The volatility swap can be viewed as a logical next step in the application of derivatives to portfolio management. Futures contracts play a key role in portfolio management, by making available derivatives that track index returns. Futures can be used both to actively gain index-like exposure, and to defensively hedge out index-like exposures. However, since futures cannot track the volatility of index returns they do not provide full control over the risk-return relationship of a portfolio.
Options can be used to modify the volatility profile of a portfolio, but options strategies produce a host of side-effects that substantially complicate risk analysis. The problem is that options do not isolate the volatility attribute of portfolios.
The dimensions of risk and return are usually thought of as tangent in the CAP-M framework, and that risk vs. reward payoffs can be manipulated indirectly by choosing different asset mixes. With volatility swaps, increased risk can be transformed directly into higher returns, so that the introduction of volatility swaps in to the CAP-M risk-reward framework is not simply the introduction of an asset that is negatively correlated with the market. Volatility changes the rules of the game by allowing for a rotation of a frontier, resulting in new possibilities that arise from a transformation of the axes.
Some concepts from portfolio theory make the value of the volatility swap clear.
Associated with any position there is an expected return, µ, and a volatility of that return, *. The owner of any position has some aversion to risk, which can be represented by a number that we can determine. Say the value of your risk aversion has a value *. The utility of the position can be written as Utility = µ - **2
The utility is the position's expected return, less a penalty that depends both on the position's variance (squared volatility), and on your tolerance for risk. The position is optimal when the utility is as large as possible, given the expected return, the volatility, and risk aversion. For example, using daily data since 1982-1996, if an investor is happy with an asset allocation consisting of a 60/40 blend of U.S. stocks and bonds, the investor most likely has a risk aversion of * of 4. Using the historical data and a risk-aversion of 2 would result in an 80/20 mix. To become totally defensive, and become concerned only about keeping the portfolio risk as low as possible, then in effect, * = ƒ. The maximum utility would then require that you hold 25% stocks and 75% bonds.
A position can become sub-optimal, as measured by its utility, either because of unexpected changes in return or volatility. If return does not change, but volatility increases, the position is less desirable from a risk-return perspective. The risk associated with changes in volatility can be controlled by owning * units of the volatility swap.
Suppose volatility increases, to produce a realized position variance of (*2+ *2) over the holding period. If * swap units were purchased, then income of **2 is received. The holding period utility is therefore unaffected by a volatility increase. If volatility decreases over the holding period, the owner of the swap pays **2, but position variance is decreased to (*2- *2). The holding period is therefore also unaffected by a volatility decrease.
This means that introducing volatility swaps into the portfolio not only introduces a new asset, but an ability to rotate the entire efficient frontier. Volatilities higher than the expected volatility result in higher returns, so the points to the right shift up. Realized volatility lower than expected volatility results in lower return, so the points to the left shift down.
Consequently, risk-reward possibilities are available with volatility swaps that were not available previously. The ability to convert risk into returns means that investors can hedge their Sharpe ratio. When volatility rises, so do returns, so that the ratio is more constant. Investors bench-marked to Sharpe ratios can reduce the variation in their annual Sharpe ratios by buying volatility swaps.
This utility hedging capability of the volatility swap has wide ranging application in risk management. Below we outline how it can be used for tracking error and value at risk.
TRACKING ERROR
Tracking error is the annualized standard deviation of the return mismatch between a benchmark and a portfolio of assets. It characterizes the relative risk of a position. The utility of a portfolio relative to its benchmark can be quantified in terms of the tradeoff between expected benchmark outperformance and benchmark tracking error. More precisely,
As expressed by this utility, an active portfolio represents a specific tradeoff between your expected return, volatility and risk aversion (*). If compensated by an increase in tracking error, the investor must be compensated by an increase in outperformance to maintain the utility of the position. The volatility swap offers a unique way to decrease the impact of an unexpected increase in tracking error.
Let the sensitivity of XYZ's returns to the S&P 500 be 'ß'. If we use a standard single factor model, we can express the return of XYZ as:
The tracking variance if:
The tracking variance consists of two components, a market component (first term), and a stock specific component (second term). Tracking variance changes if either market variance changes, or if stock-specific variance changes. A volatility swap can be used to hedge the tracking variance impact of the market. If (1 - ß)2 units of a volatility swap on the S&P 500 are purchased, the utility of the portfolio is hedged with regard to changes in realized market volatility. The idiosyncratic portion of the tracking error is not affected by the swap, so the tracking error is not totally hedged. However, if XYZ is a portfolio of stocks, then the larger the portfolio the smaller the relative contribution of the idiosyncratic component. In any event the relative value contribution of stock-specific volatility to total tracking error can be estimated.
VALUE AT RISK
The Value at Risk (VaR) of a position is the expected maximum loss over a particular time horizon, expressed with some statistical confidence. It is a measure that is more focused on downside risk, and more focused on a specific holding period, than tracking error. This measure of risk has achieved substantial prominence because of requirements of both regulators and a variety of institutions.
The idea of VaR is this: if a portfolio of assets, worth D0 dollars, has normally distributed returns with annualized volatility of *, then the VaR over a time interval t is:
For 95% statistical confidence we use * = 1.65; for 99% confidence we use * = 2.33.
For the VaR estimate to be meaningful, * must be constant over the horizon of interest. For example, if volatility increases, the dollars at risk will be greater than assumed. The volatility swap can be used to offset the uncertainty associated with changing volatility. As in the tracking error example above, the VaR of the portfolio volatility because of factor exposure to the market can be stabilized by owning a volatility swap.
Since VaR is typically used for multi-asset exposures, the correlation of assets is an important consideration. For the net volatility of a multi-asset exposure to be constant, both the volatilities and correlations of the underlying assets must be constant. Of course this is unrealistic. To characterize the time varying nature of correlations and volatilities in such a way that we can use a volatility swap we extend the factor model approach used for tracking error.
To keep it simple, say we have a domestic portfolio, P, of stocks and bonds. We can express the returns of P as:
The variance of this portfolio is:
The quantity *P is what we use in the VaR calculation above.
This week's Learning Curve was written by Leon Gross of Salomon Smith Barney's equity derivatives group.
