Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

82 Optimizing Optimization

Clearly,

PPPulmd ^1

We can construct a new density by the following shift:

PPl′′′l ΔΔ,andP Pu  u P Pmd md

where 0 Δ min( P u , P l ) and

pdf x

P pdf x

P

xx

pdf x x x x

P pdf x

P

xx

u

u

u

lu

l

l

l

′

′

′

()

()

()

()







∞

0

It is easy to check that pdf ( x ) is still a well-defined density although no
longer continuous at x  x l or x  x u. Furthermore, the above transformation
can be called optimistic in that it transfers probability form the lower tail to
the upper tail of the density. A pessimistic transformation can be defined simi-
larly. Note that since we assumed a continuous density with zero probability
mass at any point, the discontinuities induced by our transformation will not
affect the existence of the integrals.

3.7.5 Trustees’ attitude to risk

Investors ’ attitude to investment risk is generally assessed by their reaction to such
qualities as volatility, peak-to-trough drawdown, recovery time, and absolute loss.
Trustees of an infinitely lived endowment should in theory be largely indifferent
to all temporary variations to asset values (save for irrecoverable losses), except
that market volatility will influence how short-term income is achieved.
An efficiently diversified endowment ought to be delivering optimally
smoothed sources of return, which allow annual withdrawal without the neces-
sity of realizing assets that are in a temporary downturn. Of much greater con-
cern to the trustees of any long-lived endowment is the level of future returns
that might reasonably be expected. Even if volatility is disregarded and diver-
sification has indeed delivered optimally smoothed sources of return, if the
current spending policy is based on expectations of future returns that are not
achieved, the longer-term sustainable returns from the endowment will clearly
be in question. In Swensen (2000) , David Swensen, Yale University Chief
Investment Officer, establishes the connection between the rate of annual spend
and endowment returns as follows:

Spending at levels inconsistent with investment returns either diminishes or

enhances future endowment levels. Too much current spending causes future