A
Note on the Reduction of Extreme Risks
by Means of Diversification

Harry
M. Kat, Professor of Risk Management
and Director Alternative Investment
Research Centre, Cass Business School,
City University, London
Helder P. Palaro, PhD Student, Cass
Business School, City University, London

March 2006

"A new study by EDHEC shows 20% allocation
to hedge funds can halve the probability
of a big loss in a portfolio." (www.hedgefundsworld.com)

"A study by EDHEC shows that an allocation
of 20% to hedge funds can reduce a fund's
probability of extreme loss by 50%".
(www.commodities-now.com)

"Using hedge funds can halve the probability
of extreme loss in a portfolio, according
to a new study from Edhec". (www.ipe.com)

A recent EDHEC study on the use of hedge
funds in asset-liability management has
concluded that "allocating 20% to hedge
funds can reduce a fund's probability of
extreme loss by 50%". This conclusion
has received quite some attention in the
media (see the above quotes for example)
and any hedge fund marketers have already
eagerly incorporated it in their sales pitch.
The question, however, is whether it is
indeed such a remarkable finding as many
people seem to believe. After all, diversification
reduces volatility and lower volatility
means less extreme risk. What therefore
needs to be shown is that the obtained reduction
in extreme risk exceeds what one would normally
expect from allocating 20% to a new diversifier.

To shed some light on this issue, we studied
the simple case of a portfolio containing
70% stocks and 30% bonds, to which a new
arbitrary diversifier is added. Assuming
normal distributions throughout, the details
are as follows:

Stock mean = 10%, volatility = 16.5%
Bond mean = 5%, volatility = 8.5%
Diversifier mean = 7.5%, volatility = 12%
Correlation (stocks, bonds) = 0.2

Given a return distribution, the x% VaR
is the return value below which x% of the
probability mass is found, ie the probability
of a return lower than the x% VaR equals
exactly x%. Since we are dealing with normal
distributions, if we set x low enough, the
x% VaR is a good measure of extreme risk.
Under the above assumptions, we therefore
first calculated the 1% VaR of a portfolio
composed of 70% stocks and 30% bonds. Subsequently,
we added the diversifier to the portfolio
and calculated the resulting drop in the
probability of a return lower than the initial
1% VaR as derived from the 70/30 portfolio
without diversifier. The result can be found
in Figure 1, which shows the percentage
reduction in the probability of an extremely
low return as a function of the size of
the diversifier allocation and the correlation
between the existing portfolio and the diversifier
return.

Figure 1: Reduction
Extreme Risk as Function Diversifier Weight
and Correlation Between Existing Portfolio
and Diversifier Return

Figure 1 shows clearly that even for relatively
high correlation coefficients, it only takes
a relatively modest allocation to the diversifier
to obtain a very substantial reduction in
the probability of an extremely low return.
The graph also shows that a given reduction
in extreme risk can be obtained in many
different ways. This becomes especially
clear if we look at a contour plot of the
above 3D-graph, which is shown in Figure
2.

Figure 2: Contour Plot
Reduction Probability Extreme Risk (1% Var)
as Function Diversifier Weight and Correlation
Between Existing Portfolio and Diversifier
Return

From Figure 2 we see that there are many
combinations of diversifier weight and correlation
that will produce the same reduction in
extreme risk. With a correlation between
the 70/30 stock-bond portfolio and the diversifier
return of 0.6 for example, we would have
to allocate 40% to the diversifier to obtain
a 50% reduction in extreme risk. When the
correlation were only 0.42, however, a 20%
allocation would suffice.

From the above it is clear that extreme
risk is not too hard to diversify away as
the tails of the distribution are the first
place where the impact of diversification
is felt. A substantial reduction of less
extreme risks is harder to accomplish, however,
as it means moving further into the distribution.
Repeating the above procedure using for
example the 5% VaR to define extreme risk,
we would end up with the contour plot shown
in Figure 3. Comparing Figure 2 and 3, we
notice that for many combinations of weight
and correlation the percentage reduction
in 5% VaR based risk is substantially less
than for 1% VaR based risk.

Figure 3: Contour Plot Reduction Probability
Extreme Risk (5% Var) as Function Diversifier
Weight and Correlation Between Existing
Portfolio and Diversifier Return

Conclusion

Even with fairly high correlation between
the existing portfolio and the diversifier
return, it typically takes only a relatively
small allocation to substantially reduce
the risk of an extremely low return. The
reduction of extreme risks is not as hard
as many people may believe.

References

Martellini, L. and V. Ziemann,
The Benefits of Hedge Funds in Asset Liability
Management, Working Paper EDHEC Risk, 2005.

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