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Multistage stochastic optimization for private equity investments


Long-term investments such as Private Equity (PE), present timing differences in cash inflows and outflows. When allocations in PE are not planned correctly, investors can suffer liquidity problems when paying for unexpected commitments. We present a multistage stochastic optimization model that includes PE assets as free cash flows projects. This model can determine PE allocations, in relation to public equity, under different market settings. Using a factor-based model to construct public and private equity markets, the major findings are: Liquidity problems can be avoided by planning PE allocations in advance and according to market conditions. Our tool reduces commitments taken for more volatile PE market or when investor target is lower. For target returns above 20 per cent, PE allocations enhance portfolio annual returns from 2 to 3 per cent (no volatility increase) only if PE net present value volatility is below 15 per cent. Beyond this point, higher returns comes with more risk. When PE investments are less correlated with public equity, the latter threshold extends to 45 per cent. PE allocation weight changes in time and according to its age. For favorable market conditions and high investors’ appetite, PE investment value can be greater than the entire wealth at some time periods. Leverage option for PE investments decreases performance, even in low PE volatility market. Potential PE gains are offset by debt interest payments and risk becomes higher.

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  1. Societe Generale Index since 2004.


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Correspondence to Lorenzo Reus.


Appendix A

Problem size

The model has nodes. There are 2N=2(MT+1−1)/(M−1) variables for Dn and Wn, MT variables for MDDn, variables for public assets (with cash) and only I variables for PE investments. So there is no difference in complexity when considering PE investment or not. Finally there are 2MT variables for the excess y and Z used to measure the utility function.

To calculate the number of constraints, notice that each node has M childs. So the total number of constraints in (3) and (5) are The number of constraints for (4) and (6) is L+1, while (7) is MT. The number of constraints in (8) and (9) is Finally, the number of constraints in (10) and (11) is (T+1)MT and

respectively. The constraints linked to the sign of the variables sum up to (3L+2)(MT−1)/(M−1)+I+2MT.

For the setting described in the section ‘Strategy construction, settings and measurements’ (M=2, L=10, I=1, T=10), the number of variables and constraints are 39 903 and 80 853, respectively:

Appendix B

Public equity market


Private equity market



Appendix C

Using (14) and (15), Figure C1 shows the mean and volatility of public equity assets, for a particular factor matrix Ω with the setting given in the section ‘Rolling horizon procedure’. Table C1 shows the correlation between public equity returns and the single PE netflows, using different values of ρ1i given in (19).

Table C1 Correlation of public equity returns and PE net flows for different correlation between l*=1 and single PE
Figure C1
figure 10

Mean and volatility of public equity in factor-based model.

Figure C1

Table C1

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Reus, L., Mulvey, J. Multistage stochastic optimization for private equity investments. J Asset Manag 16, 342–362 (2015).

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  • investment analysis
  • private equity
  • stochastic optimization
  • asset allocation