De-Smoothing Private Credit: Risk Contribution and the Allocation Decision
A pension fund's investment committee is reviewing a 25% allocation to a private-credit fund; the remaining 75% is a global equity index tracker. Over five years the private-credit fund reports an annualised return of 8.0% and volatility of 4.0%, with a reported beta to global equities of 0.20. The equity sleeve is assumed to return 7.0% with 16% volatility. The cash rate is 4.0%. The private-credit fund's monthly returns have first-order autocorrelation of 0.35, and valuations are manager-appraised with a quarterly redemption window and a 5% gate. The committee's total risk budget is 10% annualised volatility. The committee proposes raising the allocation to 35% on the strength of the reported Sharpe ratio and what it calls the fund's low correlation with equities.
1. De-smooth the private-credit return series and recompute its volatility, beta and Sharpe ratio. What does this say about the reported 4% volatility?
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De-smoothing. Model appraisal smoothing as an AR(1): r_t = (1 - rho) r*t + rho r{t-1} with rho = 0.35. Then
Var(r) = [(1 - rho)/(1 + rho)] Var(r*), so
sigma* = sigma x sqrt((1 + rho)/(1 - rho)) = 4.0% x sqrt(1.35/0.65) = 4.0% x 1.441 = 5.76%.
(The cruder sigma/(1 - rho) adjustment would give 6.15%; the AR(1) result is the correct one for this specification and is itself a lower bound.)
Beta. A contemporaneous regression only sees the part of the equity response that has already been marked. The first-order correction is beta* = beta/(1 - rho) = 0.20/0.65 = 0.31. Roughly a third of the fund's equity sensitivity is being reported in later months.
Sharpe. Reported: (8.0 - 4.0)/4.0 = 1.00. De-smoothed: (8.0 - 4.0)/5.76 = 0.69.
What it says about the 4%. It is an artefact of the appraisal process, not a risk measurement. A quarter of the reported volatility is the smoothing itself. And 5.76% still flatters: it corrects the mechanical lag but not credit tail risk, mark discretion, or the illiquidity the investor is being paid to bear.
2. Compute the fund's total volatility and the risk contribution of the private-credit sleeve, on both reported and de-smoothed inputs. Interpret the difference.
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Let w = 0.25 private credit, 1 - w = 0.75 equity.
De-smoothed inputs: sigma_pc = 5.76%, covariance = beta* x sigma_e^2 = 0.31 x 256 = 78.9 (in %^2).
- Var = w^2 sigma_pc^2 + (1-w)^2 sigma_e^2 + 2w(1-w)cov
- = 0.0625 x 33.2 + 0.5625 x 256 + 0.375 x 78.9
- = 2.07 + 144.0 + 29.6 = 175.7 -> sigma = 13.3%
Reported inputs: sigma_pc = 4.0%, cov = 0.20 x 256 = 51.2.
- = 1.0 + 144.0 + 19.2 = 164.2 -> sigma = 12.8%
Risk contribution of the private-credit sleeve.
(Sum w)_pc = w sigma_pc^2 + (1-w)cov = 0.25 x 33.2 + 0.75 x 78.9 = 67.4
Contribution to total variance = 0.25 x 67.4 / 175.7 = 9.6% on de-smoothed inputs, versus 0.25 x (0.25 x 16 + 0.75 x 51.2)/164.2 = 6.5% on reported inputs.
Interpretation.
- The risk budget is breached on either set of numbers - 12.8% reported against a 10% target. De-smoothing is not what causes the breach; it widens it by only about 0.5pp at the total-fund level, because equity dominates total variance.
- The material effect is on contribution: the private-credit sleeve carries about 50% more of the plan's risk than reported numbers imply (9.6% versus 6.5% of variance). That is the number that should drive limits and the incentive fee.
- De-smoothing raises sigma, beta and the covariance together, so it does not make private credit look like a worse diversifier. Its correlation to equity is high in both cases - around 0.85 on de-smoothed inputs. 'Low correlation' is an artefact of the same lag.
3. The committee wants to raise the allocation to 35% because the reported Sharpe is 1.0 and correlation looks low. What is your recommendation, and what constraint actually binds?
Read the worked answer
The Sharpe is not comparable. De-smoothed it is about 0.69, and part of the residual spread over the equity sleeve is an illiquidity premium the plan is being paid for non-tradability, not for skill or inefficiency. Strip out roughly 2% for illiquidity and it falls to about 0.35 - not obviously better than the equity sleeve's 0.19 once the leverage-like effect of stacking a low-vol, smoothed asset against a high-vol one is recognised.
The correlation is an artefact. Reported beta 0.20 looks like diversification; the lag-adjusted beta of 0.31 with sigma_pc of 5.76% implies a correlation near 0.85. Diversification is real but far smaller than the reported numbers suggest.
At 35%: variance = 0.1225 x 33.2 + 0.4225 x 256 + 0.455 x 78.9 = 4.1 + 108.2 + 35.9 = 148.2 -> sigma = 12.2%; the sleeve would carry about 0.35 x (0.35 x 33.2 + 0.65 x 78.9)/148.2 = 14.9% of total variance. Still a breach of the 10% budget, made worse by the fact that the sleeve's risk is now less diversifiable in a stress.
The binding constraint is liquidity, not volatility. A quarterly redemption with a 5% gate means that in a stress the plan cannot rebalance out of the sleeve: the gate binds precisely when the marks are falling and the equity sleeve is also down, with the private-credit lag deferring rather than avoiding the loss. Sizing should be driven by the plan's cash and collateral needs over the gate period, not by a volatility budget alone.
Recommendation: hold at 25%; size on de-smoothed inputs and a liquidity stress, not reported statistics; require monthly look-through to underlying borrower credit quality and independent marks before any increase; and reprice the risk contribution in the internal limit framework so the sleeve is charged for the risk it actually carries.