A friendly reminder that IRR ≠ CAGR and that volatility laundering is real
Once upon a time, in this most festive and enchanted season of the year, a poor cobbler and his two sons were visited by the Magic Investment Return Fairy. The cobbler having wished a more modern boon than having Santa’s elves make shoes while he slept, the fairy, seeing the sons’ delight in being identical in every way, granted them each the gift of the same financial skill: market timing.[1]
To share the blessings of this great Christmas miracle with all the investors of that land, each brother started a fund. The first brother launched an exchange-traded fund, raising $1 million in invested capital. The second opened a private equity fund, with $1 million in committed capital.
In the first year of investing, the first brother divined that an evil lay over the markets, and so he invested only 10% of his ETF’s capital in stocks that year. Similarly, his brother called up only $100,000 of his investors’ commitments, buying a portfolio company with no alpha and a 1-beta. As the brothers had foreseen, it was a poor year for the stock market, which declined 5%; but due to the brothers’ wisdom in putting but little money to work, only $5,000 of their investors’ initial $1 million was lost.
As the holiday season again heralded the coming of a new year, the brothers perceived glad tidings of great markets. The first brother invested another $400,000 of his fund in public stocks, and the second brother likewise called up another $400,000 of his fund’s committed capital to purchase another private but market-tracking company. Again the brothers’ foresight was sound, as the market rallied 15% in that year, earning an additional $74,250 on the $495,000 each brother had invested in equities at the beginning of the year.
As the third year approached, sensing that their inner glow was more than just the cobbler’s eggnog, the brothers put their remaining capital to work. The first brother invested into the stock market the remaining $500,000 in his ETF. The second brother called up the remaining $500,000 from his investors and bought a third 1-beta, 0-alpha company. Again, the brothers’ wisdom prevailed, as the markets rose 5%, earning each brother’s fund an additional $53,463 on the $1,069,250 invested at the beginning of the year, resulting in a terminal value of $1,122,713 for both funds.
After three years, the investors, advisors, peasants, pensions, and journalists of every kingdom, having heard tales of the investing wisdom of the two brothers, made a yuletide pilgrimage to that land to see if the stories were true. “Show us your return figures,” said they. The first brother showed that his ETF had a compound annualized growth rate (CAGR) of 3.93% for the three years.[2] The second brother showed that he had produced for his investors an internal rate of return (IRR) of 7.43%.[3]
Seeing these figures, the investors, advisors, peasants, pensions, and journalists marveled at what they assumed must be the superior wisdom of the second brother. “Active ETF Underperforms Public Equity,”[4] said the first headline, and the article quoted the first brother’s 3.93% CAGR next to the 4.68% CAGR earned by the market itself.[5] The first brother protested—correctly, but to no avail—that he had produced positive alpha to the market; his listeners thought “alpha” just meant “higher return”,[6] so they thought he must surely be wrong. “Private Equity Fund Outperforms Public Equity and Active ETF,” said the second headline, and the article quoted the second brother’s 7.43% IRR next to the market’s 4.68% CAGR and the first brother’s 3.93% CAGR. Nobody bothered to notice that the investors in the two brothers’ funds had ended up with exactly the same amount of money after three years.[7] Since both CAGR and IRR were measures of return, and since return was surely all that mattered, everyone assumed that the second brother must be twice as smart an investor as the first brother.
Nobody required the second brother to calculate his alpha, but even if they had, he could have shown a rather impressive figure, both because he had generated true positive alpha (albeit precisely the same positive alpha produced by the first brother) and because he had enjoyed the privilege of holding an opaque portfolio of private companies, enabling him to smooth his reported quarterly NAVs, thus causing his portfolio to appear less volatile than it really was. The first brother’s ETF held publicly traded securities and traded like a stock itself, so its volatility could not be obfuscated.
Seeing these headlines, the first brother’s investors redeemed their shares. The second brother’s investors would not have been allowed to redeem their holdings even had they wanted to, due to fund lockups; but of course they didn’t want to. In fact, both the first brother’s investors and the second brother’s investors clamored to sign up for the second brother’s second-round fund. In a gesture of Christmas generosity, the second brother hired the first as the lead analyst at his firm, raised his fees, and continued to produce both genuine and specious alpha for his investors.
And they all lived happily ever after.
[1] I told you this was fairy tale!
[2] Calculation: (1,122,713 / 1,000,000)^(1/3) – 1 = 3.93%.
[3] Calculation: IRR(-100,000, -400,000, -500,000, +1,122,713) = 7.43%.
[4] Okay, look, absent holiday hocus pocus, this is a headline I’m inclined to believe. After all, there is no magic investment return fairy!
[5] Calculation: [(1-5%)*(1+15%)*(1+5%)]^(1/3) – 1 = 4.68%.
[6] I don’t care what the style guides say, that comma makes more sense outside the quotation marks.
[7] In another alternate universe, slightly closer to our own, the first fund manager would charge 1% for his services, and the second would charge a 2% management fee and 20% carry—enough to wipe out even his fake alpha, let alone the real (where, by real, we mean fairy magic) benefits of his market timing gift. Also, for those wondering why the cash earned nothing in this tale, let’s just pretend the interest and the fees offset perfectly. Or if we want to be really generous, we can say the brothers earned just enough extra alpha to cover their fees.
Image by Gemini AI
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