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

Optimal solutions for optimization in practice 81

where δ is the subjective discount rate.
The optimal drawdown depends on the entity’s subjective discount rate and
level of risk aversion. To link drawdown to asset allocation, we use the follow-
ing power utility function:

UC t(,) Ct()



(^1) 
1
1
α
α
where α  risk aversion.
The optimal drawdown is:
m^1
1
()δφα
where φEZ()^1 iα.
To estimate φ , we bootstrap the monthly portfolio returns using 120,000
random draws, and sum to get 10,000 annual real (gross) returns.
We then use these to compute the sample mean:
φ  

1
10 000
1
1
10 000
,
,
Zi
i
α
∑
Part 2 — Given the optimum drawdown as derived from the power utility
function, we distinguish between:

1. Neutrality: future equal to past


2. Optimism: future better than past


3. Pessimism: future worse than past

Starting from historic data, we calculate the portfolio return:

RwrtTP t it it

i

n

, ,,

 1

∑^1 

and split the results into good ( x %), bad ( y %), and neutral (100  x  y %).
For example, to build an optimistic scenario, we oversample from the good
and undersample from the bad.
We consider more transformations of distributions that can be described
for FSD. For an arbitrary positive continuous density pdf ( x ); we consider two
points x l and x u and the probabilities:

Pl pdf x dx P pdf x dx P pdf x dx

x

u

x

md

x

l x

ul

u

() , () ()

0

∫∫ ∫

∞

and