Harry Potter and the Enchanted Sharpe Ratio

Virginia Reynolds Parker, CFA
President
Parker Global Strategies

How does one risk-adjust returns to make possible a comparison between hedge fund investments? How does one evaluate a high volatility hedge fund program versus a lower volatility program, or a high return program versus a lower return program? Many have grappled with these issues. I’ll warn you right now that if I had a "Mirror of Desire" I would envision myself writing an article with no arithmetic involved. Unfortunately, we Muggles have no magical power to describe risk-adjusted returns without some reliance on earthbound mathematics. But if you bear with me, I think I’ve borrowed a spell from a Sorcerer friend that will make this pretty painless.

The Sharpe Ratio represents one of the fundamental tools used to compare the return of funds relative to a given unit of risk. However, when applying it to hedge fund strategies incorporating leverage, a series of mysterious outcomes result. The calculation of return per unit of risk actually decreases as leverage increases! Conversely, risk per unit of return increases. Hold onto your hats as we tumble to the conclusion that there is a law of diminishing risk-adjusted returns with increased use of leverage. We also draw the conclusion that by adding leverage to a low volatility program we can generate the same risk-adjusted returns as a core higher volatility program, but still be left in the dark as to which is the better investment.

Laying out the Ingredients for the Witches’ Brew

Here are some definitions and assumptions that will help us stir this pot:

  Rp = hedge fund program return less the risk free rate (ie, the program excess return), where program return is 20% and excess return is 15%, net of Rf of 5%
 
L = leverage multiple (eg., $200 gross trading and $100 invested capital = 2)
 
Rf = risk free rate (assume 5%)
 
Rb = borrowing rate (assumed to be 5% although normally at a spread over Rf)
 
s p = volatility of program returns (assume 10%)
 

Step 1:   Sharpe Ratio= [(L*Rp – (L-1)*Rb)]
                                                     [L*s p]

Step 2:   Sharpe Ratio= [L*Rp – L*Rb + Rb]
                                                    [L*s p]

Step 3:    Sharpe Ratio= [L* (Rp – Rb) + Rb]
                                                    [L*s p]

Step 4:    Sharpe Ratio= (Rp – Rb)     +     Rb
                                               s p                L*s p

Step 5:    Sharpe Ratio= (15% - 5%)     +       5%         =     1 + .50
                                                10%                L * 10%                   L

Put Me on a Broomstick and Fly Me out of These Formulas

Hold on to your handles for just a moment. If you look at formula (5) you’ll note that the only variable is "L", because we have assumed specific rates and percentages for all the other items in order to isolate the effect of leverage. What happens if we increase leverage? The denominator of the second quotient increases, and the Sharpe Ratio decreases: the more the increase in leverage, the greater the decline in the Sharpe Ratio. We have proved the law of diminishing risk-adjusted returns. Adding leverage reduces the return per unit of risk, and increases the risk per unit of return.

But there’s another funky thing going on. Both quotients contain s p in the denominator. If volatility decreases, the Sharpe Ratio increases, and vice versa. But what happens if we increase leverage and decrease volatility? In other words, what happens if we lever a low volatility portfolio as opposed to investing in a higher volatility portfolio? There’s a possibility that an investor could end up with a better risk-adjusted return by doing this.

Pulling The Rabbit out of the Hat

So this really leaves me bewitched, bothered and bewildered. Why don’t I just go out and lever low volatility portfolios to the hilt? If I can improve my risk-adjusted return, why not? There are two answers to this question:

First, at some point the Sharpe Ratio of the low volatility but levered program will be the same as some higher volatility program. This leads us to our second reason.

Second, the Sharpe Ratio does not define all risk. So if two programs have the same Sharpe Ratio, their risk profile may actually be very different if we consider qualitative factors. Even if we "synthetically" create a portfolio, through leveraging, with the same Sharpe Ratio as an unlevered portfolio, the risk analysis has to go much deeper than the Sharpe Ratio.

Pull Back the Veil

The Sharpe Ratio does add value in evaluating the relative performance of hedge fund strategies, and managers within those strategies. One valuable trick is to calibrate the returns from each manager to a fixed volatility level. This makes it easier to compare returns of managers. To calibrate the risk-adjusted returns, decide first what volatility level you want to make the constant. Pick any percentage as long as you use it for each manager. Create an adjustment factor by dividing the constant volatility by the actual volatility of each manager. Multiply the numerator and denominator of the two quotients in Step (4) by this adjustment factor, and you have a calibrated return number.

Once you’ve done this, it becomes apparent that managers within the same strategy may have a wide range of calibrated risk-adjusted returns. How much of the differential relates to manager skill, and how much to different risks being taken within their portfolios, and how much to pure luck? We may be beating a dead horse, but ultimately the decision about whether to invest with a manager comes down to the fundamental qualitative factors we’ve discussed in previous columns:

Transparency: How much do you know about the actual sources of the different returns each manager has shown? Do you understand the manager’s strategy, risk management techniques, etc. Do the historical returns represent manager skill or luck?

Liquidity: What happens when you lever a low volatility portfolio? At certain times, the volatility of a portfolio may appear quite low, but if the portfolio becomes more and more leveraged it is subjected to increased financing risk. This is apparent in equation 4. If L=1, the negative impact of higher borrowing (Rb increase) on the first term is entirely offset by the positive impact of higher borrowing on the second term. If L=2, the negative impact of higher borrowing on the first term is only partially offset by the positive impact of higher borrowing in the second term. This leads to the conclusion that leverage exposes a portfolio to the risk of lower risk adjusted returns due to changes in borrowing costs.

This explains part of the problem with Long Term Capital Management. As spreads narrowed in the markets they traded, the firm increased leverage. The portfolio appeared to have a low volatility, but when financiers became concerned about its level of leverage LTCM had a difficult time unloading its positions. It had become too big and considerably less nimble. Given the lack of transparency at LTCM, decisions to invest and finance had been made based mainly on reputation and quantitative performance reports. Big mistake!

Value at Risk: How is the manager controlling levels of systematic sources of risk as measured by VaR? What methods does the manager apply to controlling levels of non-systematic risk such as concentration of assets?

Leverage: Do we know how much leverage is actually being used in a portfolio? Do we have confidence that the portfolio has a stable financing source? What’s the impact of a move in interest rates on the performance of the portfolio?

Pull Back the Veil

It seems that the point of all our articles has been to warn against making uninformed decisions based on a purely quantitative approach and insufficient due diligence; that is, to warn against the Dark Forces of imprudent investment decisions. Like Harry Potter, we believe investors must fight these Dark Forces.

Parker Global Strategies, LLC

Parker Global Strategies is a Manager-of-Managers providing a broad spectrum of Alternative Investment Strategies to private and institutional investors.