Build a mean-variance (Markowitz) portfolio, trace its risk-return efficient frontier, and re-cast the problem with a different risk measure — Conditional Value-at-Risk (CVaR) — all from real return data.
The example and the bundled price data are adapted from Riskfolio-Lib (Dany Cajas) — “Tutorial 1: Classic Mean-Risk Optimization” (Cajas 2026), itself built on CVXPY. The underlying models are due to Markowitz (1952) (mean-variance) and Rockafellar and Uryasev (2000) (CVaR).
We hold \(n\) assets; \(w_i\) is the fraction of our budget in asset \(i\). Returns \(r\) have mean \(\mu\) and covariance \(\Sigma\), so a portfolio \(w\) has expected return \(\mu^\top w\) and variance (risk) \(w^\top\Sigma w\). Markowitz’s insight: trade return against risk with a risk-aversion knob \(\gamma \ge 0\),
\[ \underset{w}{\mbox{maximize}}\ \ \mu^\top w - \gamma\, w^\top\Sigma w \qquad\text{subject to}\qquad w \ge 0,\ \ \sum_i w_i = 1. \]
The constraints make it a long-only, fully-invested portfolio. Small \(\gamma\) chases return; large \(\gamma\) avoids risk.
We use ~3 years of daily prices for 12 well-known stocks (a subset of the Riskfolio-Lib dataset), and turn prices into simple returns, a mean vector, and a covariance matrix:
prices <- read.csv("data/stock_prices.csv", check.names = FALSE)
P <- as.matrix(prices[, -1]) # drop the Dates column
R <- P[-1, ] / P[-nrow(P), ] - 1 # daily simple returns
mu <- colMeans(R)
Sigma <- cov(R)
n <- ncol(R)
colnames(R) [1] "AAPL" "AMZN" "ABT" "ADBE" "AMGN" "AEP" "AFL" "AIG" "ALL" "APD"
[11] "ADM" "AKAM"For each \(\gamma\) we solve the Markowitz problem and record the realized risk and return. This is the same problem re-solved over a grid — exactly where the Parameter trick from Chapter 8 pays off. We make \(\gamma\) a Parameter, build the problem once, and each iteration just sets value(gamma) and re-solves, reusing the cached compilation instead of canonicalizing from scratch 40 times.
w <- Variable(n)
ret <- t(mu) %*% w
risk <- quad_form(w, Sigma) # w' Sigma w, a convex quadratic
cons <- list(w >= 0, sum(w) == 1)
gamma <- Parameter(nonneg = TRUE) # the risk-aversion knob, set per solve
prob <- Problem(Maximize(ret - gamma * risk), cons) # built once
gammas <- 10^seq(-1, 3, length.out = 40)
fr <- sapply(gammas, function(g) {
value(gamma) <- g # swap in the new gamma; no rebuild
psolve(prob)
c(risk = sqrt(value(risk)), ret = value(ret), w = value(w))
})Note how we read the risk and return back by directly evaluating the expressions sqrt(risk) and ret after each solve — no bookkeeping required. The frontier:
df <- data.frame(risk = fr["risk", ], ret = fr["ret", ])
ind <- data.frame(risk = sqrt(diag(Sigma)), ret = mu, asset = names(mu))
ggplot(df, aes(risk, ret)) +
geom_line(color = "steelblue", linewidth = 1) +
geom_point(data = ind, color = "red") +
geom_text(data = ind, aes(label = asset), size = 3, vjust = -0.6,
check_overlap = TRUE) +
labs(x = "Risk (standard deviation of daily return)", y = "Expected daily return") +
theme_minimal()
Every point on the blue curve is the best return achievable at that level of risk; the individual stocks (red) all sit below it — diversification at work.
The frontier shows the risk-return outcome, but not what the portfolio holds. Since fr already captured the weight vector at every \(\gamma\), we can read the composition straight back out and watch diversification happen. At a few points along the grid we stack each asset’s share of the budget:
markers <- c(1, 10, 20, 30, 40)
wmat <- fr[grep("^w", rownames(fr)), markers] # 12 assets x 5 values of gamma
rownames(wmat) <- names(mu)
comp <- data.frame(
gamma = factor(rep(signif(gammas[markers], 2), each = n),
levels = signif(gammas[markers], 2)),
asset = factor(rep(names(mu), times = length(markers)), levels = names(mu)),
weight = as.vector(wmat))
ggplot(comp, aes(gamma, weight, fill = asset)) +
geom_col() +
scale_fill_manual(values = brewer.pal(n, "Paired")) +
labs(x = expression("Risk aversion " * gamma),
y = "Fraction of budget", fill = "Asset") +
theme_minimal()
The left-most bar (greedy, low \(\gamma\)) leans on one or two names; as \(\gamma\) grows the bars fragment into the full palette. That shift is the efficient frontier, seen from the portfolio’s side.
Variance penalizes upside and downside alike. Conditional Value-at-Risk (CVaR) focuses on the bad tail: \(\text{CVaR}_\alpha\) is (roughly) the average loss in the worst \(1-\alpha\) fraction of scenarios. Rockafellar and Uryasev showed it has a beautifully simple convex (in fact linear) form using the historical return scenarios \(r_t\) directly. With losses \(L_t = -r_t^\top w\),
\[ \text{CVaR}_\alpha(w) = \min_{\tau}\ \ \tau + \frac{1}{(1-\alpha)T}\sum_{t=1}^{T}\big(L_t - \tau\big)_+ . \]
In CVXR we introduce an auxiliary VaR variable \(\tau\) and minimize CVaR subject to a return target — a clean linear program:
alpha <- 0.95
target <- mean(mu) # require at least the average asset's return
Tn <- nrow(R)
wv <- Variable(n)
VaR <- Variable(1)
loss <- -R %*% wv # vector of T scenario losses
cvar <- VaR + (1 / ((1 - alpha) * Tn)) * sum(pos(loss - VaR))
prob_cvar <- Problem(Minimize(cvar),
list(sum(wv) == 1, wv >= 0, t(mu) %*% wv >= target))
psolve(prob_cvar)[1] 0.01481885check_solver_status(prob_cvar)
w_cvar <- setNames(round(value(wv), 3), names(mu))
w_cvar[w_cvar > 0.001] # the minimum-CVaR allocation AAPL AMZN ADBE AEP AFL ALL APD ADM
0.090 0.046 0.066 0.390 0.222 0.074 0.046 0.065 Same data, same CVXR machinery, a different notion of risk — and a different, typically more tail-aware, allocation. Switching risk measures is, once again, just a change to the objective.
To see how the allocation differs, hold the return target fixed and put the two side by side. The fair mean-variance counterpart minimizes variance subject to the same return floor, so both bars sit at the same expected return and only the risk measure changes:
## mean-variance allocation at the SAME return target, for an apples-to-apples comparison
wv_mv <- Variable(n)
prob_mv <- Problem(Minimize(quad_form(wv_mv, Sigma)),
list(sum(wv_mv) == 1, wv_mv >= 0, t(mu) %*% wv_mv >= target))
psolve(prob_mv)[1] 4.198194e-05check_solver_status(prob_mv)
comp2 <- data.frame(
measure = factor(rep(c("Mean-variance", "Mean-CVaR"), each = n),
levels = c("Mean-variance", "Mean-CVaR")),
asset = factor(rep(names(mu), 2), levels = names(mu)),
weight = c(value(wv_mv), value(wv)))
ggplot(comp2, aes(measure, weight, fill = asset)) +
geom_col() +
scale_fill_manual(values = brewer.pal(n, "Paired")) +
labs(x = NULL, y = "Fraction of budget", fill = "Asset") +
theme_minimal()
Both portfolios earn the same expected return, yet CVaR — caring only about the bad tail — tilts the budget toward a different mix than variance, which penalizes upside and downside alike.
Exercise (⭐). Add a diversification rule to the mean-variance problem: no single asset may exceed 25% of the budget. Re-solve at \(\gamma = 10\) and compare with the unconstrained allocation.
A position cap is one linear constraint, w <= 0.25:
base <- Problem(Maximize(ret - 10 * risk), cons)
capd <- Problem(Maximize(ret - 10 * risk), c(cons, list(w <= 0.25)))
psolve(base); w0 <- value(w)[1] 0.0003368569psolve(capd); w1 <- value(w)[1] 0.0003300224data.frame(asset = names(mu),
uncapped = round(w0, 3),
capped = round(w1, 3))[round(w0,3) > 0 | round(w1,3) > 0, ] asset uncapped capped
1 AAPL 0.023 0.028
2 AMZN 0.162 0.164
3 ABT 0.105 0.122
4 ADBE 0.114 0.114
6 AEP 0.320 0.250
7 AFL 0.257 0.250
9 ALL 0.017 0.055
11 ADM 0.002 0.018The cap spreads weight off the assets that the unconstrained solution concentrated in — a one-line way to encode an investment policy.
Parameter sweep, Chapter 8).value().For many more portfolio models (risk parity, Black-Litterman, worst-case, drawdown measures), see Riskfolio-Lib and the CVXR portfolio example.