PS 1.2 — Expectation/Variance Algebra & Correlation Geometry

Course: ASSIP 2026 · 8-Week Empirical Finance Track · Week 1 · Problem Set 1.2 Covers: Ch 1.2 (expectation linearity, variance/covariance algebra, two-asset portfolio variance, correlation as cosine / Cauchy–Schwarz, the covariance matrix \mathbf{\Sigma} and its quadratic form, minimum-variance weights). Total: 104 points across 6 problems. Each problem is self-contained.

Ground rules. Use only tools through Chapter 1.2: linearity of expectation, the definitions of variance and covariance, scaling, bilinearity, symmetry, \operatorname{Var}(aX+bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X,Y), correlation \rho = \sigma_{XY}/(\sigma_X\sigma_Y), Cauchy–Schwarz, and the quadratic form \mathbf{w}'\mathbf{\Sigma}\mathbf{w}. No regression, no calculus of several variables beyond one-variable differentiation, no distributional assumptions you are not handed. Show every step; a correct number with no reasoning earns little. Work in fractions/decimals (a return of 0.06 means +6\%), not percent signs, to keep the algebra clean.

Problem 1 — Linearity warm-up: Devon’s three wallets (12 points)

Devon tracks three on-chain assets over one week. His research desk hands him the expected weekly returns \mathbb{E}[R_A] = 0.05, \mathbb{E}[R_B] = -0.02, \mathbb{E}[R_C] = 0.011. He is not told anything about how the three assets move together — no covariances, no independence, nothing.

(a) [4 pts] Devon builds a portfolio with weights 0.5 on A, 0.3 on B, and 0.2 on C. Compute the expected portfolio return \mathbb{E}[R_p] where R_p = 0.5\,R_A + 0.3\,R_B + 0.2\,R_C. State explicitly which property of expectation lets you answer this with no information about the joint distribution.

(b) [4 pts] A friend insists, “You can’t compute that — you don’t know if the assets are correlated.” In two or three sentences, explain precisely why the friend is wrong for the expected return but would be right if the question had instead asked for the portfolio’s variance. Tie your answer to the difference between a linear operation and a nonlinear one.

(c) [4 pts] Devon now considers a leveraged position: he borrows so that his weights become 1.5 on A and -0.5 on B (a short position in B), with nothing in C. The weights still sum to 1. Compute \mathbb{E}[R_p] for this position. Does the negative weight cause any trouble for linearity? Explain in one sentence.

Problem 2 — Covariance algebra from the definitions (16 points)

Let X and Y be any two random variables with finite variance, and let a, b, c, d be constants.

(a) [6 pts] Prove the bilinearity-plus-shift identity \operatorname{Cov}(aX + c,\ bY + d) = ab\,\operatorname{Cov}(X, Y). Start from the definition \operatorname{Cov}(U,V) = \mathbb{E}[(U - \mathbb{E}[U])(V - \mathbb{E}[V])] and use linearity of expectation. Explain in one sentence why the additive constants c and d disappear.

(b) [6 pts] Using only the result of part (a), the symmetry of covariance, and \operatorname{Cov}(Z,Z) = \operatorname{Var}(Z), derive the general weighted formula \operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\,\operatorname{Cov}(X,Y). (Hint: write \operatorname{Var}(aX+bY) = \operatorname{Cov}(aX+bY,\ aX+bY) and expand using bilinearity in both slots.)

(c) [4 pts] A classmate claims that because variance “is just a kind of average,” \operatorname{Var}(X - Y) = \operatorname{Var}(X) - \operatorname{Var}(Y). Give the correct formula for \operatorname{Var}(X - Y), then construct a one-line numerical counterexample showing the classmate’s version can even produce a negative “variance,” which is impossible. (Use any small numbers you like.)

Problem 3 — Maya’s two-asset portfolio and diversification (20 points)

Maya, who studies household finance and fair lending, is splitting $1 between two assets. As stipulated population quantities (she will learn to estimate these in Ch 1.3) — note these are a new pair of assets, with different numbers from the chapter’s worked example, so re-derive rather than quoting Ch 1.2’s results:

Quantity	Asset 1	Asset 2
\mathbb{E}[R]	0.09	0.03
\operatorname{Var}(R)	0.0400	0.0025
\operatorname{sd}(R)	0.20	0.05

She holds weight w in asset 1 and 1-w in asset 2. Let \sigma_{12} = \operatorname{Cov}(R_1, R_2).

(a) [3 pts] Write \mathbb{E}[R_p] and \operatorname{Var}(R_p) as functions of w and \sigma_{12}. Why is \mathbb{E}[R_p] a straight line in w but \operatorname{Var}(R_p) a parabola?

(b) [6 pts] Fix the even split w = 0.5. Compute \operatorname{sd}(R_p) for three cases: \sigma_{12} = +0.010 (positively related), \sigma_{12} = 0 (uncorrelated), and \sigma_{12} = -0.008 (oppositely related). Report all three volatilities to three decimals.

(c) [5 pts] In which of the three cases (if any) is the blended portfolio less volatile than asset 2 alone (whose volatility is 0.05)? For each case, state in one sentence what the sign of the covariance is doing to the cross-term.

(d) [6 pts] Maya’s professor frames a book of mortgages the same way: each loan looks identical on paper, but defaults can move together (a regional recession) or independently. Explain, in three or four sentences and referencing the cross-term 2w(1-w)\sigma_{12}, why a lender holding 100 loans whose defaults move together faces more total risk than a lender holding 100 otherwise-identical loans whose defaults move independently — even though every individual loan has the same default variance.

Problem 4 — Correlation is a cosine; prove |\rho| \le 1 (22 points)

(a) [10 pts] Prove the Cauchy–Schwarz inequality for random variables, \operatorname{Cov}(X,Y)^2 \le \operatorname{Var}(X)\,\operatorname{Var}(Y), using the one-variable-knob method from the chapter: for centered variables \tilde X = X - \mathbb{E}[X] and \tilde Y = Y - \mathbb{E}[Y] and any real t, the quantity \operatorname{Var}(X - tY) = \mathbb{E}[(\tilde X - t\tilde Y)^2] cannot be negative. Expand it as a quadratic in t, and use the fact that a downward-or-upward parabola that is never negative must have discriminant \le 0. Then state, in one sentence, the exact condition under which equality holds (i.e. |\rho| = 1) and what it means geometrically.

(b) [4 pts] Conclude that -1 \le \rho_{XY} \le 1 where \rho_{XY} = \operatorname{Cov}(X,Y) / (\operatorname{sd}(X)\operatorname{sd}(Y)). Identify which step of part (a) is the random-variable translation of |\cos\theta| \le 1.

(c) [4 pts] Sam is studying two momentum signals with \operatorname{sd}(X) = 0.15 and \operatorname{sd}(Y) = 0.04. A data vendor reports \operatorname{Cov}(X,Y) = 0.006. Using the geometric picture, find the implied correlation \rho and the angle \theta between the two signals (in degrees, \cos\theta = \rho). Then explain why a reported covariance of 0.006 is at the very edge of what is possible for these two standard deviations, and give a specific covariance value the vendor could not legitimately report.

(d) [4 pts] Sam now builds a third signal Z = X^2, where X is symmetric about zero (so \mathbb{E}[X] = \mathbb{E}[X^3] = 0). Show that \operatorname{Cov}(X, Z) = 0, so \rho_{XZ} = 0 and the two are uncorrelated — yet Z is a deterministic function of X and so the two are about as dependent as variables can be. In one sentence, name the geometric reason correlation reports “no relationship” here: what kind of relationship can a cosine-of-an-angle measure see, and what kind is it blind to?

Problem 5 — A covariance matrix: symmetry, PSD, and the quadratic form (18 points)

Priya, who works on climate and insurance risk, models three lines of business with the candidate covariance matrix (entries in squared-return units) \mathbf{\Sigma} = \begin{pmatrix} 0.04 & 0.01 & 0.000 \\ 0.01 & 0.09 & -0.018 \\ 0.000 & -0.018 & 0.01 \end{pmatrix}.

(a) [3 pts] State the two structural properties every legitimate covariance matrix must have. Verify the first (symmetry) by inspection, and read off the three line-of-business variances and their standard deviations.

(b) [5 pts] Compute the three pairwise correlations \rho_{12}, \rho_{13}, \rho_{23} from \mathbf{\Sigma}. Which pair would help Priya diversify the most, and why? Confirm each |\rho| \le 1 as a sanity check (this is the matrix being well-behaved on each pair).

(c) [6 pts] Priya proposes the equal-weight portfolio \mathbf{w} = (1/3,\ 1/3,\ 1/3)'. Compute the portfolio variance \mathbf{w}'\mathbf{\Sigma}\mathbf{w} by expanding the double sum \sum_{j}\sum_{k} w_j w_k \Sigma_{jk} (there are 9 terms; group the 3 diagonal and 6 off-diagonal). Report the portfolio standard deviation.

(d) [4 pts] Priya’s colleague suggests adding a fourth line that is exactly the sum of lines 1 and 2 (a reinsurance contract bundling them). Without computing a determinant, explain why the resulting 4\times 4 covariance matrix would be singular (not invertible) and only positive semidefinite rather than positive definite. Name the weight vector \mathbf{w} (with the fourth asset) for which \mathbf{w}'\mathbf{\Sigma}\mathbf{w} = 0, and say in one sentence why this same phenomenon will block the matrix inverse in Week 2’s OLS.

Problem 6 — Deriving the minimum-variance weight (16 points)

Return to a generic two-asset world with variances \sigma_1^2, \sigma_2^2 and covariance \sigma_{12}, weights w and 1-w.

(a) [7 pts] Starting from \operatorname{Var}(R_p) = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\sigma_{12}, use single-variable calculus to derive the variance-minimizing weight w^\star = \frac{\sigma_2^2 - \sigma_{12}}{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}}. Show your differentiation, set the derivative to zero, solve for w, and verify via the second derivative that this is a minimum (not a maximum).

(b) [5 pts] Apply the formula to Maya’s Problem 3 numbers (\sigma_1^2 = 0.04, \sigma_2^2 = 0.0025) in the uncorrelated case \sigma_{12} = 0. Find w^\star, the resulting \operatorname{Var}(R_p) and \operatorname{sd}(R_p), and the expected return \mathbb{E}[R_p] at that weight (using \mathbb{E}[R_1]=0.09, \mathbb{E}[R_2]=0.03). Confirm that this volatility is lower than the even-split volatility you found in Problem 3(b).

(c) [4 pts] Examine the formula’s two extremes. (i) If the assets are perfectly positively correlated and \sigma_{12} = \sigma_1\sigma_2, what does w^\star simplify to, and why does this make economic sense? (You may assume \sigma_1 \neq \sigma_2.) (ii) Explain in one sentence why the chapter says this same “parabola with a genuine bottom” reappears when OLS chooses its coefficients in Week 2.

Solutions: see book/appendices/E-solutions-manual/E-w1-ps1.2-solutions.md.

# PS 1.2 — Expectation/Variance Algebra & Correlation Geometry **Course:** ASSIP 2026 · 8-Week Empirical Finance Track · Week 1 · Problem Set 1.2 **Covers:** Ch 1.2 (expectation linearity, variance/covariance algebra, two-asset portfolio variance, correlation as cosine / Cauchy–Schwarz, the covariance matrix $\mathbf{\Sigma}$ and its quadratic form, minimum-variance weights). **Total:** 104 points across 6 problems. Each problem is self-contained. **Ground rules.** Use only tools through Chapter 1.2: linearity of expectation, the definitions of variance and covariance, scaling, bilinearity, symmetry, $\operatorname{Var}(aX+bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X,Y)$, correlation $\rho = \sigma_{XY}/(\sigma_X\sigma_Y)$, Cauchy–Schwarz, and the quadratic form $\mathbf{w}'\mathbf{\Sigma}\mathbf{w}$. No regression, no calculus of several variables beyond one-variable differentiation, no distributional assumptions you are not handed. Show every step; a correct number with no reasoning earns little. Work in fractions/decimals (a return of $0.06$ means $+6\%$), not percent signs, to keep the algebra clean. --- ## Problem 1 — Linearity warm-up: Devon's three wallets (12 points) Devon tracks three on-chain assets over one week. His research desk hands him the expected weekly returns $\mathbb{E}[R_A] = 0.05$, $\mathbb{E}[R_B] = -0.02$, $\mathbb{E}[R_C] = 0.011$. He is not told *anything* about how the three assets move together — no covariances, no independence, nothing. **(a) [4 pts]** Devon builds a portfolio with weights $0.5$ on $A$, $0.3$ on $B$, and $0.2$ on $C$. Compute the expected portfolio return $\mathbb{E}[R_p]$ where $R_p = 0.5\,R_A + 0.3\,R_B + 0.2\,R_C$. State explicitly which property of expectation lets you answer this with no information about the joint distribution. **(b) [4 pts]** A friend insists, "You can't compute that — you don't know if the assets are correlated." In two or three sentences, explain precisely why the friend is wrong for the *expected return* but would be right if the question had instead asked for the portfolio's *variance*. Tie your answer to the difference between a linear operation and a nonlinear one. **(c) [4 pts]** Devon now considers a leveraged position: he borrows so that his weights become $1.5$ on $A$ and $-0.5$ on $B$ (a short position in $B$), with nothing in $C$. The weights still sum to $1$. Compute $\mathbb{E}[R_p]$ for this position. Does the negative weight cause any trouble for linearity? Explain in one sentence. --- ## Problem 2 — Covariance algebra from the definitions (16 points) Let $X$ and $Y$ be any two random variables with finite variance, and let $a, b, c, d$ be constants. **(a) [6 pts]** Prove the bilinearity-plus-shift identity $$ \operatorname{Cov}(aX + c,\ bY + d) = ab\,\operatorname{Cov}(X, Y). $$ Start from the definition $\operatorname{Cov}(U,V) = \mathbb{E}[(U - \mathbb{E}[U])(V - \mathbb{E}[V])]$ and use linearity of expectation. Explain in one sentence *why* the additive constants $c$ and $d$ disappear. **(b) [6 pts]** Using only the result of part (a), the symmetry of covariance, and $\operatorname{Cov}(Z,Z) = \operatorname{Var}(Z)$, derive the general weighted formula $$ \operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\,\operatorname{Cov}(X,Y). $$ (Hint: write $\operatorname{Var}(aX+bY) = \operatorname{Cov}(aX+bY,\ aX+bY)$ and expand using bilinearity in *both* slots.) **(c) [4 pts]** A classmate claims that because variance "is just a kind of average," $\operatorname{Var}(X - Y) = \operatorname{Var}(X) - \operatorname{Var}(Y)$. Give the correct formula for $\operatorname{Var}(X - Y)$, then construct a one-line numerical counterexample showing the classmate's version can even produce a *negative* "variance," which is impossible. (Use any small numbers you like.) --- ## Problem 3 — Maya's two-asset portfolio and diversification (20 points) Maya, who studies household finance and fair lending, is splitting \$1 between two assets. As stipulated population quantities (she will learn to estimate these in Ch 1.3) — note these are a *new* pair of assets, with different numbers from the chapter's worked example, so re-derive rather than quoting Ch 1.2's results: | Quantity | Asset 1 | Asset 2 | |---|---|---| | $\mathbb{E}[R]$ | $0.09$ | $0.03$ | | $\operatorname{Var}(R)$ | $0.0400$ | $0.0025$ | | $\operatorname{sd}(R)$ | $0.20$ | $0.05$ | She holds weight $w$ in asset 1 and $1-w$ in asset 2. Let $\sigma_{12} = \operatorname{Cov}(R_1, R_2)$. **(a) [3 pts]** Write $\mathbb{E}[R_p]$ and $\operatorname{Var}(R_p)$ as functions of $w$ and $\sigma_{12}$. Why is $\mathbb{E}[R_p]$ a straight line in $w$ but $\operatorname{Var}(R_p)$ a parabola? **(b) [6 pts]** Fix the even split $w = 0.5$. Compute $\operatorname{sd}(R_p)$ for three cases: $\sigma_{12} = +0.010$ (positively related), $\sigma_{12} = 0$ (uncorrelated), and $\sigma_{12} = -0.008$ (oppositely related). Report all three volatilities to three decimals. **(c) [5 pts]** In which of the three cases (if any) is the *blended* portfolio less volatile than asset 2 alone (whose volatility is $0.05$)? For each case, state in one sentence what the sign of the covariance is doing to the cross-term. **(d) [6 pts]** Maya's professor frames a book of mortgages the same way: each loan looks identical on paper, but defaults can move together (a regional recession) or independently. Explain, in three or four sentences and referencing the cross-term $2w(1-w)\sigma_{12}$, why a lender holding 100 loans whose defaults move *together* faces more total risk than a lender holding 100 otherwise-identical loans whose defaults move *independently* — even though every individual loan has the same default variance. --- ## Problem 4 — Correlation is a cosine; prove $|\rho| \le 1$ (22 points) **(a) [10 pts]** Prove the Cauchy–Schwarz inequality for random variables, $$ \operatorname{Cov}(X,Y)^2 \le \operatorname{Var}(X)\,\operatorname{Var}(Y), $$ using the one-variable-knob method from the chapter: for centered variables $\tilde X = X - \mathbb{E}[X]$ and $\tilde Y = Y - \mathbb{E}[Y]$ and any real $t$, the quantity $\operatorname{Var}(X - tY) = \mathbb{E}[(\tilde X - t\tilde Y)^2]$ cannot be negative. Expand it as a quadratic in $t$, and use the fact that a downward-or-upward parabola that is never negative must have discriminant $\le 0$. Then state, in one sentence, the exact condition under which equality holds (i.e. $|\rho| = 1$) and what it means geometrically. **(b) [4 pts]** Conclude that $-1 \le \rho_{XY} \le 1$ where $\rho_{XY} = \operatorname{Cov}(X,Y) / (\operatorname{sd}(X)\operatorname{sd}(Y))$. Identify which step of part (a) is the random-variable translation of $|\cos\theta| \le 1$. **(c) [4 pts]** Sam is studying two momentum signals with $\operatorname{sd}(X) = 0.15$ and $\operatorname{sd}(Y) = 0.04$. A data vendor reports $\operatorname{Cov}(X,Y) = 0.006$. Using the geometric picture, find the implied correlation $\rho$ and the angle $\theta$ between the two signals (in degrees, $\cos\theta = \rho$). Then explain why a *reported* covariance of $0.006$ is at the very edge of what is possible for these two standard deviations, and give a specific covariance value the vendor could *not* legitimately report. **(d) [4 pts]** Sam now builds a third signal $Z = X^2$, where $X$ is symmetric about zero (so $\mathbb{E}[X] = \mathbb{E}[X^3] = 0$). Show that $\operatorname{Cov}(X, Z) = 0$, so $\rho_{XZ} = 0$ and the two are *uncorrelated* — yet $Z$ is a deterministic function of $X$ and so the two are about as dependent as variables can be. In one sentence, name the geometric reason correlation reports "no relationship" here: what kind of relationship can a cosine-of-an-angle measure see, and what kind is it blind to? --- ## Problem 5 — A covariance matrix: symmetry, PSD, and the quadratic form (18 points) Priya, who works on climate and insurance risk, models three lines of business with the candidate covariance matrix (entries in squared-return units) $$ \mathbf{\Sigma} = \begin{pmatrix} 0.04 & 0.01 & 0.000 \\ 0.01 & 0.09 & -0.018 \\ 0.000 & -0.018 & 0.01 \end{pmatrix}. $$ **(a) [3 pts]** State the two structural properties every legitimate covariance matrix must have. Verify the first (symmetry) by inspection, and read off the three line-of-business *variances* and their standard deviations. **(b) [5 pts]** Compute the three pairwise correlations $\rho_{12}, \rho_{13}, \rho_{23}$ from $\mathbf{\Sigma}$. Which pair would help Priya diversify the most, and why? Confirm each $|\rho| \le 1$ as a sanity check (this is the matrix being well-behaved on each pair). **(c) [6 pts]** Priya proposes the equal-weight portfolio $\mathbf{w} = (1/3,\ 1/3,\ 1/3)'$. Compute the portfolio variance $\mathbf{w}'\mathbf{\Sigma}\mathbf{w}$ by expanding the double sum $\sum_{j}\sum_{k} w_j w_k \Sigma_{jk}$ (there are 9 terms; group the 3 diagonal and 6 off-diagonal). Report the portfolio standard deviation. **(d) [4 pts]** Priya's colleague suggests adding a fourth line that is *exactly* the sum of lines 1 and 2 (a reinsurance contract bundling them). Without computing a determinant, explain why the resulting $4\times 4$ covariance matrix would be **singular** (not invertible) and only positive *semi*definite rather than positive definite. Name the weight vector $\mathbf{w}$ (with the fourth asset) for which $\mathbf{w}'\mathbf{\Sigma}\mathbf{w} = 0$, and say in one sentence why this same phenomenon will block the matrix inverse in Week 2's OLS. --- ## Problem 6 — Deriving the minimum-variance weight (16 points) Return to a generic two-asset world with variances $\sigma_1^2$, $\sigma_2^2$ and covariance $\sigma_{12}$, weights $w$ and $1-w$. **(a) [7 pts]** Starting from $\operatorname{Var}(R_p) = w^2\sigma_1^2 + (1-w)^2\sigma_2^2 + 2w(1-w)\sigma_{12}$, use single-variable calculus to derive the variance-minimizing weight $$ w^\star = \frac{\sigma_2^2 - \sigma_{12}}{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}}. $$ Show your differentiation, set the derivative to zero, solve for $w$, and verify via the second derivative that this is a minimum (not a maximum). **(b) [5 pts]** Apply the formula to Maya's Problem 3 numbers ($\sigma_1^2 = 0.04$, $\sigma_2^2 = 0.0025$) in the uncorrelated case $\sigma_{12} = 0$. Find $w^\star$, the resulting $\operatorname{Var}(R_p)$ and $\operatorname{sd}(R_p)$, and the expected return $\mathbb{E}[R_p]$ at that weight (using $\mathbb{E}[R_1]=0.09$, $\mathbb{E}[R_2]=0.03$). Confirm that this volatility is lower than the even-split volatility you found in Problem 3(b). **(c) [4 pts]** Examine the formula's two extremes. (i) If the assets are perfectly positively correlated and $\sigma_{12} = \sigma_1\sigma_2$, what does $w^\star$ simplify to, and why does this make economic sense? (You may assume $\sigma_1 \neq \sigma_2$.) (ii) Explain in one sentence why the chapter says this same "parabola with a genuine bottom" reappears when OLS chooses its coefficients in Week 2. --- *Solutions: see `book/appendices/E-solutions-manual/E-w1-ps1.2-solutions.md`.*