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Measuring Bets With Relative Value At Risk

07 Jan 2002

The use of sophisticated risk management tools is being rapidly adopted in the investment management industry. We view this trend as natural given that asset managers are in the business of taking risk. In other words, whether their mandate is to fund contingent liabilities, such as pension funds, produce excess returns over a benchmark, such as traditional asset managers, or generate exceptional absolute returns, such as hedge funds, asset managers need to assume risk in order to meet their objectives.

Risk management tools can help asset managers determine whether the risks they are actually taking are the right ones. We can quantify risk across decision areas to ensure that we take more risk in areas where we have more skill. Relative marginal VaR (relative MVaR) and relative incremental VaR (relative IVaR) are risk statistics that permit asset managers to allocate risk as desired within certain guidelines. For example, an asset manager with more skill in asset allocation than security selection should minimize the security specific risk and earn their excess returns through bets at the asset class level. In addition, if they are not prepared to take currency risk they should eliminate it or outsource its management through currency overlay. MVaR and IVaR are useful tools for currency overlay managers to identify strategies that enhance returns and control risk.


Relative VaR and Tracking Error

The most popular measure of risk is VaR. It measures the smallest loss that would be incurred with a certain probability over a given time horizon. Relative VaR measures the smallest level of underperformance, relative to a benchmark, that would occur with a certain probability over a given time horizon. Note that VaR is just a special case of relative VaR where the benchmark is cash in the base currency. Therefore, relative VaR can be used by pension plans (benchmark = liabilities), traditional asset managers (benchmark = investment benchmark), and hedge funds (benchmark = cash).

Relative VaR can also be interpreted as forward looking tracking error. To be consistent with tracking error we can calculate relative VaR at the 84% confidence level to reflect a one standard deviation event.


Allocating Risk with MVaR and IVaR

In addition to the total risk of the portfolio and standalone risk of each position, we need to measure the correlation effects in the portfolio. MVaR and IVaR are statistics that measure the contribution of a position or a group of positions to the total risk of the portfolio.

We define MVaR for a group of positions as the difference between the VaR of the total portfolio and the VaR of the portfolio after selling the positions in the group and investing the proceeds in cash. In other words, MVaR = VaR(Portfolio) - VaR(Portfolio - Group), so that if MVaR is positive the group is adding risk to the portfolio (i.e., the group is a risk contributor), and if it is negative the group decreases risk (i.e., the group acts like a hedge or a risk diversifier).

MVaR can also be measured relative to a benchmark. In this case, we subtract, from the total relative VaR, the relative VaR of the portfolio after removing a group of securities from both the portfolio and the benchmark. Hence, relative MVaR answers the question: how much would our relative VaR change if we perfectly match the benchmark in a certain group? Just like VaR is a special case of relative VaR, MVaR is a special case of relative MVaR. That is, relative MVaR measures the impact on the portfolio's relative VaR of dropping a group of instruments to neutral weight, that is, either zero or benchmark weight.

Relative IVaR measures the change in relative VaR when we decrease the over/underweight, relative to a benchmark, of a group of positions by a small amount. IVaR is similar to MVaR except that instead of matching the benchmark perfectly in a certain group of positions, we simply reduce its relative weight by a small amount. IVaR can be interpreted as the sensitivity of relative VaR to changes in the weighting of a group. Another important characteristic of relative IVaR is that the sum of the IVaRs of any set of groups representing a mutually exclusive partition of the portfolio is equal to the total relative VaR. Thus, relative IVaR can be interpreted as the risk contribution of each of these groups.


Risk Measures and Asset Allocation

Suppose we are given a mandate to manage an equity portfolio against a U.S. dollar currency-hedged benchmark. The benchmark consists of 65% Standard & Poor's 500, 10% FTSE 100, 10% Nikkei 300, 7.5% DAX, and 7.5% CAC 40. Furthermore, we consider that our main skill lies in the asset allocation process and believe that European equities will overperform relative to U.S. and Japanese equities. We determine that the asset mix will be 60% S&P 500, 12% FTSE 100, 8% Nikkei 300, 10% DAX, and 10% CAC 40. In other words, the over/underweight relative to the benchmark is -5% S&P 500, 2% FTSE 100, -2% Nikkei 300, 2.5% DAX, and 2.5% CAC 40. Note that since the benchmark is currency hedged, we are also taking on foreign exchange risk, that is, we are overweight 12% GBP, 8% JPY, and 20% EUR relative to the benchmark.

Table 1 shows relative weights, 84% annual MVaR, and 84% annual IVaR for the benchmark and currency allocations. Relative VaR is 315 basis points, which means that we have a 16% chance of underperforming by more than 315bps in one year. By looking at the MVaR and IVaR figures we can see that most of the risk is being contributed by currency fluctuations. The biggest contributor to relative VaR is the euro exposure. The MVaR for our euro exposure is 143bps, which means that if we eliminated it, possibly through a euro/dollar forward or option, the relative VaR for the total portfolio would drop by 143bps. If we wanted to decrease, but not eliminate, our euro exposure, we could use IVaR to get an indication of the potential risk reduction achieved with a certain transaction. For example, if we use euro/dollar forwards to decrease the euro relative weight from 20% to 18% (a 10% reduction in weight), we expect relative VaR to decrease by 19bps or 10% of our 190bps euro IVaR.


Currency Strategies

We want to budget most of the risk for the equity allocation decisions in which we are more skilled, but we also want to leave some room to implement a currency strategy.

Let us say that we wanted to reduce relative VaR to 150bps. Since we have a high currency IVaR, we could successfully implement a currency hedging strategy to decrease total risk. A first option would be to eliminate our currency exposure entirely. If we did this, the relative VaR of the portfolio would drop to 147bps. Another alternative would be to implement an active strategy where we express a directional view. Table 2 shows an example of a view in which sterling overperforms (relative weight is 8%), we stay neutral on yen (relative weight is 0%), and the euro underperforms (relative weight is ­3%). The total relative VaR of our new allocation is 147bps, so we are not only within the risk limit, but we also did not incur additional risk by implementing our view. Moreover, the currency contribution to total risk is only 9% (the sum of IVaRs for the currencies), which corresponds to 6% of the total relative VaR.

Table 1
.Relative WeightsRelative MVaR (bp)Relative IVaR (bp)
S&P 500-5.0%119
FTSE 1002.0%25
Nikkei 300-2.0%36
CAC 402.5%-33
table 2
.Relative WeightsRelative MVaR (bp)Relative IVaR (bp)
S&P 500-5.0%-831
FTSE 1002.0%1722
Nikkei 300-2.0%28
CAC 402.5%2637

This week's Learning Curve was written byJorge Mina, senior researcher atRiskMetrics Groupin New York.

07 Jan 2002