3.10 Case studies

3.10.1Long-Term Capital Management, 1998: leverage meets correlation#

Context LTCM was a Greenwich, Connecticut hedge fund founded in 1994 by John Meriwether, with partners including Myron Scholes and Robert Merton. It ran relative-value trades (convergence trades between closely related bonds, swap spreads, equity pairs and volatility positions), each with small expected profit and, historically, low volatility, made worthwhile by leverage. Primary source: President's Working Group on Financial Markets, Hedge Funds, Leverage, and the Lessons of Long-Term Capital Management (April 1999), https://home.treasury.gov/system/files/236/hedgfund.pdf (PWG below).

Timeline (PWG unless stated)

• End-1997: about $2.7 billion returned to investors; capital of $4.8 billion at 1 Jan 1998 (Lowenstein's account gives $4.67 billion; sources differ). Total assets $129 billion at end-1997, leverage about 28 to 1. Off-balance-sheet derivatives with notional value of about $1.3 trillion at end-1997.
• 17 Aug 1998: Russia devalued the rouble and declared a debt moratorium.
• 31 Jul to 31 Aug 1998: capital fell from $4.1 billion to $2.3 billion, a loss of $1.8 billion (about 44%) in August. Balance-sheet assets at 31 Aug exceeded $125 billion, leverage above 25 to 1 on the original capital and far higher on the remaining capital; futures notional above $500 billion, swaps above $750 billion and other OTC derivatives above $150 billion.
• 23 Sep 1998: 14 banks and broker-dealers, in a recapitalisation organised by the Federal Reserve Bank of New York, put in about $3.6 billion for 90% of the fund ($3.625 billion per Federal Reserve History, "Near Failure of Long-Term Capital Management"). Total 1998 losses of about $4.6 billion are reported by Lowenstein (reported by Wikipedia); the PWG report does not state a total.

Analysis Three mechanisms from this module combined.

1. Kelly with small $\sigma$. For a convergence trade with excess return $\mu$ and volatility $\sigma$, $f^* = \mu/\sigma^2$. A trade earning 1% a year with 2% volatility has $f^* = 25$. LTCM's leverage was in the range a Kelly calculation on calm-period volatility would endorse. But Proposition 3.9's companion result applies: if the true volatility is $\lambda$ times the estimate, the real multiple is $c\lambda^2$. A doubling of volatility in a crisis turns full Kelly into $c = 4$, deep in the negative-growth region; at 25:1 leverage a 4% fall in asset value removes all equity.
2. Correlation to one. The PWG report says the shocks of August 1998 "confounded expectations of relatively low correlations between market prices", showing that global portfolios like LTCM's were less diversified than assumed. Proposition 3.10: positions that are independent are each sized near their own Kelly fraction; if they become perfectly correlated, the book is $n$ times Kelly on one bet. The common factor was a global flight to liquidity: in every trade LTCM was long the less liquid instrument and short the more liquid one.
3. Fat tails and the drawdown constraint. A 44% monthly loss on a strategy with a historical volatility in the teens is many standard deviations under a normal model (M02 · 2.10.2). And unlike a theoretical Kelly bettor, a leveraged fund faces margin calls: wealth falling to a fraction $x$ is ruin when lenders withdraw. With $c \ge 2$ the drift $m_c$ of Proposition 3.8(a) is non-positive, so the probability of reaching any $x < 1$ is 1 (the limit of (3.12) as $c \uparrow 2$).

3.10.2Archegos Capital Management, 2021: concentration hidden in swaps#

Context Archegos was the family office of Bill Hwang. It held concentrated long positions in a small number of US-listed stocks (including ViacomCBS and several Chinese ADRs) mostly through total return swaps (derivatives in which a bank holds the shares and pays the client the shares' return in exchange for a financing fee), with several prime brokers, none of which saw the whole position.

Timeline

• According to the SEC's complaint (press release 2022-70, 27 Apr 2022, https://www.sec.gov/news/press-release/2022-70), Archegos grew from about $1.5 billion of capital and $10 billion of exposure in March 2020 to over $36 billion of capital and $160 billion of exposure in March 2021.
• Week of 22 Mar 2021: ViacomCBS shares fell sharply after a share offering by the company (widely reported; the exact announcement date is not independently confirmed here). Archegos missed margin calls on 25–26 Mar, and from 26 Mar its prime brokers sold the positions in large blocks. ViacomCBS fell about 27% on 26 Mar (reported by multiple outlets via Wikipedia).
• Bank losses (reported by FT, Reuters and Barron's via Wikipedia): Credit Suisse about $5.5 billion, Nomura about $2.85 billion, Morgan Stanley about $911 million, UBS $774 million in Q1 2021, MUFG about $300 million.
• Hwang was convicted on 10 Jul 2024 on 10 of 11 counts and sentenced on 20 Nov 2024 to 18 years (reported by the NYT and Reuters via Wikipedia).

Analysis Gross exposure of $160 billion on $36 billion of capital is leverage of about 4.4. That is modest by LTCM's standard; the problem was what it was applied to. A portfolio concentrated in a handful of volatile stocks in two clusters (US media, Chinese internet ADRs) has a covariance matrix close to rank one, so by Proposition 3.10 it is approximately a single bet. Assume, for illustration, that such a portfolio has 50% annual volatility. Then (3.9) gives a growth rate of $4.4\mu - \frac12(4.4)^2(0.25) = 4.4\mu - 2.42$: the portfolio needed an arithmetic excess return of 55% a year just to have zero expected log growth. Equivalently, 4.4 times leverage on a 50%-volatility asset is Kelly-optimal only if $\mu = f\sigma^2 = 1.1$, an expected return of 110% a year. Whatever the actual volatility, the leverage implied an extreme belief about the edge, while Proposition 3.9 says the evidence for any edge in five stocks is a handful of years of data.

A second mechanism is outside the Kelly model: position size relative to market liquidity. Archegos's holdings were large fractions of the free float of some stocks, so forced selling moved prices against the seller (market impact, M05). The model assumes the price process does not depend on your trades; at that size it does, and the effective volatility in liquidation is far higher than the historical one.

3.10.3Ed Thorp: Kelly in practice#

Context Edward Thorp, a mathematician, used the Kelly criterion (Kelly 1956) to size bets in blackjack after proving that card counting gives the player an edge in some deck states. Beat the Dealer (Random House, 1962) published the method. He then applied the same reasoning to warrants and convertible bonds, and from 3 Nov 1969 ran an investment partnership, later Princeton–Newport Partners, on hedged convertible and option arbitrage.

What the record shows In his 2006 chapter, Thorp reports that the partnership and its successors compounded at about 20% a year with an annualised standard deviation of about 6% and roughly zero correlation with the market, through May 1998, on about $80 billion of trades (Thorp 2006, self-reported; no independent audit is cited here). The same chapter gives the full and half Kelly numbers of §3.4 (half Kelly has three-quarters of the growth; probability of ever halving 1/2 and 1/8), and works a Kelly example on Berkshire Hathaway ($m = 0.20$, $s = 0.15$, $r = 0.06$, giving $f^* = 6.22$).

Analysis Apply (3.10) to the reported figures, assuming a risk-free rate of 6%: $S = (0.20 - 0.06)/0.06 = 2.3$ and $f^* = 0.14/0.0036 \approx 39$. Nobody ran these strategies at 39 times leverage, and Thorp's own writing stresses betting below full Kelly. The reasons are exactly those of this module: the 6% volatility is measured in calm conditions and would rise in a crisis (3.5, LTCM above); hedged arbitrage returns have fat left tails from deal breaks and funding squeezes (M02 · 2.6); and the realised Sharpe ratio is an estimate whose error, while small at $t \approx S\sqrt{29} \approx 12$, is not the main risk. For blackjack, by contrast, the edge in each deck state is computable from combinatorics, the outcomes are bounded, and full or near-full Kelly on the computed edge is defensible. Kelly works best when the probabilities come from a model you can prove, and worst when they come from a backtest.

3.10.4The retail F&O pattern in SEBI's studies#

Context SEBI's studies of individual traders in equity derivatives are tabulated, with sources, in M01 · 1.11 Case 1 (93% of individuals lost money over FY22–FY24; 87.7% in FY26). This case uses only the four findings that bear on bet sizing. From the FY26 study (SEBI PR No. 50/2026, 20 Aug 2026): "higher turnover relative to capital employed or equity portfolio was associated with higher loss rates"; about 90% of traders who lost in two consecutive years and kept trading lost again; in 79% of cases average gains in profitable quarters were smaller than average losses in losing quarters; and options were about 92% of individual losses.

Analysis Each finding maps to a result in this module.

1. Negative expectancy after costs. If the average retail options trade has $E \le 0$ in R terms before costs (option buyers pay the variance risk premium, M09) and costs are added, then $E < 0$. With $E < 0$ no positive bet size has positive growth (Proposition 3.2), and fixed-size betting leads to ruin with probability 1, after an expected $a/(q - p)$ bets, with $a$ the capital in bet units (Proposition 3.12 with $N \to \infty$).
2. Turnover relative to capital. Turnover relative to capital is bet size relative to wealth, $f$. With $E < 0$, $g(f)$ is decreasing in $f$, so higher turnover per unit of capital means faster expected decline. The SEBI finding is what the theory predicts.
3. Small wins, large losses. "Average gains smaller than average losses" with a win rate below the break-even level is $p\bar W < (1-p)\bar L$: negative expectancy expressed in Tharp's terms. It is consistent with the disposition effect (selling winners early, holding losers; M14), which mechanically shrinks $\bar W$ and grows $\bar L$.
4. Persistence. Ninety per cent of two-year losers losing again is what sampling theory predicts for traders whose true $E$ is negative: their losses are not bad luck, and Proposition 3.3(c) says time makes them more certain, not less.