A Picture’s Worth a thousand Words
As actuaries, we’re often like Markopolos. We’re the ones for
whom the challenge lies not in understanding the mathematics
but in explaining it to others. The frustration Markopolos felt
as he penned his memo to the SEC is palpable in his writing.
He understood that a Beta of 6 percent for a fund with Madoff’s
described investment strategy was unreasonable, and was able
to list 28 other red flags as well. But what he couldn’t do was
communicate the unreasonableness of Madoff’s returns to the
SEC in a way that captured its attention.
In the aftermath of Madoff’s confession, I and others have
independently performed a quantitative analysis of the return
distribution Madoff reported to Fairfield Sentry, comparing it
with the distribution that should have resulted if Madoff had indeed been, as he claimed, investing in a basket of stocks highly
correlated to the S&P 100 and using a split-strike conversion
strategy. The comparison is alarming.
To understand why, it’s important to know the basic principles behind the investment strategy Madoff said allowed him
to return 8 to 12 percent per annum to his investors, regardless
of which direction the broader market was heading.
One of the problems Markopolos encountered in convincing
regulators of Madoff’s malfeasance was his difficulty in explaining how the split-strike conversion strategy worked. While
split-strike has many moving parts and can be difficult to execute well, the basic principles behind it are relatively simple.
Sometimes called a “collar” by traders, a split-strike conversion approach to investing makes use of put options (financial
contracts that allow, but don’t require, the holder to sell a basket
of stocks at a specified strike price to the contractual counterparty) to impose limits on downside volatility. The purchase of
the put options is financed by the sale of call options (financial
contracts that allow, but don’t require, the holder to purchase a
basket of stocks at a specified strike price from the contractual
counterparty), which results in limits to upside gains as well.
According to “Don’t Ask, Don’t Tell,” a 2001 Barron’s article
by Erin E. Arvedlund, one of Madoff’s feeder funds described
the strategy as follows:
Typically, a position will consist of the ownership of 30-35
S&P 100 stocks, most correlated to that index, the sale of
out-of-the-money calls on the index and the purchase of
out-of-the-money puts on the index. The sale of the calls is
designed to increase the rate of return while allowing upward
movement of the stock portfolio to the strike price of the calls.
The puts, funded in large part by the sale of the calls, limit the
Chart 1 illustrates how the strategy is meant to work. The diagonal line represents the interval between the established put
option and call option strike prices, where portfolio value (
excluding the value of any held options) is equal to the market
index one is attempting to replicate—in Madoff’s stated case,
this would be the S&P 100.
Theoretically, if an investor weren’t buying any put options
or selling any call options, the diagonal line would extend from
zero to infinity. To limit the downside risk inherent in any index
or basket of securities, the investor purchases put options, represented by the lower left horizontal line. This is the put option
strike price for the index—the price at which the counterparty
has agreed to purchase the basket of stocks. If the value of the
index drops below the strike price, the investor exercises his or
her option and sells the underlying stocks at the pre-established
strike price. This creates a downside limit on losses. Selling the
call options, represented by the right-hand side of the graph,
generates the income necessary to finance the purchase of the
put options. Selling call options imposes some limitation on
the upside, but that’s the price the investor pays to limit the
Returns are earned in three ways with a split-strike conversion approach:
When stock prices rise; 1.
When dividends are distributed; 2.
When income is earned through the sale of call options. 3.
Portfolio value decreases when stock prices decline and also
when funds are expended to purchase the put options needed
to limit the downside. Put options essentially function as insurance, and purchasing insurance imposes a cost that must be
factored into the portfolio’s total return.
The value of the portfolio is also affected negatively when
the value of the index rises above the strike price of the call options, limiting upside potential.
the S&P 100 and Downside Volatility
Chart 2 shows the distribution of monthly returns for the
S&P 100 during the 15-year period when Madoff was supposedly using a split-strike conversion strategy to manage the
Fairfield Sentry Ltd. feeder fund.
As illustrated, the distribution of monthly returns for the
S&P 100 follows an approximately normal distribution pattern
(although the tails of the distribution are typically thought to be
slightly thicker than a normal distribution would suggest).
Data points have been fit to a normal curve, with a mean of
0.85 percent and marked volatility on each side. The monthly returns of the S&P 100 during this time ranged from - 14. 5
percent to + 10. 8 percent, although Chart 2 shows a slight restriction of this range. The standard deviation was measured
as 4. 1 percent.
The distribution of returns resulting from a split-strike conversion strategy would find its basis in this distribution but have
less volatility and some modification to the tails. That modification will depend on the strike prices for the put and call options
on the portfolio and how frequently those prices are changed,
relative to the value of the underlying index. The put and call
options should also have a significant impact on the mean of