Dewynne, J.

Howison, S.D.

English

Mathematics in Industry Information Service Eprints Archive

Risk/Reward
UBS
UBS want to consider ways in which a client can raise money by using various kinds
of debt (fixed rate, floating rate, etc). Debt of type i is issued in a proportion ai, where
The client may also independently use options to hedge revenues from a natural resource
(eg oil), with a hedge ratio 6. (0 ~ 6. ~ 1). UBS have a (confidential) model which takes
a strategy (aI, ... ,aN; 6.) and assigns to it a point in a risk-reward diagram; see figure l.
(Unlike in classical portfolio theory, the UBS model does not require the efficient frontier
to be convex.)
The curve joining A and B in this diagram is the "efficient frontier" on which the
portfolio should ideally be situated. A client might at present be at P, ie exposed to
excessive risk, for a given return and want to move to Q.
There were two aspects to the problem:
(i) find the strategies corresponding to Q and the most robust of these (in terms of
sensitivity to small changes, either in the model or in the strategy);
(ii) for the benefit of clients, interpret the pictures graphically (more below).
For point (i), two possibilities were suggested. The first is to search directly for the
require minimum risk point; simulated annealing seems a good algorithm given that the
map from strategies to the x-y plane is only known numerically. The second is, as at
present, to cover the strategy space reasonably uniformly with randomly chosen points
and to plot the images in the x-y plane. After this, it is not difficult to locate the
strategies leading to point Q.
B
Figure 1:
1
Figure 2:
>
Figure 3:
In both cases, robustness can simply be tested by looking at the effect of changing the
strategy parameters by small amounts (either systematically, one at a time, to evaluate
partial derivatives of y (which is minimised at Q for given x), or randomly).
Concerning graphics, at present the risk-reward diagram is computed for four types
of debt (hence three independent types) and one hedge ratio. The debts lie in a tetra-
hedron and the hedge ratio is independent, see figure 2. For graphical purposes, a two-
dimensional slice of the tetrahedron is taken and its image in the x-y plane is shown.
Often, the map from the slice (parametrised by (rI, 1'2): (a,13) in the problem state-
ment) is many-to-one. For visualization purposes it would be helpful to "uncouple" this
multiple-valued map in some way by expanding into a third dimension, see figure 3. It
was suggested that a good way to do this would be via an "equiareal" map from the
('"Y1,1'2) plane to R3 via (rI, 1'2) H (X(rl,1'2),y(rl,1'2),Z(X,y)) so that elements of area
in the (1'1,1'2) plane give the same area on the image surface. This entails solving an
equation rather similar to the equation of geometric optics and the following difficulties
become apparent:
(i) the equation is nonlinear and in general its solutions have singularities (caustics,
etc). It is not at all clear (although an interesting question) whether there is any
smooth solution at all;
(ii) the boundary conditions to be applied are far from obvious;
(iii) even if there is a smooth solution, it may still self-intersect.
2
Figure 4:
There is a large literature on representation of surfaces. One simple alternative
suggestion follows. Suppose, for simplicity, we have only three types of debt: QI,
Q2 and Q3 = 1 - QI - Q2 (rather than a 2-dimensional subset of the 3-dimensional
space above). Given QI, Q2 we can calculate (X(QI' (2), Y(QI' (2)) in the risk-reward di-
agram is multi-valued, as noted above. Open the multi-valued map up by looking at
(X(QI' (2), Y(QI' (2), QI + (2) This has the advantage that level curves of the surface are
lines Q3 =constant; see figure 4. The map is not equiareal and it, too, may self-intersect;
but it is easy to compute.
Participants
Sam Howison, Jeff Dewynne, Pat Hagan, Colin Please, Richard Stratford.
Report written by Sam Howison
3


Risk/Reward

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