Finance

% Invest 10 000 now, receive growing cash flows for 6 years
cf = [-10000 1800 2300 2700 3000 3100 3200];
rates = linspace(0, 0.25, 200);
v = arrayfun(@(r) pvvar(cf, r), rates);
r_irr = irr(cf);

subplot(2, 1, 1)
bar(0:6, cf), xlabel('year'), ylabel('cash flow'), title('Cash flows')
subplot(2, 1, 2)
plot(100*rates, v, 'LineWidth', 2), hold on
yline(0, 'k'), xline(100*r_irr, '--', sprintf('IRR = %.1f %%', 100*r_irr))
plot(8, pvvar(cf, 0.08), 'o', 'MarkerSize', 7), hold off
xlabel('discount rate (%)'), ylabel('net present value')
title(sprintf('NPV at 8 %%: %.0f  →  invest if the cost of capital is below the IRR', pvvar(cf, 0.08)))

% 300 000 over 30 years at 4.5 %: the standard schedule vs paying 200 extra each month
rate = 0.045/12; n = 360; P = 300000;
[principal, interest, balance, pay] = amortize(rate, n, P);

b = P; bal2 = []; int2 = 0;                       % with extra payments
while b > 0
  i = b * rate; b = b - (pay + 200 - i); int2 = int2 + i;
  bal2(end+1) = max(b, 0);
end

plot((1:n)/12, balance/1000, (1:numel(bal2))/12, bal2/1000, 'LineWidth', 2)
legend('standard', '+200 per month'), xlabel('years'), ylabel('balance (thousands)')
title(sprintf('Payment %.2f/month. Extra 200: %.1f years sooner, %.0f less interest', pay, (n - numel(bal2))/12, sum(interest) - int2))

% European options on a stock: strike 100, 6 months, 3 % rate, 25 % volatility
K = 100; r = 0.03; T = 0.5; sig = 0.25;
S = linspace(60, 140, 200);
[c, p] = blsprice(S, K, r, T, sig);
[dc, dp] = blsdelta(S, K, r, T, sig);
g = blsgamma(S, K, r, T, sig);

subplot(2, 2, 1)
plot(S, c, S, max(S - K, 0), 'k:', 'LineWidth', 1.8), title('Call: price vs payoff at expiry'), xlabel('stock price')
subplot(2, 2, 2)
plot(S, p, S, max(K - S, 0), 'k:', 'LineWidth', 1.8), title('Put'), xlabel('stock price')
subplot(2, 2, 3)
plot(S, dc, S, dp, 'LineWidth', 1.8), yline(0, 'k'), title('Delta'), legend('call', 'put'), xlabel('stock price')
subplot(2, 2, 4)
[SS, TT] = meshgrid(linspace(60, 140, 40), linspace(0.02, 1, 40));
contourf(SS, TT, blsprice(SS, K, r, TT, sig), 14), colorbar
title('Call price by stock price and time left'), xlabel('stock price'), ylabel('years to expiry')

% Geometric Brownian motion (simulated, not a forecast): 8 % drift, 20 % volatility, one year
rng(11)
S0 = 100; mu = 0.08; sig = 0.2; T = 1; steps = 252; N = 4000; dt = T/steps;
Z = randn(N, steps);
paths = S0 * exp(cumsum((mu - sig^2/2)*dt + sig*sqrt(dt)*Z, 2));
ST = paths(:, end);

loss = S0 - ST;                                   % loss over the year
VaR = prctile(loss, 95); CVaR = mean(loss(loss >= VaR));

% price a call by Monte Carlo under the risk-neutral drift, and compare with Black–Scholes
r = 0.03; K = 105;
STq = S0 * exp((r - sig^2/2)*T + sig*sqrt(T)*randn(N, 1));
mc = exp(-r*T) * mean(max(STq - K, 0));
bsc = blsprice(S0, K, r, T, sig);

subplot(1, 2, 1)
plot((0:steps)/steps, [S0*ones(40, 1), paths(1:40, :)]', 'LineWidth', 0.6)
xlabel('years'), ylabel('price'), title('40 of 4000 simulated paths')
subplot(1, 2, 2)
histogram(ST, 50), xline(S0 - VaR, 'r--', '95 % VaR')
xlabel('price after one year'), title(sprintf('VaR %.1f, CVaR %.1f · call: MC %.2f vs BS %.2f', VaR, CVaR, mc, bsc))

% Four asset classes with illustrative yearly returns, volatilities and correlations (not market data)
names = {'Bonds', 'Equity', 'Property', 'Commod.'};
mu  = [0.03 0.08 0.06 0.05];
vol = [0.05 0.17 0.13 0.20];
R = [1 0.1 0.2 0; 0.1 1 0.6 0.3; 0.2 0.6 1 0.2; 0 0.3 0.2 1];
Sigma = (vol' * vol) .* R;
rf = 0.02;

% long-only frontier: for each target return, the least risky weights w >= 0 that sum to 1 (quadprog)
targets = linspace(min(mu), max(mu), 30);
risk = zeros(size(targets)); W = zeros(numel(targets), 4);
for k = 1:numel(targets)
  w = quadprog(2*Sigma, zeros(4, 1), [], [], [mu; ones(1, 4)], [targets(k); 1], zeros(4, 1), []);
  W(k, :) = w'; risk(k) = sqrt(w' * Sigma * w);
end
[fr, fm] = frontier(mu, Sigma, 40);                % short selling allowed (closed form), for comparison
[~, k] = max((targets - rf) ./ risk);              % best Sharpe ratio among long-only portfolios

subplot(1, 2, 1)
plot(100*fr, 100*fm, 'k--', 100*risk, 100*targets, 'LineWidth', 2), hold on
scatter(100*vol, 100*mu, 40, 'filled')
plot(100*risk(k), 100*targets(k), 'rp', 'MarkerSize', 12), hold off
legend('short selling allowed', 'long-only', 'single assets', 'best Sharpe'), xlim([0 22])
xlabel('risk (% per year)'), ylabel('expected return (%)'), title('Efficient frontiers')
subplot(1, 2, 2)
bar(100*W(k, :)), xticklabels(names), ylabel('weight (%)'), ylim([0 100])
title(sprintf('Best long-only mix: Sharpe %.2f', (targets(k) - rf) / risk(k)))

% A 10-year 5 % bond (face 100, semi-annual coupons) and a 30-year one
y = linspace(0.005, 0.12, 120);
p10 = arrayfun(@(x) bndprice(x, 0.05, 10), y);
p30 = arrayfun(@(x) bndprice(x, 0.05, 30), y);
[p0, D, Cx] = bndprice(0.05, 0.05, 10);           % price, Macaulay duration, convexity
% duration + convexity approximation around 5 %
dy = y - 0.05; approx = p0 * (1 - D/(1 + 0.05/2)*dy + 0.5*Cx*dy.^2);

plot(100*y, p10, 100*y, p30, 'LineWidth', 2), hold on
plot(100*y, approx, 'k:', 'LineWidth', 1.5), xline(5, ':'), hold off
legend('10-year', '30-year', 'duration + convexity (10y)')
xlabel('yield (%)'), ylabel('price (face 100)')
[~, D30] = bndprice(0.05, 0.05, 30);
title(sprintf('Longer bonds react more: duration %.1f vs %.1f years', D, D30))

% A simple trend rule tested on simulated prices — an illustration of backtesting, not advice
rng(21)
days = 1000;
r = 0.0003 + 0.012*randn(1, days);
r(400:520) = r(400:520) - 0.004;                        % a simulated bear market
price = ret2tick(r, 100);
fast = movavg(price, 'exponential', 20);
slow = movavg(price, 'simple', 100);
inMarket = [0, fast(1:end-1) > slow(1:end-1)];           % yesterday's signal, today's position
strat = ret2tick(r .* inMarket(2:end), 100);

subplot(2, 1, 1)
plot(price, 'Color', [0.6 0.6 0.6]), hold on, plot(fast, 'LineWidth', 1.4), plot(slow, 'LineWidth', 1.4), hold off
legend('price', '20-day EMA', '100-day SMA'), title('Signals')
subplot(2, 1, 2)
plot(price, 'LineWidth', 1.4), hold on, plot(strat, 'LineWidth', 1.4), hold off
legend('buy and hold', 'crossover'), xlabel('trading day')
title(sprintf('Sharpe (yearly) %.2f vs %.2f · max drawdown %.0f %% vs %.0f %%', ...
  sharpe(r)*sqrt(252), sharpe(r .* inMarket(2:end))*sqrt(252), 100*maxdrawdown(price), 100*maxdrawdown(strat)))

% GARCH(1,1)-style returns (simulated): calm and stormy periods, more extreme days than a normal curve
rng(8)
n = 2500; w = 0.000002; a = 0.09; b = 0.9;
r = zeros(1, n); v = w/(1 - a - b) * ones(1, n);
for t = 2:n
  v(t) = w + a*r(t-1)^2 + b*v(t-1);
  r(t) = sqrt(v(t)) * randn;
end
subplot(2, 1, 1)
plot(100*r, 'LineWidth', 0.5), hold on
plot(100*movstd(r, 40)*sqrt(252)/10, 'r', 'LineWidth', 1.5), hold off
legend('daily return (%)', 'rolling volatility (yearly %, ÷10)'), xlabel('day')
title('Volatility comes in clusters')
subplot(2, 1, 2)
z = zscore(r);
histogram(z, 80), hold on
x = linspace(-6, 6, 200); plot(x, normpdf(x) * n * (max(z) - min(z))/80, 'k', 'LineWidth', 1.5), hold off
xlim([-6 6]), xlabel('standardised return')
title(sprintf('kurtosis %.1f (normal: 3) · days beyond 4σ: %d (normal: ≈%.2f)', kurtosis(r), sum(abs(z) > 4), n*2*(1 - normcdf(4))))