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CO 372: Portfolio Optimization Models

Fall 2021

Problem Set 7

Due: Tuesday 2021-12-7 at 5 pm EDT. Late assignments are not accepted. Papers must be handed in on-line using the labeled dropbox on Crowdmark. Each question is handed in as a separate upload. You can either prepare your solutions electronically using, e.g., LaTeX, or else you can hand-write them and submit a scan. In the latter case, please take care that the scan is of good quality with a white background.

Collaboration policy.

● Students are allowed to discuss question with each other in general terms including helping each other on Piazza. Do not post solutions or partial solutions on Piazza.

● No student should hand in work that entirely represents someone else’s eﬀort.  Coding should be done entirely on one’s own.

● Students who work together privately on homework should list their teammates in their submission. Teams of size up to 5 are allowed.

Please explain/justify your answers. (Total score is 45 with 15 BONUS marks. The BONUS 15 marks mean that a mark can be greater than 100% and count for the ﬁnal grade.)

 Contents 2 2 2 2 3 3

List of Figures

1 Ecient Portfolios (score 5+5=10)

Recall the quadratic problem (Pr3) for an optimal portfolio from M3: min   一tr¯T x + x HxT

s.t. Ax = b

where we assume H > 0, t ≥ 0, A e 皿m ×n , m < n and A is full row rank. Let x(t) denote an optimal solution (portfolio).

1. Show that x(t) is unique for each t ≥ 0. Does an optimum x(t) exist for t < 0? Is it unique?

2. With the assumption t ≥ 0, deﬁne: x(t) is an ecient portfolio. Show that the optimal x(t) is an ecient portfolio

2 Rates of Return and Risks (score 10+10=20, BONUS score 5

2.1 Properties (score 3+3+4=10)

Recall that, for i = 1, . . . , n, we model the rates of return of the ith security as a random variable ri with given mean i  = 匝[ri]. Each portfolio x = (xi ) e 皿n has a rate of return that is represented by rx = ri xi , and rx  = ri xi .

1. Show that rx is a random variable.

2. What is the mean rate of return 匝[rx].

3. Recall that the risk is measured by the variance with H > 0: σ 2  = 匝[(rT xr¯T x)2] = x Hx. Can the risk of a portfolio be reduced toT  0?

2.2 Two Assets (score 4+3+3 = 10, BONUS score 5

Recall the variance and covariance

σi(2)  = var(ri ) = 匝(ri 一 i )2 ,    σij  = 匝[(ri 一 i )(rj  一 j )],

that lead to the correlation coecients σij

ρ¯ij  =

Recall that the correlation matrix H satises Hij  = 匝[(ri 一 i )(rj  一 j )].

1. Show that for the two asset case, the variance

σ 2 (x) = σ 1(2)x1(2) + σ2(2)x2(2) + 2ρ12 σ 1 σ2 x1 x2 .

(2.1)

2. Consider (Pr1), minimizing risk subject to a lower bound on return (or exact return), a wealth constraint, and no shorting.

min x  HxT s.t. T x ≥ r0 , eT x = 1, x ≥ 0.

We consider the ecient risk (σ)/returns ( T x) frontier.

Fix:

σ 1  = .08; σ2  = .18; 1  = .03; 2  = .14; with ﬁve values ρ¯12  e (一1, 一0.5, 0, 0.5, 1}.

(a) Consider the extreme case ρ 12   =  1 of perfectly positively correlated

securities. Show that the ecient risk (σ)/returns ( T x) is a straight line joining points associated with ones security only.

(b) Consider the extreme case ρ 12  = 一1 of perfectly negatively correlated

securities. Show that the ecient risk (σ)/returns ( T x) consists of two lines for ecient and inecient frontiers that intersect on the expected return axis.

(c) BONUS: Draw the eecient (and inecient) frontiers for the remaining three points for ρ 12 given above. You can do this with a for loop and with cvx/MATLAB.

Conclude that for a given level of expected return, the standard devi- ation of an ecient portfolio decreases when the correlation coecient decreases from 1 ↓ 一1.

3 Active Set for QP (BONUS score 10)

Combine the code from Assign 6 (that handled Case 1) with a code that handles Case 2, and complete an active set code for solving the general QP. Recall that in Case 2b, the key is to use the sign of the KKT multiplier to decide which active set element to remove from the active set and thus obtain a search direction. Illustrate that the code works on a random example.

4 Finite Dierence Derivatives (score 5+5+5=15)

1. In M48, the optimal choice offor (FD1) is worked out on Slide 12 for a general function f : 皿 → 皿. Determine the optimal value of ∈ > 0 for (FD2) and the error (sum of truncation and cancellation) for this choice. However, instead of balancing the two errors, take a derivative of the sum and set the derivative to 0 to ﬁnd the best.

2. Apply the analysis in Item 1 specically to the function f (x) = lnx at x = 1.5 to ﬁnd the optimalin terms of ∈mach .

3. Check your analysis in Matlab by running some computations as in Slide

9 of M48. Note that eps in Matlab gives the exact value of ∈mach . Write a sentence to describe what the computation showed.