A MARKOVIAN APPROXIMATED SOLUTION TO A PORTFOLIO MANAGEMENT PROBLEM

4. A CALIBRATED MODEL

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x 10⁴
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Portfolio Selection. Expected Values

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Suppose an agent with an original wealth of ar₀ =
$100,000 wants to maximise their satisfaction
during the coming T = 10 years. The instanta-
neous satisfaction is measured by ʌ/ɛ^(t)- The
risk free asset price drift is r = .05 and that of the
risky asset is a = .11 with the volatility σ = .4.
The agent’s discount rate is ρ = .11.

For these parameter values, the agent’s expected
discounted total utility is J = ρ(0)√100000 =
723.09. Figure 2 presents the optimal strategies;
the expected wealth and consumption rate time
profiles are shown in Figure 3.

Wealth

Consumption Rate

1

2

9         10

0

3         4         5         6         7         8

years

Fig. 3. Optimal expected wealth and consumption
rate time profiles.

Optimal Strategies

1

0.8

0.6

0.4

0.2

0

0              2              4              6              8             10             12

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x 10⁴

Fig. 2. Optimal (reference) strategies.

Optimal Realisations. Average Utility = 751

x 10⁴

iF

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0         1         2         3         4         5         6         7         8         9         10

□ 0.5-

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Figure 4 shows ten wealth and strategy time
profiles, which correspond to ten noise realisations
dw(t), t ∈ [0,T], obtained from a random number
generator. The average total discounted utility of
these 10 portfolios is 751, which is more than the
theoretical J = 729. However, it is evident that
the optimal portfolios’ performance has a large
variance ⁸ .

0∣-------------------------------1-------------------------------1-------------------------------1--------------------------------1--------------------------------1--------------------------------1-------------------------------₁-------------------------------1-------------------------------1-------------------------------

0         1         2         3         4         5         6         7         8         9         10

x 10⁴

4

2

0         1         2         3         4         5         6         7         8         9         10

years

Fig. 4. Optimal wealth, investment and consump-
tion rate realisations.

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