Solutions and Proofs: Optimizing Portfolios An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan
Covariance Proof: Cov(X, Y) = E [XY Y E [X] XE [Y] + E [X] E [Y]] = E [XY] E [Y] E [X] E [X] E [Y] + E [X] E [Y] = E [XY] E [X] E [Y]
Properties of Covariance (1 of 2) Proof: Let X, Y, and Z be random variables, then Cov (X + Y, Z) = E [(X + Y)Z] E [X + Y]E [Z] = E [XZ] + E [YZ] E [X] E [Z] E [Y]E [Z] = E [XZ] E [X] E [Z] + E [YZ] E [Y]E [Z] = Cov (X, Z) + Cov(Y, Z)
Properties of Covariance (2 of 2) I Proof: Let {X 1, X 2,..., X k, X k+1 } be random variables. Cov ( k+1 ) X i, Y 1 ( k ) = Cov X i, Y 1 + Cov(X k+1, Y 1 ) = = k Cov (X i, Y 1 ) + Cov(X k+1, Y 1 ) k+1 Cov(X i, Y 1 ) Therefore by induction we may show that the result is true for any finite, integer value of n and m = 1.
Properties of Covariance (2 of 2) II When m is an integer larger than 1 we can argue that m Cov X i, = m Cov X i, Y j j=1 = = = j=1 Y j m Cov Y j, X i j=1 j=1 j=1 m Cov ( ) Y j, X i m Cov ( ) X i, Y j.
Properties of Covariance (2 of 2) III Proof: Let Y = n X i, then Var(Y) = Cov(Y, Y) ( ) Var X i = Cov X i, = = = j=1 j=1 X j Cov ( ) X i, X j Cov(X i, X i ) + Var(X i ) + Cov ( ) X i, X j j i Cov ( ) X i, X j j i
Properties of the Correlation Proof: Cov(X, Y) = Cov (X, ax + b) = E [X(aX + b)] E [X] E [ax + b] [ ] = E ax 2 + bx) E [X] (ae [X] + b) [ = ae X 2] + be [X] ae [X] E [X] be [X] ( = a E [X 2] E [X] 2) = avar (X) avar(x) ρ(x, Y) = Var (X) a 2 Var(X) = a a
Schwarz Inequality I Proof: If a and b are real numbers then the following two inequalities hold: [ 0 E (ax + by) 2] [ = a 2 E X 2] + 2abE [XY] + b 2 E [Y 2] [ 0 E (ax by) 2] [ = a 2 E X 2] 2abE [XY] + b 2 E [Y 2] If we let a 2 = E [ Y 2] and b 2 = E [ X 2] then the first inequality above yields [ 2E X 2] E [ 0 2E [ Y 2] 2 E [ X 2] E [ Y 2] X 2] E [ Y 2] + 2 E [ Y 2] E [ X 2] E [XY] E [ Y 2] E [ X 2] E [XY] E [XY].
Schwarz Inequality II Similarly the second inequality produces E [XY] E [ X 2] E [ Y 2]. Therefore, since E [ X 2] E [ Y 2] E [XY] E [ X 2] E [ Y 2] [ (E [XY]) 2 E X 2] [ E Y 2].
Range of the Correlation Proof: (Cov (X, Y)) 2 = (E [(X E [X])(Y E [Y])]) 2 [ E (X E [X]) 2] E [(Y E [Y]) 2] = Var(X) Var(Y) Thus Cov(X, Y) Var(X) Var(Y), which is equivalent to the inequality, 1 Cov (X, Y) Var (X) Var (Y) 1 1 ρ(x, Y) 1.
Technical Result I Proof: E [ X(X K) +] = = = 1 x(x K) + 1 2πσ x e (ln x µ)2 /2σ 2 dx 0 1 (x K)e (ln x µ)2 /2σ 2 dx 2πσ K 1 2π (ln K µ)/σ (e σz+µ K)e σz+µ e z2 /2 dz
Technical Result II Make the substitution σz = ln x µ. E [ X(X K) +] = e2(µ+σ2 ) 2π Keµ+σ2 /2 2π (ln K µ)/σ (ln K µ)/σ e (z 2σ)2 /2 dz e (z σ)2 /2 dz ( ) µ ln K = e 2(µ+σ2) φ + 2σ σ ( ) µ ln K Ke µ+σ2 /2 φ + σ. σ
Concavity and Derivatives I Proof: If f is concave on (a, b) then by definition f satisfies λf(x) + (1 λ)f(y) f(λx + (1 λ)y) for every x, y (a, b) and every λ [0, 1]. Assume x < y. If w = λx + (1 λ)y and if 0 < λ < 1 then a < x < w < y < b. By the definition of w, (1 λ)[f(y) f(w)] λ[f(w) f(x)] 1 λ = w x y x and λ = y w y x. Substituting these expressions yields f(y) f(w) y w f(w) f(x) w x 0
Concavity and Derivatives II Applying the Mean Value Theorem to each of the difference quotients of implies that for some α and β satisfying with x < α < w < β < y, f (β) f (α) 0 Using the Mean Value Theorem once more proves that for some t with α < t < β which implies f (t) 0. f (t)(β α) 0
Jensen s Inequality (Discrete Version) Proof: Let µ = n λ ix i and note that since λ i [0, 1] for i = 1, 2,...,n and n λ i = 1, then a < µ < b. The equation of the line tangent to the graph of f at the point (µ, f(µ)) is y = f (µ)(x µ) + f(µ). Since f is concave on (a, b) then Therefore f(x i ) f (µ)(x i µ) + f(µ) for i = 1, 2,..., n. λ i f(x i ) ( [ λi f (µ)(x i µ) + f(µ) ]) = f (µ) (λ i x i λ i µ) + f(µ) ( ) = f(µ) = f λ i x i. λ i
Jensen s Inequality (Continuous Version) Proof: For the sake of compactness of notation let α = 1 0 φ(t) dt, and let y = f (α)(x α) + f(α), the equation of the tangent line passing through the point with coordinates (α, f(α)). Since f is concave then which implies that f(φ(t)) f (α)(φ(t) α) + f(α), 1 0 f(φ(t)) dt 1 0 [ f (α)(φ(t) α) + f(α) ] dt 1 = f(α) + f (α) (φ(t) α) dt 0 ( 1 ) = f(α) = f φ(t) dt. 0
Expected Utility Solution: A rational investor will select the investment with the greater expected utility. The expected utility for investment A is 1 2 u(10) + 1 2 u(0) = 1 2 ) (10 102 = 3. 25 The expected utility for B is u(m) = M M 2 /25. Thus the investor will choose the coin flip whenever 3 > M M2 25 Thus investment A is preferable to B whenever M < $3.49.
Certainty Equivalent Solution: The certainty equivalent and payoffs of investment A must satisfy the following equation. C C2 25 = 1 (X X 2 2 25 + Y Y 2 ) 25 ) [ 2 ( = 1 X 25 2 2 ( C 25 2 ) 2 + ( Y 25 2 ) 2 ] ( C = 25 2 1 X 25 ) 2 ( + Y 25 ) 2 2 2 2
Minimum Variance Analysis I Proof: Since the rates of return are uncorrelated, the variance of the returned wealth W is Var (W) = α 2 i σ2 i, and is subject to the constraint that 1 = n α i. Applying the technique of finding the minimum using Lagrange Multipliers yields the following system of equations. ( ) ( ) = λ α i α 2 i σ2 i α i = 1
Minimum Variance Analysis II These equations are equivalent to respectively: 2α i σi 2 = λ for i = 1, 2,...,n, and α i = 1. Solving for α i in the first equation and substituting into the second equation determines that λ = 2 n j=1 1. σj 2 Substituting this expression for λ into the first equation yields α i = 1 σ 2 i n j=1 1 σ 2 j for i = 1, 2,...,n.
Portfolio Separation Theorem Proof: Suppose x is a portfolio for which r(x) = b, then ( ) 1 1 b r(x) = r b x = 1. Thus 1 bx is a portfolio with unit expected rate of return. For the portfolio w, σ 2 (bw ) = b 2 σ 2 (w ) b 2 σ 2 ( 1 b x ) = σ 2 (x).