*Originally publised on Pulse: https://www.linkedin.com/pulse/challenging-validation-contrario-massimiliano-saccone-cfa/*

Over the past two weeks, the conversation and the exchange of emails with a highly reputable and quant skilled professional in the private equity industry have posed an interesting intellectual challenge and created a very useful opportunity for testing “*a contrario”* the DaRC methodology and for discussing the relation between duration and time horizon.

My authoritative interlocutor purported that, for the same PE fund under analysis, the theoretical case of “maximum recycling” (i.e. the possibility of netting the cash flows sometimes admitted in the limited partnership agreements) that was simulated, based on the simplified annualized example of my patent submission, would have brought to different results than the case of the patent example (where contributions and distributions are treated according to the way they are tagged in the relevant general partners’ notices).

If the results were to be different between the two cases, the stated implication was that “the DaRC metrics should be treated with caution, and would not make a good basis (for indices and) derivative transactions”.

Unpleasant outcome, wouldn’t it be, after having invested researching and developing the DaRC over the last 6 years exactly for indexation and synthetization purposes?

But the competence of the source of the challenge deserved a careful review of the spreadsheet and the calculations that were provided vis-à-vis the formulas and the spirit of the DaRC methodology.

In spite of my hammering throughout this blog, what the challenge was bringing about was that a pretty high IRR number seemed the only stable performance reference. Instead, the challenged DaRC metrics (like the TVPI multiple, but on a significantly lower variance scale) looked actually different, apparently proving my interlocutor’s thesis.

Just looking at percentage figures, I risked to be fooled by IRRs.

Strange feeling. Because the simulations, structured according to the logic of the proposed challenge but performed using our QY analytics suite on actual market data, were delivering identical DaRC results for the two cases.

Hold on!

There’s a missing element in the spreadsheet. That is the proper consideration of the time horizon, that is instead the fundamental constraint of our QY analytics suite. That’s where the differences in the spreadsheet come from! I can reconcile them – it’s not in the DaRC metrics’ development spirit that the identical wealth growth underlying the two cases ended up not being represented by an identical time-weighted rate of return.

Returns, be it in percentage or dollar terms, should never be evaluated without the proper, explicit consideration of time.

And time, in a financial framework, matters both as:

- time horizon (hence the notions of since inception, contractual maturity, annualized rates, compounding that have all to do with the calendar time and the 1
^{st}of Jan. – 31^{st}of Dec. periodicity associated with time-weighted return conventions of our debentures); - duration, whose explicit consideration the DaRC methodology imposes preliminarily to the calculation of the rate of returns – under the deterministic constraint both financing and reinvestment subjective assumptions should not be part of performance calculation, hence diluted (which implies that the investor is on the capital market line tangent to the efficient frontier and his/her only assets are the fund and the risk free rate, at which he can lend or borrow).

But back to the challenge to put the numbers in perspective. The spreadsheet showed the durations of the DaRC metrics in two different tabs so that it was easy to lose sight of the slight difference (5.98 versus 6.35 years) between the two cases, noticing and “worrying” instead about the difference of the returns (respectively 16% versus 15%).

In reality, what seemed to be different results for the total wealth of the hypothetical investors of the two cases were simply different representations at different time horizons – i.e. contractual maturities.

If the two cases had been recalculated on the same time horizon the performances would have been identical – demonstrable, for the interested readers, with daily frequency at any point in time over the life of the funds (or any derivative or indexed matching replication).

“But there’s a duration issue then!” My interlocutor’s doubts hadn’t been dissipated yet….

Actually, there is no issue. Instead and I believe, coherently, given the different contractual agreements represented by the two cases, the derived durations are instead supposed to be different because the two funds’ contractual risk profile that generates them is different.

In fact the possibility of recycling capital may reflect the fact that, of the total capital initially committed, more capital remains available for future investment commanding a longer duration, hence the possibility of higher (longer term) risk.

Therefore, for the same net cash flow profile, investors in the two cases, investing directly or in derivative form, will have at any point in time the same performance and the same wealth – but the recycling option will imply more risk. Both the risk and return characteristics will be unambiguously and deterministically captured by the DaRC metrics.

To conclude and draw a clear line of distinction between the DaRC and other performance analysis methodologies that were recalled in the discussion, I’d like to go back to the example.

Assuming a 6.35 years’ cut-off, I deterministically reconciled the figures in the example with the NAV plus the residual cash as if the investors were on the capital market line – with final wealth equivalent to 2.43x the amount initially committed and an annualized since inception time –weighted performance of 15%.

For completeness, for the same cases, the IRRs were 43% (which compounded over 6.35 years would imply a 10.04x wealth growth) and I am not sure what else (because compounding is clearly not appropriate) and the TVPI multiples were 2.3x or 4.25x.

There may be ways to reconcile IRRs and TVPIs (and possibly other methods) to the year 6.35 2.43x deterministic wealth of the investors in the two cases (and their subjective choices of financing, leverage and reinvestment).

But the interesting outcome of this exercise is that they’ll need to produce a time weighted performance of 15%, identical to the figure produced by the DaRC metrics – otherwise it will offer the possibility of a risk-free arbitrage profit.

Which makes all this now a “relieving a contrario DaRC validation”.