Sat, 01/10/2005 - 22:00
Three reasons come to mind if asked why a risk management and monitoring framework should be put in place:
External reporting needs
Hedge funds have often few regulatory reporting needs. At times they choose to carry out some form of risk or exposure reporting for their investors. The extent of such reporting varies according to the relationship between the fund and the investors, and is often carried out as part of a fund's marketing strategy. In which case, the reporting should be done keeping the target investor's profile in mind. The type of investor being sought influences the type of values being reported. For example, some private investors may be mostly concerned with potential loss and expected volatility estimates.
Companies managing funds of hedge funds may instead be keen on receiving information of a different type. Every decision to allocate capital to a hedge fund depends among other things on the overall portfolio (fund of funds) structure. The fund of funds manager will want to monitor that each fund he has invested in remains the best fit, given the other components, to the portfolio. The choice of, and the allocations to, the various funds making up the whole portfolio may be the result of a (more or less formalised) process of optimisation. Such a process is based on a set of elements or characteristics such as style, expected volatility, or correlation with the other portfolio components. Consequently the fund of funds manager will be concerned that the characteristics of the various funds relevant for the allocation decision remain 'stable', similar to what they were when the allocation decision was taken.
So - everything considered, a fund of funds manager is likely to be very keen on receiving:
Finally, a fund of fund manager is likely to desire raw data, which can be manipulated and re-aggregated, rather than static reports.
I will now show in more detail, using examples, how the hedge fund manager could, with a single risk management framework, satisfy at once both internal and external risk monitoring needs.
Portfolio risk management
The most obvious purpose of managing portfolio risk is to keep the volatility of the returns down to acceptable levels and avoid unexpected losses beyond a certain size. But at the same time, 'managing risk' is carried out to make sure the invested capital is allocated efficiently to the various instruments/strategies making up the portfolio. In order to carry this out, it is necessary to perform two simple steps:
The two direct benefits of this activity are:
The steps needed to carry this out are, broadly:
Identify the risk factors
This means identifying those events which affect (and ultimately generate) the return of the portfolio, and decomposing them in measurable quantities. In some cases these are easy to identify, for example when a fund plays yield curve spreads or FX rate movements. In other cases these are more complex, for example if the investment strategy is built around the relative mispricing of closely related stocks, or the differences between the correlation of certain assets and the correlation implied by the prices of derivatives written on those same assets.
Normally, we would expect that the portfolio management platform used for the daily trading is capable of measuring this. In addition, those strategies that are built around exploiting pricing differentials could not be implemented without the aid of a sound quantitative model to measure such differentials. Moreover, their success is actually based on the ability to measure and capture 'mispricings' which are invisible to the untrained eye. Nevertheless, this will address those factors that are actively traded. It is worth noting that a complex portfolio may happen to assume 'undesired' exposure ie, place involuntary bets on market events. This again can be very easy to identify and hedge away (for instance, the undesired FX exposure which may be assumed while trading assets denominated in different currencies), or rather complex to identify and quantify (undesired exposure to price convexity, exposure to potential lack of liquidity). In any case, the point here is to understand what causes the fund's value to change, and quantify how much of the net asset value (NAV) can be caused by changes in these factors.
Some meaningful measures can be achieved at the end of this process: mainly, market exposures and sensitivities to market factor movements.
Identifying a model to describe possible future outcome
Once most of the factors influencing the value of the fund are identified, we need to understand how and by how much we should expect them to move.
This is commonly achieved in two ways:
The first of these approaches is usually implemented by assuming that the factors influencing a fund's value are jointly Normally distributed, each with an expected return of 0, and volatilities and cross correlations as measured historically over a certain time-span.
The fact that this approach relies on past volatilities and correlations means that some investment strategies are not very suited to it. This is true every time the return is (to put it roughly) 'a bet' on certain one-off events happening (this is he case for the appropriately named 'event driven strategies'), since past market behaviour has no explanatory value to predict the likelihood of unprecedented events. These strategies need to be addressed with a scenario-based approach.
Verify that capital allocation is optimal
Once that the potential P&L associated to changes in the risk factor has been calculated, together with either:
These outcomes should be analysed to make sure the 'hot spots' ie, the exposures with the biggest potential contribution to the overall expected volatility are in line with the investment and capital allocation objectives of the fund manager. Normally this 'potential contribution' is calculated marginally, highlighting the impact on the overall potential loss/expected volatility of eliminating the exposure to a factor/instrument/strategy/book, or the impact of incrementing such exposure by a unit.
In order to show how this process may look like in practice, I will examine two examples: an event driven strategy, and an arbitrage strategy.
Example of an event driven strategy: merger arbitrage
Let us consider a hyper-simplified approach: we assume that an existing portfolio is in place, with a number of positions in it. Each position is made up of various legs (usually two), defined as long or short holdings in stocks involved in a potential merger.
We will try to calculate potential losses associated to it by identifying the risk factors and 'modelling' the market's events that may cause changes in the fund's value.
Let us distinguish two potential market events: mergers resulting in cash offers, and mergers resulting in paper offers.
Cash offers
The easiest case: A stock trades below the price offered by a potential buyer. The possible profit is the difference between actual price and offered price, the possible loss the difference between actual price and the price before the offer was known in the market.
The expected loss is the product between the probability of the acquisition failing and the possible loss.
The open issue is calculating such probability. We will return to this.
Paper (share exchange) offer
The merger implies a certain exchange ratio between the shares of the two companies. For instance, it implies a ratio 'r', such that for each share of company A, r shares of company B will be received.
A portfolio manager witnesses, implied in the current share prices, a ratio r_{c}, lower than r. As a consequence, he engineers a trade to exploit such a difference: the manager sells shares in B and buys a number of shares in A correspondent to the final exchange ratio r. The value of our position is then:
Where E is the number of shares in consideration: the ratio implied in
the current prices is:
so the value of our position can be expressed as:
If the merger succeeds, the ratio will become r, higher than r_{c}, and we will unwind position with a profit of:
Note that the price P_{A} will have changed in the meanwhile, so the profit is positive (since r > r_{c}) but its magnitude is uncertain, and function of P_{A}.
On the other side, if the merger fails we assume that the ratio will fall to the level r_{f}, the 'long run' pre-merge announcement (or rumour) level, giving way to a loss equal to:
The risk factors at are:
The possible loss due to r changing depends on the probability of a merger being completed - such probability changes with time, with press announcements and, critically, with the reception of antitrust approval, and with the emergence of other bidders, etc.
The effect of the change in price (one of the two stocks, since the two prices are linked by r) determines the magnitude of the profit or loss.
In order to calculate potential portfolio losses, one can simply assign to each bet a probability of the merger failing. The product of the losses, calculated as shown above, times these probabilities will yield an estimate of the overall potential loss of the fund. Clearly, the value assigned to these probabilities is crucial. This can be done by setting up a model in which these probabilities are a function of some observable market variables, or by establishing arbitrary values based on one's personal experience and feeling. In order to avoid one's daily mood influencing the estimate of a fund's potential loss, it would be better to formally model these probabilities according to some clear criteria once and for all.
An equivalent way is to set up a model based on scenarios, where the loss is defined by a certain number of deals breaking.
Conclusions
Complex pricing models have not been used to analyse this strategy. Not all the tools needed to build a successful merger arbitrage portfolio can be implemented in a model. These will include analytical skills, market experience, sector and industry knowledge, legal and regulatory skills, financial analysis skills etc.
It is also worth noting that we have dealt with a very simple case here: long and short positions were directly taken on equity stocks. In reality, it may be more efficient to achieve long or short exposure via other instruments, like options written on the relevant stocks, futures, etc. In this case we do need some formally implemented pricing models.
Data on the past market activity can give useful insights when building scenarios, but the estimation of risk should not be heavily reliant on historical data. For this strategy, it is likely that each deal has its own specifics. In general, the fact that this style is based on one-off events tends to make scenario-based approaches rather popular in such cases.
The purposes of all this activity would be:
Finally, we began our analysis assuming that an existing portfolio was in place. An extreme scenario should be considered when calculating the risk of the strategy: one where the number of mergers in the market dramatically decreases, decimating the moneymaking opportunities of the fund.
Example of an arbitrage strategy: Convertible Arbitrage
For this strategy we must examine some pricing models and historical market information will be more useful than for the previous strategy.
In this example, we also consider a hyper-simplified approach. A convertible arbitrage manager will be buying convertible bonds, and hedging away some of the exposure to various market factors arising from these securities.
Let us first of all consider a general formula to express the price of a convertible bond as a function of market variables:
Each type of variable highlights exposure to a certain 'asset class'. The expression means that the market value of a convertible bond depends on the level of the following market variables:
One of the ways to give the generic formula for the convertible bond price a more explicit shape is:
or
where
- the present value of the cash-flows expected from the bond
and
- the value of the option on the stock.
Identify the risk factors:
The convertible arbitrage manager will decide which factor to remain exposed to, and which one to hedge away, in order to generate a positive return. The cautious risk manager will observe the strategy and calculate potential losses due to adverse movement in the
market, keeping track of:
Stock prices movements
First of all, the exposure to stock price movements may need to be hedged, which is achieved by delta-hedging the stock option implied in the convertible bond price. This hedge is usually fine for 'small' equity price movements. For 'bigger' ones, we need to consider at a higher order effect, the gamma exposure.
This option is usually priced using numerical pricing techniques (numerical implementation of tree-based models) for which closed form analytical expressions do not exist. Sometimes a Black-Sholes formula is used to approximate it. In any case, given a pricing model, it is possible to calculate how much of the equity exposure is hedged (delta) and how stable this hedge is (gamma).
Let us look briefly at the stability of the hedge. We can consider the expression for the gamma in the Black Sholes case (for a call option) is:
(P is the stock price, X the strike)
Two simple considerations are:
Implied volatility
The other major factor to consider here is the implied volatility. The prudent risk manager needs to assess the sensitivity to changes in the implied volatility of the underlying stock's price, universally called vega.
In order to simply visualise the vega we can again use the Black-Sholes formula:
It is easy to see that (for long calls) vega will always be positive, shaped like a Gaussian distribution and maximum when the option is in the money, and directly proportional to the time left to expiry.
Traditionally, convertible bond managers tend to hold convertible bonds. I can reasonably expect a long vega exposure.
Our manager may not want to hedge it at all. If for any reason a convertible bond looks 'cheaply' priced compared with their theoretical value, of all the elements that make up the value of a convertible, there will be at least one whose value is understated: the implied volatility is the most likely candidate for this. It tends to be used as a 'residual' category in the option pricing activity - ie, tends to be reverse-engineered from the observed market value of an option given the other variables, which are more 'commoditised' and observable (interest rate, price of the underlying). Therefore, buying a cheaply priced convertible usually implies taking a long exposure on its implied volatility.
It is also not easy to hedge this exposure. Hedging the implied volatility of a particular instrument isn't straightforward. It may be done by writing another option, or by selling an exchange-traded option, if available, on the same stock. It may be done by writing another option, or by selling an exchange-traded option, if available, on the same stock, a call option as our convertible bond positions are essentially long call options. Otherwise, a short vega exposure can be achieved selling options of a different type, more or less closely related to the actual stock we are trading. Stock index options, or futures options, can be used. In order to 'sell' volatility we can create option strategies like 'short straddles' or 'short strangles' (short call, short put on the same instrument at the same or differing, strikes) - a short straddle creates a short volatility exposure.
We need to consider that it may not always be possible/practical to sell options on the most appropriate instrument. Considerations on the availability of exchange traded stock options, liquidity of the OTC market etc. must be made.
The vega of each instrument will tell us by how much the NAV of the fund will change if the implied volatility of each instrument moves. If we are hedging the volatility of an option with another option, we need to know also how likely the implied volatility levels of the various instruments are to move together by similar amounts.
There are a few points the prudent risk manager needs to keep in mind:
In conclusion, we must consider:
- the vega by instrument
- the level of correlation between the implied volatility levels of each instruments
- the degree of approximation of our systems when calculating an overall expected loss based on the vega of all our positions and the level of volatility and correlation of each implied volatility^{1}
Interest rates and credit spreads
Now let's move to the 'other' component of the convertible bond price, represented by the expression
The sum of the cash-flows expected from the bond, each discounted to back to today.
I have expressed discount factors as ie, functions of continuously compounding 'rates'. In this case, the rate for the date j is made up of a certain market base rate r_{j} plus a spread s.
So - the variables affecting the value of this portion are:
The sensitivity of the bond's price to changes to these variables can be calculated, and these exposures can be hedged away with common interest rate instruments or credit derivatives (usually credit default swaps).
Putting it all together
Let us consider the 'complete' expression:
or
It is made up of two terms. The credit spread shows in the first, but not in the second. Implied volatility shows in the second, but not in the first.
The level by which the convertible bonds are in (or out of) the money will cause the first or the second term to be the bigger.
For in the money convertibles, the Equity portion will dominate. The convertible will behave 'like' a stock. This is truer as the gamma will be relatively low, as shown above.
For deep out of the money convertibles, the bond portion will dominate. The credit spread exposure may be the most significant bet of these positions.
At the money convertible, a combination of the two effects will take place. Vega and gamma are at their peak.
In general, we may be able to find ways to calculate the changes in value of the portfolio. We need a technically demanding system to capture the way correlations across exposure hedge each other out, in an environment in which most of the instruments we care about have non-linear behaviour. Ideally, we should use a Monte-Carlo based Value at Risk (VaR) with full revaluation of the portfolio (at least at various discrete nodes), incorporating exposure to equity, interest rate, credit spread, implied volatility. Moreover, in order to make the results readable, such system needs to give us a good level of detail on the various exposures, and the contribution to the overall VaR. We must also be able to carry out some stress testing by shocking the most relevant variables.
Conclusions
The process we had to follow in order to identify the risk factors may be considered tedious, but can be defined with a good degree of accuracy. The first step is to define and implement accurate pricing models for the instruments we will be handling. These models will allow us to calculate the changes in the value of the various positions due to changes in the risk factors.
Also, the factors we have identified are observable market variables: stock prices, interest rates, FX rates, implied volatility levels, credit spreads, etc. It is possible to gather a significant history of values for each one of them, which allows us to calculate synthetic indicators of these market variables: average values, historical volatilities and cross correlations. Historical information is commonly used to estimate these factors' possible future behaviour.
Once again, the purposes of all this activity would be:
General observations
The two strategies analysed are different, but there are considerations common to both. For example:
Satisfying external needs
I began this article by discussing potential external reporting needs. If a formalised risk management environment is implemented, most external reporting needs can be addressed as a by-product of the actual risk management activity.
For example, it would be possible, if ever needed, to describe one's investment strategy in terms of exposure to risk factors. As we have seen, the identification of risk factors and the calculation of their contribution to the fund's value are the first step to setting up a riskmonitoring framework.
For example, I said above that a fund of funds manager could be interested in receiving:
No. 1 above can be achieved by showing that the exposure to certain risk factor is stable and consistent with the investment strategy, and that there is no exposure to other factors. The main problem here is in defining meaningful factors and agreeing how to calculate exposure to them.
No. 2 may be easily achieved by reporting the VaR, where appropriate, or another potential loss estimate.
No. 3 should be achieved as no. 1 - by re-describing the portfolio's 'bets' in terms of aggregated factor exposure, an investor could check whether exposure to these same or to different factors is recurrent within their fund of funds.
In general, once a risk-monitoring framework is implemented, some of the sensitivity and exposure figures can be used to carry out some investor reporting activity, if desired. This may be the way to turn some of the costs associated with risk monitoring into marketing investments.
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^{1} If a VaR figure is calculated, proxies are likely to be used to as indications of the volatility of various instruments' implied volatility levels and their cross correlation.
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