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.
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 (2022–2024) of daily adjusted prices for the 12 largest U.S. companies by market capitalization, 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] "NVDA" "GOOGL" "AAPL" "MSFT" "AMZN" "AVGO" "META" "TSLA" "BRK-B"
[10] "JPM" "LLY" "WMT" 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 weight spreads across a broader, more defensive set of names. 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] 0.02056038check_solver_status(prob_cvar)
w_cvar <- setNames(round(value(wv), 3), names(mu))
w_cvar[w_cvar > 0.001] # the minimum-CVaR allocation AVGO BRK-B JPM LLY WMT
0.063 0.320 0.086 0.283 0.248 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] 9.208972e-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] 0.0001773069psolve(capd); w1 <- value(w)[1] 0.0001396685data.frame(asset = names(mu),
uncapped = round(w0, 3),
capped = round(w1, 3))[round(w0,3) > 0 | round(w1,3) > 0, ] asset uncapped capped
1 NVDA 0.023 0.019
6 AVGO 0.085 0.104
9 BRK-B 0.205 0.250
10 JPM 0.068 0.127
11 LLY 0.243 0.250
12 WMT 0.377 0.250The 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 more portfolio models and risk measures, see the CVXR portfolio example.