Chapter 4 - Holt Flashcards
(24 cards)
what are the einvestment pricniples in this chapter based on
purely mean and variance of asset returns. And budget restrictions
elaborate on the goals of investment
We want to maximize expected returns, but we also want to minimize uncertainty. There is a tradeoff here.
what is S_t^k
Spot price for asset k at time t
What is B_0
The price of a bond, a risk free asset, at time 0
How do we represent the posiiton in risky assets?
We use a vector, h
h = (h1, h2, …, hn)^T
where h_k is the number of units of asset k we hold.
can h_k be negative?
Yes, it represent short selling
Give the price of a portfolio, at time 0 and time 1
time 0:
V_0 = h_0 B_0 + ∑h_k S_0^k
Time 1:
V_1 = h_0 + ∑h_k S_1^k
epxress V uisng monetary weights
V_0 = w_0 + ∑w_k
V_1 = w_0 1/B_0 + ∑w_k S_1^k /S_0^k
what is the vector of returns
R = (S_1^1/S_0^1 , …)
use matrix/vector notaiton to write the value V_1 of the portfolio
V_1 = w_0 R_0 + w^T R
Therefore, E[V_1] = w_0 R_0 + w^T mu
what is the variance of V_1
It is shown that it can be decomposed into:
Var(V_1) = w^T ∑ w
what properties does the covariance matrix hold
Always symmetric and positive semi-definite
elaborate on the 3 cases of investment problems investors might want to solve
1) The tradeoff problem.
The tradeoff problem is about maximizinbg the expected payoff of the portfolio, less some dollar-normalized variance of the same payoff multiplied by a constant (inivestor specific, represents the tradeoff) all udner the single constriant of that the initial portfolio must not cost more than some fixed amount of cash.
2) Maximization of expectation problem.
Maximize payoff of portfolio based on 2 contraitns:
1: Variance of portfolio at time 1 must be samller than some fixed boundary
2: Initial pric/cost must be witihn bounds
3) minimization of variance
Minimize the variance.
constraints includes that the epxcted payoff must be above some threshold, and of course the cost must sastisfy our cash positiuon.
express teh tradeoff problem using matrix/vector notation
maximize w^T mu - c/(2V_0) w^T ∑ w
subject to w^T 1 <= V_0
key about the tradeoff problem
It is a convex problem. It has a solution (w, lambda) if lambda is greater than 0, and we consider the system of equations:
grad[c/(2V_0) w^T ∑w - w^T mu] + lambda grad[w^T 1]
and
w^T 1 = V_0
how can we determine c, the tradeoff parameter?
No straightforward way.
However, one possibility is to find two investment that appear equally attractive, and then solve the equation
in the formula for the solution to the tradeopff problem, what is the “+” sign?
represent the max operator: max(expression, 0)
what happens here?
It represent that if we do not have a risk free asset to absorb.
Essentially, the optimizer will try to reduce variance, and the only way to do this when there is no risk free asset is to take offsetting positions in the risky assets.
therefore, the problem only make sense if we have risk free asset as a choice.
The key in this image is that the wegihts and solution does not satisfy the constraint of w^T 1 <= V_0
What ends up happening is that the optimizer claim that the optimal position in this case is ot short more than we invest. This cause us to sit on a positive cash position that remains unallocated. This makes no sense in finance. Therefore, the conslusion is that we need risk free asset to make these problems be applicable.
name an issue with Minimum-variance portfolio when there is no risk free asset
Idk, the point is that if you are a mean-variance investor, you would never choose the minimum-variance portfolio
What is “1/B_0”?
1 if the FV
B_0 is hte price
Therefore, this ratio is teh return on the bond
Give teh expected payoff and variance of the portfolio at time 1
E[V_1] = E[w0 R0 + w^T R] = w0 R0 + w^T mu
Var[V_1] = var[w0 R0 + w^T R] = Var[w^T R] = w^T ∑ w
give a reason for using the minimum variance portfolio
the mean mu (expected value/return) of stocks is notoriuosuly difficult to measure. It usually varies quite a bit.
However, the covariance vary a lot less. Therefore, while we might not be comfortable with assigning expected returns, we can assign covariances.
elaborate on the sharpe ratiop
does not depend on C.
It represent a risk-adjusted measure. Expected excess return divided by the standard deviation of the asset.
