Single Period Goal-Based Wealth Management

Load Data

Load the data and then annualize the asset returns and the variance-covariance matrix.

% Load data
load BlueChipStockMoments.mat
AssetMean = 12*AssetMean;
AssetCovar = 12*AssetCovar;

Plot Efficient Frontier

Use Portfolio to create a Portfolio object and estimateMaxSharpeRatio to compute the maximum Sharpe ratio. Then, use plotFrontier to plot the efficient frontier for long-only, fully-invested portfolios and mark the maximum Sharpe ratio portfolio.

% Create Portfolio object
p = Portfolio(AssetList=AssetList,AssetMean=AssetMean,...
    AssetCovar=AssetCovar);

% Add long-only, fully-invested constraints
p = setDefaultConstraints(p);

% Compute maximum Sharpe ratio portfolio
wSR = estimateMaxSharpeRatio(p);

% Plot efficient frontier
numPorts = 20;
figure;
[pret,prisk] = plotFrontier(p,20);
hold on
plot(estimatePortRisk(p,wSR),estimatePortReturn(p,wSR),'b.',...
    'MarkerSize',20);
legend('Efficient frontier','Max Sharpe Ratio',...
    'Location','southeast')
hold off

Choose Single Portfolio Using GBWM

GBWM provides a way to choose a single portfolio on the efficient frontier that allows you to incorporate the investor's preference in a straightforward way. In this case, the investor's goal is to reach a target wealth $\mathit{G}$ from their initial wealth ${\mathit{W}}_0$ at the end of the investment period $\mathit{T}$. Therefore, you can formulate the problem as choosing the portfolio that maximizes the probability of having a wealth at time $\mathit{T}$ that surpasses the goal $\mathit{G}$, while making sure that the chosen weights are on the efficient frontier.

% Define investor's preferences
W0 = 100;
G = 200;
T = 10;

For this example, assume that the wealth evolves following a Brownian motion [1]

$${\mathit{W}}_{\mathit{T}}={\mathit{W}}_0\exp \left\{\left(\mu -\frac{{\sigma }^2}{2}\right)\mathit{T}+\sigma \sqrt{\mathit{T}}\mathit{Z}\right\}$$,

where $\mathit{Z}$ is a standard normal random variable.

The approach in this example also assumes that the returns follow a stationary distribution. In other words, the approach assumes that the returns have the same expectation and covariance throughout the investment period. The term “single-period” refers to the fact that the optimal allocation is computed only once at the beginning of the investment horizon assuming that the allocations will remain constant. This is a “set-it-and-forget-it” strategy. Although you can recompute the allocation weights at each reinvestment period by updating the time horizon and initial wealth, a single-period problem does not consider possible allocation changes in the middle of the investment horizon. In contrast, the optimal solution for a "multi-period" approach takes into account possible changes to the weights at intermediate investment periods. Thus, instead of just selecting the initial portfolio weights, multi-period problems compute the optimal strategy that should be followed given the changes in wealth at each of the reinvestment periods. The Dynamic Portfolio Allocation in Goal-Based Wealth Management for Multiple Time Periods example shows a multi-period version of the problem in this example.

Define Objective Function and Solve Problem

Find the portfolio that maximizes the probability of achieving the investor's goal given the prior assumptions. Maximizing $\mathit{P}\left({\mathit{W}}_{\mathit{T}}\ge \mathit{G}\right)$ is equivalent to minimizing $\mathit{P}\left({\mathit{W}}_{\mathit{T}}\le \mathit{G}\right)$, so the objective function is rewritten as

$$\underset{\mathit{w},\mu ,\sigma }{\max }\mathit{P}\left({\mathit{W}}_{\mathit{T}}\ge \mathit{G}\right)=\underset{\mathit{w},\mu ,\sigma }{\min }\mathit{P}\left({\mathit{W}}_{\mathit{T}}\le \mathit{G}\right)=\underset{\mathit{w},\mu ,\sigma }{\min }\Phi \left(\frac{1}{\sigma \sqrt{\mathit{T}}}\left[\ln \left(\frac{\mathit{G}}{{\mathit{W}}_0}\right)-\left(\mu -\frac{{\sigma }^2}{2}\right)\mathit{T}\right]\right)$$.

Since the relationship of $\mu$ and $\sigma$ to the weights $\mathit{w}$ is represented by equality constraints in the problem, you can substitute these values in the objective function and remove the equality constraints that show the relationship between these three variables.

Define the objective function handle using the definition above and then use estimateCustomObjectivePortfolio to solve the problem.

% Define objective function
probability = @(w) normcdf(1/sqrt(T*(w'*p.AssetCovar*w)) * ...
    (log(G/W0) - (p.AssetMean'*w - (w'*p.AssetCovar*w)/2)*T));

% Solve problem
wGBWM = estimateCustomObjectivePortfolio(p,probability);

Plot Efficient Frontier

Use plotFrontier to plot the efficient frontier, the maximum Sharpe ratio portfolio, and the portfolio that maximizes the probability of achieving the goal (called the GBWM portfolio).

% Plot efficient frontier
figure;
plotFrontier(p,pret,prisk);
hold on
plot(estimatePortRisk(p,wSR),estimatePortReturn(p,wSR),'b.', ...
    'MarkerSize',20);
plot(estimatePortRisk(p,wGBWM),estimatePortReturn(p,wGBWM),'r.', ...
    'MarkerSize',20);
legend('Efficient frontier','Maximum Sharpe ratio', ...
    'GBWM Portfolio','Location','southeast')
hold off

The GBWM portfolio is riskier than the maximum Sharpe ratio portfolio. This extra risk is necessary to maximize the probability of attaining the investor's wealth goal $\mathit{G}$ by time $\mathit{T}$.

Compare GBWM Portfolio to Maximum Sharpe Ratio Portfolio

Compare the probabilities of achieving the wealth goal $\mathit{G}$ by the end of the investment period $\mathit{T}$ for the maximum Sharpe ratio portfolio against the GBWM portfolio.

Tprobabilities = table(1-probability(wSR),1-probability(wGBWM),...
    RowNames={'P(W_T >= G)'},VariableNames={'SR','GBWM'})

Tprobabilities=1×2 table
                     SR        GBWM  
                   _______    _______

    P(W_T >= G)    0.94592    0.95347

The probability of successfully meeting the investor's wealth goal does not change much between the maximum Sharpe ratio portfolio and the GBWM portfolio. Using this information, an investor can understand the trade-off between achieving their wealth goal $\mathit{G}$ by time $\mathit{T}$ compared to choosing a less risky portfolio. For example, an investor might choose a minimum variance portfolio that has a much lower probability of achieving their wealth goal $\mathit{G}$ by time $\mathit{T}$. Find the minimum variance (MV) portfolio, and compare the probability of achieving the wealth goal using this portfolio with the other two approaches.

wMV = estimateFrontierLimits(p,'min');
Tprobabilities.MV = 1-probability(wMV)

Tprobabilities=1×3 table
                     SR        GBWM        MV  
                   _______    _______    ______

    P(W_T >= G)    0.94592    0.95347    0.8252

References

[1] Das, Sanriv R., Daniel Ostrov, Anand Radhakrishnan, and Deep Srivastav. “A New Approach to Goals-Based Wealth Management.” Journal of Investment Management. 16, no. 3 (2018): 1–27.