --> 00:14:38.325
and p 500 over fund outflows, also discounted

326
00:14:38.325 --> 00:14:39.605
with the s and p 500.

327
00:14:40.585 --> 00:14:42.965
So a fund that has a public market equivalent

328
00:14:42.975 --> 00:14:46.765
above one is a fund that outperformed the s and p 500.

329
00:14:46.945 --> 00:14:50.485
If it's below one, it underperformed for each fund.

330
00:14:50.505 --> 00:14:52.605
We calculate this public market equivalent.

331
00:14:52.705 --> 00:14:55.765
So we have a distribution of public market equivalents,

332
00:14:56.015 --> 00:15:00.165
which we match to, model implied distributions

333
00:15:00.475 --> 00:15:02.805
that we obtain using simulations.

334
00:15:03.825 --> 00:15:07.485
So using this procedure, we calibrate the parameters,

335
00:15:07.585 --> 00:15:11.325
ca driving returns of private equity funds.

336
00:15:11.705 --> 00:15:16.325
We obtain expected returns on private equity at,

337
00:15:16.685 --> 00:15:19.165
14% against 8% for stocks.

338
00:15:19.585 --> 00:15:23.965
The volatility of private equity is calibrated at 33.5%

339
00:15:24.435 --> 00:15:27.045
against, 15% for stocks.

340
00:15:27.065 --> 00:15:30.965
So these values are in line with adult values that are used,

341
00:15:31.145 --> 00:15:32.445
in, in, in papers.

342
00:15:32.625 --> 00:15:36.565
Um, we have a beta for private equity at 1.47.

343
00:15:36.745 --> 00:15:39.325
That's a beta that's really close to the values

344
00:15:39.395 --> 00:15:41.325
that was obtained, in the paper

345
00:15:41.425 --> 00:15:43.645
by GaN co-authors in 2018

346
00:15:44.415 --> 00:15:46.045
using a completely different method.

347
00:15:46.145 --> 00:15:50.885
Um, we have a cap alpha for private equity at 3.6%,

348
00:15:51.225 --> 00:15:56.205
and that's also very close to a paper by Ang Azman in 2018.

349
00:15:56.265 --> 00:15:58.605
So our value makes sense, they're in line

350
00:15:58.605 --> 00:16:00.805
with the literature, they're in line with intuition,

351
00:16:01.105 --> 00:16:03.445
and it allow us to match distributions

352
00:16:03.505 --> 00:16:07.525
of performance measures and inflows remarkably well.

353
00:16:07.695 --> 00:16:11.605
Commitment, quantity risk is very costly to the investor,

354
00:16:11.715 --> 00:16:14.485
whereas commitment, timing risk is not.

355
00:16:14.485 --> 00:16:16.125
So what this means is

356
00:16:16.125 --> 00:16:19.645
that the investors care very much about,

357
00:16:19.675 --> 00:16:23.085
what the commitment to wealth ratio is going

358
00:16:23.085 --> 00:16:26.765
to be at the time of, capital call.

359
00:16:26.765 --> 00:16:28.885
They care very much about not knowing

360
00:16:29.235 --> 00:16:30.405
what this ratio will be,

361
00:16:30.545 --> 00:16:33.525
but they don't care so much about not knowing when

362
00:16:33.595 --> 00:16:35.205
capital call will happen.

363
00:16:35.905 --> 00:16:37.645
So why do they care so much about the

364
00:16:37.645 --> 00:16:38.725
commitment wealth ratio?

365
00:16:39.105 --> 00:16:42.725
The reason is that if this commitment towards ratio becomes

366
00:16:42.865 --> 00:16:45.165
too large at the time of capital Cola,

367
00:16:45.355 --> 00:16:49.485
investors basically have the option to either default,

368
00:16:49.745 --> 00:16:52.125
but we've seen that default is pretty costly for them

369
00:16:52.705 --> 00:16:54.045
or to transfer that amount

370
00:16:54.045 --> 00:16:55.685
and to have a, a large portion

371
00:16:55.705 --> 00:16:58.205
of their portfolio invested in private equity

372
00:16:58.545 --> 00:17:00.405
for unknown period of time.

373
00:17:01.585 --> 00:17:04.885
So in order to avoid reaching these high values

374
00:17:05.185 --> 00:17:08.445
of commitment to wealth ratios, what investors do is

375
00:17:08.445 --> 00:17:09.605
that they under allocate,

376
00:17:09.635 --> 00:17:11.965
they under commit to private equity.

377
00:17:12.905 --> 00:17:14.925
So because they under commit, they would like

378
00:17:15.065 --> 00:17:18.005
to adjust their commitment at the time of capital call.

379
00:17:18.225 --> 00:17:21.205
And in fact, they would be willing, they would like

380
00:17:21.545 --> 00:17:25.165
to nearly double the commitment at time of capital call.

381
00:17:25.865 --> 00:17:28.285
They'd be willing to pay a large amount of money

382
00:17:28.385 --> 00:17:29.925
to have this possibility.

383
00:17:29.945 --> 00:17:32.445
They'd be willing to pay 1.2%

384
00:17:32.785 --> 00:17:36.725
of their initial wealth in order to have this possibility.

385
00:17:36.865 --> 00:17:41.365
And this represents a quarter of their P commitments.

386
00:17:41.365 --> 00:17:43.205
So that's, that's a large amount of money.

387
00:17:44.825 --> 00:17:46.125
So now the question is,

388
00:17:46.145 --> 00:17:49.045
can we actually diversify this risk in order

389
00:17:49.185 --> 00:17:50.405
to address this question?

390
00:17:50.405 --> 00:17:51.805
What we did is that we looked at

391
00:17:51.805 --> 00:17:54.605
what happened when extending the model to a model

392
00:17:54.635 --> 00:17:57.645
with two funds and a model with an infin model of false.

393
00:17:58.505 --> 00:18:00.245
And the answer is no,

394
00:18:00.245 --> 00:18:03.605
we actually can't diversify commitment, quantity risk.

395
00:18:03.605 --> 00:18:05.125
There's two reasons for that.

396
00:18:06.185 --> 00:18:10.325
The first reason is that commitment, quantity risk is driven

397
00:18:10.385 --> 00:18:12.205
by the commitment wealth ratio.

398
00:18:12.865 --> 00:18:14.725
If there's a drop in the stock market,

399
00:18:15.425 --> 00:18:18.685
the commitment wealth ratio will increase

400
00:18:18.865 --> 00:18:20.605
for all funds at the same time.

401
00:18:21.345 --> 00:18:23.485
So there will be no diversification effect.

402
00:18:24.075 --> 00:18:27.125
There's no way to diversify commitment, quantity risk

403
00:18:27.145 --> 00:18:29.045
by increasing the amount of funds.

404
00:18:29.825 --> 00:18:30.885
The second reason is

405
00:18:30.885 --> 00:18:32.685
that there's a possible funding mismatch.

406
00:18:32.945 --> 00:18:35.325
If you have many funds, then you'll finance

407
00:18:36.105 --> 00:18:40.485
the capital calls of the later funds using the distributions

408
00:18:40.545 --> 00:18:41.565
of the earlier funds.

409
00:18:41.905 --> 00:18:44.005
But remember that the times of capital, coal

410
00:18:44.005 --> 00:18:46.285
and capital distribution are stochastic.

411
00:18:46.985 --> 00:18:49.645
So if capital calls happen a little too early,

412
00:18:49.865 --> 00:18:53.005
or capital distributions happen a little too late,

413
00:18:53.355 --> 00:18:56.405
then you won't be able to finance later funds

414
00:18:56.555 --> 00:18:57.845
with the proceeds of earlier

415
00:18:57.855 --> 00:18:59.165
funds, and there will be a mismatch.

416
00:18:59.825 --> 00:19:01.365
So basically, commitment,

417
00:19:01.845 --> 00:19:04.245
quantity risk is a big deal, for investors.

418
00:19:04.915 --> 00:19:06.765
It's, a bigger deal than the other

419
00:19:06.765 --> 00:19:07.965
illiquidity frictions.

420
00:19:07.965 --> 00:19:09.845
When we compare the cost, it's larger.

421
00:19:10.585 --> 00:19:14.685
And this is not a risk that can be diversified by,

422
00:19:14.685 --> 00:19:16.165
investing in, in many funds.

423
00:19:16.265 --> 00:19:18.365
Um, the strategic default would be like, you know, you,

424
00:19:18.365 --> 00:19:20.085
you committed money, but you are, you are allowed

425
00:19:20.085 --> 00:19:22.525
to say no when it, when, when, when it comes in, right?

426
00:19:22.585 --> 00:19:25.685
So, so then it's a, it's, it's a, if you take it literally

427
00:19:25.685 --> 00:19:28.005
and there will be no penalty, then, then, then that's it.

428
00:19:28.005 --> 00:19:29.125
The commitment doesn't matter

429
00:19:29.125 --> 00:19:31.445
because you can just renege whenever you want.

430
00:19:32.065 --> 00:19:34.705
Um, but in practice you cannot really

431
00:19:34.905 --> 00:19:36.025
renege on, on, on a commitment.

432
00:19:36.025 --> 00:19:37.385
So you, you, you, you have to do it.

433
00:19:37.925 --> 00:19:40.665
Uh, nonetheless, you could, you, you, you,

434
00:19:40.685 --> 00:19:42.865
you could have some strategic de de default

435
00:19:42.865 --> 00:19:44.505
or so during the life of a deal and so on.

436
00:19:44.505 --> 00:19:46.145
And we, we, we, we, we model that.

437
00:19:46.445 --> 00:19:50.345
It doesn't have like a major, IM impact,

438
00:19:50.375 --> 00:19:52.825
because if you default, you need to restart again.

439
00:19:52.845 --> 00:19:54.105
So you, you have committed

440
00:19:54.105 --> 00:19:55.185
and then you say, okay, I changed my mind.

441
00:19:55.225 --> 00:19:57.665
I want to be out now. But then you need to recommit

442
00:19:57.665 --> 00:20:00.025
and you need to wait again the entire amount of time, right?

443
00:20:00.025 --> 00:20:02.465
So it's a bit like you go to the doctor, you've been waiting

444
00:20:02.485 --> 00:20:03.825
for an hour, you're very annoyed.

445
00:20:03.845 --> 00:20:06.025
And so you say, okay, I'm, I'm gonna default on this.

446
00:20:06.085 --> 00:20:08.505
I'm gonna leave, and, but you still need to go to the doctor

447
00:20:08.565 --> 00:20:09.705
and then you're gonna start again.

448
00:20:09.705 --> 00:20:11.865
But like, well, you know, you might as well have stayed

449
00:20:11.865 --> 00:20:13.545
because it, it doesn't change everything.

450
00:20:13.765 --> 00:20:15.425
So it, it is slightly different than that

451
00:20:15.425 --> 00:20:17.025
because the world has changed during

452
00:20:17.025 --> 00:20:18.065
that hour you were waiting.

453
00:20:18.205 --> 00:20:21.225
So you may, adjust a few parameters, but it,

454
00:20:21.225 --> 00:20:23.425
but it's not massively useful

455
00:20:23.685 --> 00:20:25.625
Is that neither the option to default

456
00:20:26.085 --> 00:20:28.465
nor the secondary markets actually alleviate

457
00:20:28.475 --> 00:20:29.905
commitment, quantity risk.

458
00:20:30.285 --> 00:20:33.385
So remember that in order to manage commitment, quantity,

459
00:20:33.535 --> 00:20:36.745
risk investor under allocates, so it's rarely

460
00:20:37.265 --> 00:20:39.385
optimal given this under location,

461
00:20:39.685 --> 00:20:43.785
to default at the time of capital, call, especially

462
00:20:43.785 --> 00:20:46.985
because default is so costly for, the investor.

463
00:20:48.155 --> 00:20:51.125
Similarly, the secondary market is very rarely used.

464
00:20:51.425 --> 00:20:55.085
And the reason for that is that in a model with one funder,

465
00:20:55.085 --> 00:20:56.845
partial sales are not allowed,

466
00:20:56.845 --> 00:20:58.605
which is in line with practice.

467
00:20:59.265 --> 00:21:02.005
And so it's rarely optimal for the investor

468
00:21:02.005 --> 00:21:04.245
to sell all their stakes in private equity.

469
00:21:05.065 --> 00:21:08.645
So neither the secondary market nor,

470
00:21:09.025 --> 00:21:11.725
nor default are actually options that are really worth it

471
00:21:11.725 --> 00:21:13.045
for the, for the investor.

472
00:21:14.145 --> 00:21:15.725
The second result is

473
00:21:15.725 --> 00:21:18.245
that the liquidity on the secondary market

474
00:21:18.905 --> 00:21:22.245
and commitment, quantity risk are not substitutes.

475
00:21:22.245 --> 00:21:23.365
There are actually compliments.

476
00:21:23.365 --> 00:21:26.525
So let me explain that one good thing, that by

477
00:21:27.145 --> 00:21:30.485
making the secondary market more liquid, so by,

478
00:21:30.485 --> 00:21:33.085
letting the discount on the secondary market go to zero

479
00:21:33.815 --> 00:21:35.885
commitment, quantity risk would disappear.

480
00:21:36.075 --> 00:21:37.365
This is actually not the case.

481
00:21:37.985 --> 00:21:41.365
If you pull the discount up to zero, what's gonna happen is

482
00:21:41.365 --> 00:21:44.685
that the investor is going to invest more in private equity,

483
00:21:44.945 --> 00:21:48.845
and this increased allocation will make them even more

484
00:21:48.845 --> 00:21:52.205
willing to pay in order to get rid of,

485
00:21:52.215 --> 00:21:53.845
commitment, quantity risk.

486
00:21:54.345 --> 00:21:57.005
So basically commitment, quantity, risk matters.

487
00:21:57.465 --> 00:21:59.765
And it's not, it's neither the option to default

488
00:21:59.905 --> 00:22:03.525
nor the secondary market that, can alleviate it.

489
00:22:03.655 --> 00:22:05.725
Often when people insure investors and,

490
00:22:05.725 --> 00:22:08.045
and always think about liquidity risk management in private

491
00:22:08.045 --> 00:22:10.965
equity, they are thinking about the fact that they need

492
00:22:10.965 --> 00:22:13.885
to keep some money in cash or bonds or liquid assets,

493
00:22:13.885 --> 00:22:15.885
because in case money gets called in private equity,

494
00:22:15.885 --> 00:22:17.485
they need to have it readily available.

495
00:22:18.145 --> 00:22:21.765
Um, and we, what we show again in our paper is

496
00:22:21.765 --> 00:22:24.205
that this is not such a big deal if you are being reasonable

497
00:22:24.205 --> 00:22:25.445
with your private equity allocation.

498
00:22:25.585 --> 00:22:29.645
So if you are being reasonable, if you calibrate the model,

499
00:22:29.645 --> 00:22:32.365
right, in any case, you don't want to have more than 15,

500
00:22:32.365 --> 00:22:33.645
20% in private equity.

501
00:22:34.025 --> 00:22:37.245
And if it's only 15, 20% of your portfolio that is illiquid,

502
00:22:37.465 --> 00:22:38.765
it is not really a big deal.

503
00:22:38.765 --> 00:22:40.805
You will always have the money to really pay

504
00:22:40.805 --> 00:22:41.885
for, for the capital call.

505
00:22:41.885 --> 00:22:43.605
You can sell some shares, et cetera.

506
00:22:44.225 --> 00:22:47.845
Um, the people get in trouble in practice if people commit

507
00:22:47.845 --> 00:22:50.005
50, 60 or 80% of their portfolios,

508
00:22:50.005 --> 00:22:52.085
but our model will never tell you to do that.

509
00:22:52.105 --> 00:22:55.125
So then, you know, that, that, that liquidity

510
00:22:56.025 --> 00:22:57.445
per se is never an issue.

511
00:22:58.065 --> 00:22:59.325
What's an issue,

512
00:22:59.325 --> 00:23:03.325
however, for, then insur investors is that the, the fact

513
00:23:03.325 --> 00:23:05.125
that again, when you commit money,

514
00:23:05.435 --> 00:23:07.645
when money will be called, you don't know how much

515
00:23:07.645 --> 00:23:09.165
of your portfolio this will represent.

516
00:23:09.465 --> 00:23:11.205
And so the key insight from, from the model,

517
00:23:11.205 --> 00:23:14.565
which we actually hadn't anticipated, is that the model,

518
00:23:14.625 --> 00:23:17.965
says that you should under commit to private equity.

519
00:23:18.465 --> 00:23:22.125
And then if you can top up at the moment, the money is cold.

520
00:23:22.345 --> 00:23:23.565
And that's very interesting

521
00:23:23.565 --> 00:23:25.645
because it's actually what people are doing,

522
00:23:26.305 --> 00:23:29.005
but then they don't seem to be doing it for that reason.

523
00:23:29.225 --> 00:23:31.445
Uh, but, but, but, so it gives them one more reason

524
00:23:31.505 --> 00:23:33.805
for doing this, this thing.

525
00:23:33.905 --> 00:23:37.605
So to be more concrete, the idea is that imagine you want

526
00:23:37.605 --> 00:23:39.045
to be 20% in private equity

527
00:23:39.545 --> 00:23:41.325
and imagine that on expectation, indeed,

528
00:23:41.325 --> 00:23:43.405
it would be 20% given how usually the rest

529
00:23:43.405 --> 00:23:46.085
of your portfolio grows over time, and you will be correct,

530
00:23:47.185 --> 00:23:50.005
but you do have a chance that you may end up with 25

531
00:23:50.005 --> 00:23:53.325
or 30% if, if, if your portfolio, went down in value.

532
00:23:54.365 --> 00:23:56.625
So what you do is that you say, okay, I'm gonna commit so

533
00:23:56.625 --> 00:23:58.665
that on expectation I will be at 10%

534
00:23:58.775 --> 00:23:59.785
when my money is called.

535
00:24:00.205 --> 00:24:02.585
Now, it's annoying because I usually want to be at 20%,

536
00:24:03.045 --> 00:24:06.345
but by, by committing so that I'm at 10% in expectation,

537
00:24:06.825 --> 00:24:09.265
I will hardly ever be at 20% or more.

538
00:24:09.485 --> 00:24:11.825
Uh, and so in that zone, I don't wanna be,

539
00:24:12.685 --> 00:24:16.285
but then most of the time when the money is called, I'm

540
00:24:16.285 --> 00:24:18.365
around 10 and I don't wanna be around 10, I wanna be a 20.

541
00:24:19.305 --> 00:24:22.205
So it'll be very handy if I can top up at don't moment.

542
00:24:22.345 --> 00:24:23.765
The, the money is called with something

543
00:24:23.765 --> 00:24:26.005
that gives you me immediately the kick if I

544
00:24:26.005 --> 00:24:27.245
want to, to be a 20.

545
00:24:27.905 --> 00:24:30.405
And that will be in, in a, in, in a world where the, the,

546
00:24:30.425 --> 00:24:32.245
the rest of the market has increasing value.

547
00:24:32.305 --> 00:24:33.445
So it's in booming times.

548
00:24:33.825 --> 00:24:35.885
In good times when my money is called,

549
00:24:36.005 --> 00:24:37.285
I will typically want to top up.

550
00:24:38.065 --> 00:24:40.005
And we do see that in practice it's called

551
00:24:40.005 --> 00:24:41.245
co-investment programs.

552
00:24:41.245 --> 00:24:43.885
So when money is called from investors, they are invited

553
00:24:43.905 --> 00:24:45.605
to co-invest and they tend

554
00:24:45.605 --> 00:24:48.205
to co-invest much more when we are in good time

555
00:24:48.205 --> 00:24:49.405
when the markets are booming.

556
00:24:49.705 --> 00:24:50.725
So they are doing that,

557
00:24:51.065 --> 00:24:52.645
but usually they're motivated more

558
00:24:52.645 --> 00:24:55.365
by like things like saving fees and things like that.

559
00:24:55.585 --> 00:24:58.085
But they never think about co-investments as a tool

560
00:24:58.185 --> 00:24:59.725
to manage their liquidity.

561
00:25:00.065 --> 00:25:01.645
So under committing prior equity,

562
00:25:01.785 --> 00:25:03.685
and if it turns out that things are going okay

563
00:25:03.685 --> 00:25:06.365
and you are under committ when the capital call is coming,

564
00:25:06.595 --> 00:25:08.525
then you use the co-investment to top up.

565
00:25:09.185 --> 00:25:11.405
Um, so that's like an, an insight

566
00:25:11.405 --> 00:25:13.685
that we haven't thought about that, it makes a lot

567
00:25:13.685 --> 00:25:15.525
of sense once you see the solution.

568
00:25:15.765 --> 00:25:18.805
'cause of course it's math, so it's logical one is come out,

569
00:25:18.825 --> 00:25:21.565
but, but not always intuitively like ex we, we,

570
00:25:21.565 --> 00:25:22.565
we haven't thought about that

571
00:25:22.865 --> 00:25:26.205
In order to manage, commitment, quantity risk

572
00:25:26.205 --> 00:25:27.605
because that's what really matters

573
00:25:27.825 --> 00:25:29.165
for private equity investors.

574
00:25:29.345 --> 00:25:32.605
Uh, investors under allocate to private equity.

575
00:25:32.625 --> 00:25:34.965
And this under allocation is costly.

576
00:25:34.985 --> 00:25:38.005
It has impact in terms of portfolio allocation

577
00:25:38.225 --> 00:25:39.645
and welfare for investors.

578
00:25:40.595 --> 00:25:43.645
This result actually rationalizes something we've been

579
00:25:43.675 --> 00:25:46.245
observing over the last, 10 to 15 years,

580
00:25:46.735 --> 00:25:49.165
which is the increased offering in

581
00:25:49.165 --> 00:25:50.525
co-investment opportunities.

582
00:25:51.145 --> 00:25:53.845
So co-investment opportunities are offered

583
00:25:53.905 --> 00:25:57.445
by general partners to some limited partners,

584
00:25:57.445 --> 00:25:58.645
so their favorite ones,

585
00:25:59.265 --> 00:26:02.645
and they basically allow the limited partners to get rid

586
00:26:02.645 --> 00:26:03.925
of commitment, quantity risk,

587
00:26:04.305 --> 00:26:07.205
to adjust the commitment at the time of capital call.

588
00:26:07.985 --> 00:26:09.525
So what our model

589
00:26:09.545 --> 00:26:10.965
and our resource,

590
00:26:11.155 --> 00:26:15.045
explain is why this co-investment opportunities have been

591
00:26:15.045 --> 00:26:17.645
so popular and why there's so much demand for them.

592
00:26:17.795 --> 00:26:21.445
Because there are a way to, to get rid of, of commitment,

593
00:26:21.805 --> 00:26:22.205
quantity, risk.

Modeling Capital Commitment Risks in Asset Allocation

00:00:00 - So when you go to any investment course, you are basically taught, a vari, a variation of mean variance, which was created 50, 60 years ago, or more, 70. And, it's the same in practice. And what you do then is that you say, well, how do I decide how to allocate between different type of assets? And you say how much you expect in terms of return and that kind of risk, and then they tell you, okay, you should put 20% in real estate and 10% on, on gold and whatever.

00:00:34 - that has happened on in recent years is that for a number of, of of assets, now you need to commit in advance, the capital. And so if somebody says, you should put 10% in private equity, it's like, I cannot just put 10% in private equity. I have to commit, an amount of money. And then I don't know when they will call this money. It may represent 10% of my portfolio, but it depends what my portfolio will do in terms of evolution, right? So for example, if, if I have a hundred million right now

00:01:06 - and they call it in two, three years time, if the stock market went very well and my portfolio is worth 200, then my 10 million committed, it is just 5% of my portfolio. but if the market tanked and went to like 50 million, my portfolio tanked and went to 50 million, then my 10 million now is 20% of my portfolio and it's probably too much. and so, so this creates a new type of risk. And, and, and there's a need for modeling this, this, this issue.

Modeling Commitment and Illiquidity in Private Equity Funds

00:01:37 - let's just model the, the fact that you have to commit in advance and have to wait until it's deployed. But it was more with the idea that then it creates a sorts of like illiquidity and, and, a, a sort of debt that you have hanging on you like an unfunded commitment, we say. So, but in the end, it turned out that most of the action came from the fact that I like what I just described, but the rest of your portfolio value is gonna fluctuate a lot. And when the money is gonna be deployed,

00:02:10 - And this is in fact what really annoys you as an investor, not so much the fact that your money is not really accessible or the fact that, you have made the, you, you have this debt hanging on you. So most private equity funds are so-called blind pooled funds. The way it works, is that you have a company which acts as general partner. You can think about KKR. So those are the gps. They manage the funds. The capital is provided by the limited partners

00:02:39 - So those are the LPs. So when the fund is raised, there's an initial fundraising period of about 18 months during which the limited partners will commit. So ante to providing cash to the GPS on demand over a pre-specified investment period of about five years. So if you're the LP, and I'm the gp, if you commit to providing, $50 million to the funds, then you basically trust me

Understanding Capital Commitment Risks in Private Equity

00:03:11 - You won't give me your money immediately. I will go looking for these opportunities. And then once I will have identified them, I will get back to you and ask for the money. So this where this can happen years after the fundraising period. So there's two risks that are associated with capital commitment. The first risk, is a risk that the limited partner does not know when the capital will be called. They don't know when the general partner will get back

00:03:42 - This risk, we refer to it as commitment, timing, risk. The second risk arises because at the time of capital commitment, the limited partner chooses the amount to commit. So they commit a positive amount of money, but their wealth is going to fluctuate between the time of commitment and the time at which the capital is invested. So if you invested this $50 million, and if that was 10% of your wealth at the time

00:04:14 - If there's a stock market crash between capital commitment and capital call, imagine that the 10% of $500 million have dropped to $250 million. this means that the capital commitment is now 20% of your wealth at the time of capital core. We call this risk commitment, quantity risk. So these two risks are very specific to private equity funds. And then there are different liquidity fee,

Challenges and Risks in Private Equity Investment

00:04:44 - to private equity funds. So there's the, illiquidity of the secondary market. If you're the LP during the holding period, you might want to sell stakes in your private, equity funds. You can go and secondary market, in which case you sell these stakes at a discount. So there's also uncertainty on the time of distribution of, investment proceeds. So as a gp, I, I invested your money, this can last years,

00:05:16 - and to return you the investment proceeds. So you have no control over that. So this, a liquidity risk was studied in a paper by Ang Pap Elau and Westerfield in 2014. And they have shown that it has effect significant effects on portfolio allocation and on welfare. What we focus on is capital commitment. So, risks that are focused on on commit commitment risk. So commitment, timing and commitment, quantity risk.

00:05:48 - and the liquidity of the second of, of the secondary market. And the other risks, the, The, the model was very complex, so it, it, it took us more than 10 years to solve it. And, and it, it, it, it, it was really, yeah, a bit crazy. yeah, it's mathematically quite demanding. We, at, at one point, we even hired a, a PhD students in math here at Oxford to like, who specialize in like partial differential equation systems. And then he failed because I think what was,

Modeling Capital Commitment Processes in Investments

00:06:19 - but you understand the economics so that you, you can really like set up things correctly to to, to solve them. so the model is quite complicated, in, in, in the details of it. But the, the overall idea is relatively simple. You, you have, when you commit capital, you start a, a pass, some process, we call it. It's, it's, it's, it's a process that has no memory. Every period your money may be called or not called. And, it has a certain expected duration

00:06:50 - to be called, and if a model would be at year frequency, it would mean you can expect to be called after three years. But if you could be called after one year, because the, the flip may, may, may be, you know, on the right side, as soon as the first year, so you have this person process that you start and you commit a certain amount of money to it, you wait until it kicks. When it ticks, it calls your money in, your money is deployed, the post on process starts again. When it ticks again, your money comes back to you and, and, and you don't have access to that money when, when, when invested and then you commit again and, and so on.

00:07:23 - but with just one process, it's quite okay. But when you start getting to two, three or more, then it gets like really hard. and in practice, people have 20 or 30 funds. So if you are doing it on 2030 funds, it'll be like technically almost impossible to program. I mean, conceptually it's feasible, but, but, but managing to program more these PDs you'd get lost in them. Like it is nearly impossible to, to, to program. and so we have a limit case where you have an infinite number of funds because things simplify a little bit. but we, we, we solved it only

Modeling Private Equity Investments: Data and Dynamics

00:07:53 - and so that, that, that's, that's a model, that we had to write. And the data, we used data that are now publicly available or relatively easily available with, with frequent. so by now, practically fund cash flow detailed data are, are widely available. And so we had like 5,000 of them to calibrate the model on. So we could do some, some things that when we started the project like 10, 15 years ago, it, it would've been harder to, to do in terms of, yeah, calibration. So it's

00:08:25 - different features of private equity. We have a dynamic, portfolio model. so it's a model, with which features, risk averse investor with infinite horizon and with access to liquid stocks, bonds, and private equity thoughts. So we model investments from the perspective of the lp. So at times zero, the LP commits a positive amount of money to the private equity fund.

00:08:56 - but this amount, this amount of money is not transferred yes, yet, between the time of capital commitment and the time of capital call, the LP is going to pay a management fee to the GP in order to model the fact that capital caller is a random time, it's a time that is unknown to the limited partner. We use a stochastic process, a hitting time, a so-called process in order to model the time of capital call when this hitting process hits on,

Investment Commitment and Liquidity Frictions

00:09:28 - And the limited partner then has the option between defaulting on their commitment or transferring the money to the private equity fund. Default is however very costly for the investor because it means that the investor is gonna lose forever access to the private, private equity markets. So if the investor chooses not to default, then they transfer the amount, which is then gonna grow at the private equity fund returner. But this amount, during the whole holding period,

00:10:00 - this amount is gonna be stuck in private equity. So basically the investor is not gonna be able to consume out of this amount. The investor is only able to consume out of their liquid wells. So the ones that is invested in liquid stocks and bonds with the, for the amount that is invested in private equity, there's no way to consume out of that. We have a second question process, a set a second hitting processor to model the fact that the distribution of the investment proceeds,

00:10:33 - So when this personal process hits, then the investment proceeds are, provided back to the investor. The investor can commit to a new fund, and basically the process repeat. So the investor maximizes the utility of consumption and gets the consumption out of investments in those liquid stocks, bonds, and private equity funds. And in this setup, we can basically study the effect of the different liquidity frictions

Modeling Commitment Risks in Private Equity Investments

00:11:04 - at a time, one at a time. So for example, we shut down, we shut down, commitment, timing, risk, yeah, by letting, the investor know when the request process is gonna hit. So by, by letting the investor know when the capital is going to be called, we shut down commitment, quantity risk by letting the investor adjust the amount that is allocated to private equity at the time of capital call. We can also study the interactions

00:11:36 - and liquidity by varying the discounts on the secondary market. So basically, if you put the discount to zero, you have a secondary market that is perfectly lake quota. So we have also some extensions of our model. We have an extension to a model in which two font can be be traded simultaneously. We have another extension in which an infinite amount of fonts can be traded simultaneously. And those extensions allow us to study whether commitment risk can be diversified

00:12:07 - So we actually spent a fair amount of time on, calibration. so the calibration of the parameters driving the dynamics of liquid stocks and bonds is very standard. It's based on s and p 500 log returns, and on three months treasury bond, what's new is the calibration of the parameters driving the dynamics of private equity funds. So for that, we use frequent data. This is data on, fund cash flows. So we have, data as of 2020,

Calibration of Capital Call Parameters in Private Equity Funds

00:12:42 - between 1991 and, to 2015. So these funds include venture capital, growth, equity, and leverage buyout funds. So we have a very large amount of, of wso. The first parameter that we calibrate is a parameter that drives the process that triggers capital core. So this parameter is intensity of capital core.

00:13:13 - that we calculate the empirical distribution of cash inflows. So the data distribution, those inflows, they contain management fees and code capital, we have them for each fund. So we have the whole distribution, and we match this distribution to the model implied distribution, and we obtain an intensity of capital call, which basically, means that the, the, the commitments, period is going to be on average three years.

00:13:46 - of capital distribution, we could be doing the same thing with, the distributions in the data. The problem with that is that we would need about 15 years of data for each funder in order to have the whole time series of distributions. So that would mean that we had, we would have to limit the exercise to funds that were raised before 2005 in order to work around that issue. What we do is that we compute the distribution

Evaluating Private Equity Performance via Public Market Equivalent

00:14:18 - of computing the distribution of outlets, we choose a very standard measure of performance, which was a public market equivalent, which was introduced in 2005 by Kaplan insurer, and which basically calculates the ratio of funds inflows discounted with the s and p 500 over fund outflows, also discounted with the s and p 500. So a fund that has a public market equivalent above one is a fund that outperformed the s and p 500.

00:14:50 - We calculate this public market equivalent. So we have a distribution of public market equivalents, which we match to, model implied distributions that we obtain using simulations. So using this procedure, we calibrate the parameters, ca driving returns of private equity funds. We obtain expected returns on private equity at, 14% against 8% for stocks.

00:15:24 - against, 15% for stocks. So these values are in line with adult values that are used, in, in, in papers. we have a beta for private equity at 1.47. That's a beta that's really close to the values that was obtained, in the paper by GaN co-authors in 2018 using a completely different method. we have a cap alpha for private equity at 3.6%,

Understanding Commitment and Quantity Risks in Private Equity

00:15:56 - So our value makes sense, they're in line with the literature, they're in line with intuition, and it allow us to match distributions of performance measures and inflows remarkably well. Commitment, quantity risk is very costly to the investor, whereas commitment, timing risk is not. So what this means is that the investors care very much about, what the commitment to wealth ratio is going

00:16:26 - They care very much about not knowing what this ratio will be, but they don't care so much about not knowing when capital call will happen. So why do they care so much about the commitment wealth ratio? The reason is that if this commitment towards ratio becomes too large at the time of capital Cola, investors basically have the option to either default, but we've seen that default is pretty costly for them or to transfer that amount

00:16:55 - of their portfolio invested in private equity for unknown period of time. So in order to avoid reaching these high values of commitment to wealth ratios, what investors do is that they under allocate, they under commit to private equity. So because they under commit, they would like to adjust their commitment at the time of capital call. And in fact, they would be willing, they would like to nearly double the commitment at time of capital call.

Challenges in Diversifying Commitment Quantity Risk

00:17:28 - to have this possibility. They'd be willing to pay 1.2% of their initial wealth in order to have this possibility. And this represents a quarter of their P commitments. So that's, that's a large amount of money. So now the question is, can we actually diversify this risk in order to address this question? What we did is that we looked at what happened when extending the model to a model with two funds and a model with an infin model of false.

00:18:00 - we actually can't diversify commitment, quantity risk. There's two reasons for that. The first reason is that commitment, quantity risk is driven by the commitment wealth ratio. If there's a drop in the stock market, the commitment wealth ratio will increase for all funds at the same time. So there will be no diversification effect. There's no way to diversify commitment, quantity risk by increasing the amount of funds.

00:18:30 - that there's a possible funding mismatch. If you have many funds, then you'll finance the capital calls of the later funds using the distributions of the earlier funds. But remember that the times of capital, coal and capital distribution are stochastic. So if capital calls happen a little too early, or capital distributions happen a little too late, then you won't be able to finance later funds with the proceeds of earlier funds, and there will be a mismatch.

Understanding Commitment Quantity Risk in Investment

00:19:01 - quantity risk is a big deal, for investors. It's, a bigger deal than the other illiquidity frictions. When we compare the cost, it's larger. And this is not a risk that can be diversified by, investing in, in many funds. the strategic default would be like, you know, you, you committed money, but you are, you are allowed to say no when it, when, when, when it comes in, right? So, so then it's a, it's, it's a, if you take it literally and there will be no penalty, then, then, then that's it. The commitment doesn't matter because you can just renege whenever you want.

00:19:34 - renege on, on, on a commitment. So you, you, you, you have to do it. nonetheless, you could, you, you, you, you could have some strategic de de default or so during the life of a deal and so on. And we, we, we, we, we model that. It doesn't have like a major, IM impact, because if you default, you need to restart again. So you, you have committed and then you say, okay, I changed my mind. I want to be out now. But then you need to recommit and you need to wait again the entire amount of time, right? So it's a bit like you go to the doctor, you've been waiting

00:20:03 - And so you say, okay, I'm, I'm gonna default on this. I'm gonna leave, and, but you still need to go to the doctor and then you're gonna start again. But like, well, you know, you might as well have stayed because it, it doesn't change everything. So it, it is slightly different than that because the world has changed during that hour you were waiting. So you may, adjust a few parameters, but it, but it's not massively useful Is that neither the option to default nor the secondary markets actually alleviate commitment, quantity risk. So remember that in order to manage commitment, quantity,

Understanding Commitment Quantity Risk in Private Equity

00:20:37 - optimal given this under location, to default at the time of capital, call, especially because default is so costly for, the investor. Similarly, the secondary market is very rarely used. And the reason for that is that in a model with one funder, partial sales are not allowed, which is in line with practice. And so it's rarely optimal for the investor to sell all their stakes in private equity.

00:21:09 - nor default are actually options that are really worth it for the, for the investor. The second result is that the liquidity on the secondary market and commitment, quantity risk are not substitutes. There are actually compliments. So let me explain that one good thing, that by making the secondary market more liquid, so by, letting the discount on the secondary market go to zero commitment, quantity risk would disappear.

00:21:37 - If you pull the discount up to zero, what's gonna happen is that the investor is going to invest more in private equity, and this increased allocation will make them even more willing to pay in order to get rid of, commitment, quantity risk. So basically commitment, quantity, risk matters. And it's not, it's neither the option to default nor the secondary market that, can alleviate it. Often when people insure investors and, and always think about liquidity risk management in private

Balancing Liquidity and Private Equity Commitments

00:22:10 - to keep some money in cash or bonds or liquid assets, because in case money gets called in private equity, they need to have it readily available. and we, what we show again in our paper is that this is not such a big deal if you are being reasonable with your private equity allocation. So if you are being reasonable, if you calibrate the model, right, in any case, you don't want to have more than 15, 20% in private equity. And if it's only 15, 20% of your portfolio that is illiquid, it is not really a big deal.

00:22:40 - for, for the capital call. You can sell some shares, et cetera. the people get in trouble in practice if people commit 50, 60 or 80% of their portfolios, but our model will never tell you to do that. So then, you know, that, that, that liquidity per se is never an issue. What's an issue, however, for, then insur investors is that the, the fact that again, when you commit money, when money will be called, you don't know how much of your portfolio this will represent.

00:23:11 - which we actually hadn't anticipated, is that the model, says that you should under commit to private equity. And then if you can top up at the moment, the money is cold. And that's very interesting because it's actually what people are doing, but then they don't seem to be doing it for that reason. but, but, but, so it gives them one more reason for doing this, this thing. So to be more concrete, the idea is that imagine you want to be 20% in private equity

Strategic Capital Commitment and Co-Investment Insights

00:23:41 - it would be 20% given how usually the rest of your portfolio grows over time, and you will be correct, but you do have a chance that you may end up with 25 or 30% if, if, if your portfolio, went down in value. So what you do is that you say, okay, I'm gonna commit so that on expectation I will be at 10% when my money is called. Now, it's annoying because I usually want to be at 20%, but by, by committing so that I'm at 10% in expectation, I will hardly ever be at 20% or more. and so in that zone, I don't wanna be,

00:24:16 - around 10 and I don't wanna be around 10, I wanna be a 20. So it'll be very handy if I can top up at don't moment. The, the money is called with something that gives you me immediately the kick if I want to, to be a 20. And that will be in, in a, in, in a world where the, the, the rest of the market has increasing value. So it's in booming times. In good times when my money is called, I will typically want to top up. And we do see that in practice it's called co-investment programs. So when money is called from investors, they are invited

00:24:45 - to co-invest much more when we are in good time when the markets are booming. So they are doing that, but usually they're motivated more by like things like saving fees and things like that. But they never think about co-investments as a tool to manage their liquidity. So under committing prior equity, and if it turns out that things are going okay and you are under committ when the capital call is coming, then you use the co-investment to top up. so that's like an, an insight that we haven't thought about that, it makes a lot of sense once you see the solution.

Understanding Commitment Quantity Risk in Private Equity

00:25:18 - but, but not always intuitively like ex we, we, we haven't thought about that In order to manage, commitment, quantity risk because that's what really matters for private equity investors. investors under allocate to private equity. And this under allocation is costly. It has impact in terms of portfolio allocation and welfare for investors. This result actually rationalizes something we've been observing over the last, 10 to 15 years,

00:25:49 - co-investment opportunities. So co-investment opportunities are offered by general partners to some limited partners, so their favorite ones, and they basically allow the limited partners to get rid of commitment, quantity risk, to adjust the commitment at the time of capital call. So what our model and our resource, explain is why this co-investment opportunities have been so popular and why there's so much demand for them.

00:26:21 - quantity, risk.

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