On December 31, 1997, Long-Term Capital Management L $\sigma$ , Where E(R) Is the Expected Value And

Essay

On December 31, 1997, Long-Term Capital Management L.P. ("LTCM" or "Management Company") distributed approximately $2.7 billion of capital to investors in the fund that LTCM managed. Following a high return for 1997 (25.3% gross of fees and 17.1% net of fees) the fund's NAV (Net Asset Value) had swelled to $7.5 billion, and the share of the fund held by LTCM principals and employees had grown to $1.9 billion.
LTCM had notified investors in September, 1997 that it would be reducing investor capital in the fund at year-end. In a letter to investors, LTCM explained that it had decided to return to "non-strategic" investors the capital that was contributed between 1995 and 1997. The firm explained its reasoning in a letter to shareholders dated:
"As previously reported, the Fund has for some time been closed to new investment from non-Management Company investors with the exception of compelling cases of strategic value that would accrue from additional investment. From inception, the Fund has implemented its investment strategies subject to a constraint on its level of risk and subject to the requirement of maintaining adequate liquidity capital. The Management Company believes that these two constraints are not binding currently and the Fund has excess capital. This has occurred, primarily, because of a substantial increase in the capital base from the larger-than-expected, past realized rates of return, and high reinvestment rates elected by the Fund's investors. Therefore, it has become necessary to reduce the amount of capital significantly to bring the Fund's capital base more in line with its risk and liquidity needs."
The fund had set a risk target of an annual standard deviation (sigma) of NAV equal to 20% of NAV (approximately that of a U.S. stock portfolio). The expected profit (increase in NAV) for 1998 is $750 million (a daily expected profit of $3 million multiplied by 250 trading days per year).
Looking back, the daily sigma of NAV was $45 million, which translated in an annual sigma of
$710 million. This figure was estimated over past daily movements in NAV. The annual sigma is computed by multiplying the daily sigma by the square root of the number of trading days in a year (approximately 250 trading days). The mathematical arbitrage models used give a slightly higher theoretical figure equal to a daily sigma of $60 million, or annual sigma of $950 million. So the annual risk was well below the target of 20%. The empirical measure was less than 10% of NAV (710/7500) and the theoretical measure was 12.8% (950/7500). Because new arbitrage opportunities were hard to find, it was decided to get back to the risk target on NAV by reducing the capital base rather than expanding the arbitrage portfolio.
a. How much capital could LTCM reimburse to satisfy its 20% risk target (use the data from the mathematical arbitrage model)?
LTCM did calculate VaR. You will base your calculation on the theoretical sigma measures (daily sigma of $60 and annual sigma of $950). Remember that in a normal distribution, there is 5/100 chances to be below E(R) - 1.645 $\sigma$ , where E(R) is the expected value and $\sigma$ is the sigma. There is 1/100 chances to be below E(R) - 2.326 $\sigma$ and 1/1000 chances to be below E(R) - 3.10 $\sigma$ .
b. What is the daily VaR at 5% and 1%?
c. What is the annual VaR at 5% and 1%? Interpret your results.
d. Assume that the portfolio of LTCM is made up of 10 arbitrages, which each have a daily sigma of $19 million. They are all uncorrelated. What is the daily sigma for the fund?
e. It turns out that in periods of crisis the return on all arbitrages are perfectly correlated. Also the expected return becomes zero (instead of 750 million). What is the new daily sigma for the fund? What is the new annual sigma?
f. Under the assumptions of Question 5, what is the annual VaR at 1%?