Portfolio value under scenarios and Monte Carlo VaR¶
Problem¶
A portfolio holds five stocks, with current prices \(P_0\) and share quantities \(q\). Given a matrix of three return scenarios, compute the portfolio value and return in each one. Then replace the three scenarios with 100 simulated ones and estimate the 95% Value at Risk.
Method¶
The portfolio value is the price vector times the quantities, today and in each scenario \(s\):
\[
V_0 = P_0\, q^\top, \qquad V_s = \big(P_0 \circ (1 + R_s)\big)\, q^\top, \qquad R^{p}_s = \frac{V_s}{V_0} - 1
\]
For the simulation, each stock's return is drawn from \(\mathcal{N}(0,\, 0.1^2)\). The 95% VaR is the 5th percentile of the 100 simulated portfolio returns.
Code¶
clear all, clc
% Point 1
% input of daily return matrix R
R = [ 0.03, 0.05, 0.04, 0.02, 0.05;
0.05, 0.06, 0.05, 0.03, 0.08;
-0.04, -0.02, 0.01, -0.01, -0.10];
[n, m] = size(R); % n (rows) are the scenarios, m (columns) are the stocks
P_0 = [10, 20, 15, 20, 25]; % current price vector (at time 0)
P = repmat(P_0, n, 1) .* (1 + R); % calculation of asset price simulation matrix
% new price = current price * (1 + rate of return)
% repmat replicates the row vector so it can be multiplied by each scenario in R
q = [1000, 800, 1100, 700, 500]; % input of the stock share vector (number of shares held for each stock)
V_q = P * q'; % value of the portfolio for each scenario
V_0 = P_0 * q'; % current portfolio value
R_q = (V_q / V_0) - 1; % portfolio return for each scenario
% (final value / initial value) - 1: one return per scenario (the three rows of R)
% Point 2 - Monte carlo simulation
% given the previously introduced:
% m number of stocks
% q stock shares
% P_0 current price vector
% V_0 current portfolio value
% Instead of three given scenarios, generate 100 random ones and see
% how the portfolio behaves across them. The last three steps are the
% same as in Point 1.
n_MC = 100; % n_MC are the scenarios of the Monte Carlo simulation
R_MC = randn(n_MC, m) / 10; % simulation of a multivariate normal rv
% 100 x 5 matrix of normal returns; dividing by 10 gives a 10% volatility
P_MC = repmat(P_0, n_MC, 1) .* (1 + R_MC); % calculation of asset price simulation matrix
V_MC_q = P_MC * q'; % value of the portfolio for each scenario
R_MC_q = (V_MC_q / V_0) - 1; % portfolio return for each scenario
% extra: histogram of the simulated returns
figure;
histogram(R_MC_q, 15); % 15 bins
title('Portfolio return distribution (Monte Carlo simulation)');
xlabel('Portfolio return (e.g. 0.05 = 5%)');
ylabel('Frequency (number of scenarios out of 100)');
grid on;
% extra: 95% Value at Risk
% "With 95% confidence, the portfolio will not lose more than X"
% 1. confidence level
livello_confidenza = 95;
percentile_var = 100 - livello_confidenza; % the 5% threshold
% 2. VaR as a return, from the percentile of the simulated returns
VaR_95_perc = prctile(R_MC_q, percentile_var);
% 3. print the result as a percentage
fprintf('95%% VaR of the portfolio: %.2f%%\n', VaR_95_perc * 100);
% 4. the same loss in currency
VaR_95_valore = V_0 * VaR_95_perc;
fprintf('Estimated maximum loss at 95%%: %.2f EUR\n', VaR_95_valore);
import numpy as np
import matplotlib.pyplot as plt
# %% Point 1
# input of daily return matrix R
R = np.array([[ 0.03, 0.05, 0.04, 0.02, 0.05],
[ 0.05, 0.06, 0.05, 0.03, 0.08],
[-0.04, -0.02, 0.01, -0.01, -0.10]])
n, m = R.shape # n (rows) are the scenarios, m (columns) are the stocks
P_0 = np.array([10, 20, 15, 20, 25]) # current price vector (at time 0)
P = np.tile(P_0, (n, 1)) * (1 + R) # MATLAB: repmat(P_0, n, 1) .* (1 + R) (* is element by element)
q = np.array([1000, 800, 1100, 700, 500]) # number of shares held for each stock
V_q = P @ q # value of the portfolio for each scenario (@ is the matrix product, MATLAB's *)
V_0 = P_0 @ q # current portfolio value
R_q = (V_q / V_0) - 1 # portfolio return for each scenario
# %% Point 2 - Monte Carlo simulation
n_MC = 100 # scenarios of the Monte Carlo simulation
R_MC = np.random.randn(n_MC, m) / 10 # 100 x 5 matrix of normal returns, 10% volatility
P_MC = np.tile(P_0, (n_MC, 1)) * (1 + R_MC)
V_MC_q = P_MC @ q
R_MC_q = (V_MC_q / V_0) - 1
# %% extra: histogram of the simulated returns
plt.figure()
plt.hist(R_MC_q, 15) # 15 bins
plt.title('Portfolio return distribution (Monte Carlo simulation)')
plt.xlabel('Portfolio return (e.g. 0.05 = 5%)')
plt.ylabel('Frequency (number of scenarios out of 100)')
plt.grid(True)
# %% extra: 95% Value at Risk
livello_confidenza = 95
percentile_var = 100 - livello_confidenza
VaR_95_perc = np.percentile(R_MC_q, percentile_var) # MATLAB: prctile
print(f"95% VaR of the portfolio: {VaR_95_perc * 100:.2f}%") # f"...{x:.2f}..." is MATLAB's fprintf('%.2f', x)
VaR_95_valore = V_0 * VaR_95_perc
print(f"Estimated maximum loss at 95%: {VaR_95_valore:.2f} EUR")
plt.show()
Results¶
Today's value is \(V_0 = 69{,}000\). In the three given scenarios:
| Scenario | Portfolio value | Return |
|---|---|---|
| 1 | 71,665 | +3.86% |
| 2 | 72,705 | +5.37% |
| 3 | 67,055 | −2.82% |
On one simulated run (random seed fixed at 0 when generating this page), the 95% VaR is −7.10%, a loss of 4,901.54 EUR.
Takeaways¶
- The portfolio return is the change in total value, so each stock weighs by price times quantity, not by number of shares.
- The matrix form (\(P\,q^\top\)) values every scenario in one line: no loop over scenarios.
- A VaR estimated from 100 scenarios is noisy: a different seed gives a visibly different number. More scenarios, or a closed form, give a stable one.