Optimization Flashcards
(21 cards)
what is the assumptions of this chapter?
We use only mean, varaince and budget contraints.
What are the primary goals of portfolio optimization
Investors want to maximize expected return, but also minimize variance. This is the tradeoff between returns and uncertainty.
notation for an asset
S_t^k
Stock/asset ‘k’ at time ‘t’.
Notation for a position
We use a vector, where each element in the vector is the number of units we have invested in some asset k. Therefore, the size of the vector is ‘n’, the same as the total number of assets in our world.
what is h_0?
If specified, it is our position in the risk free asset
interpreation of negative values in h
Negative posiitons indicate short selling.
give the budget constraint at tiem 0
h_0 B_0 + ∑h_k S_0^k [k=1, n] <= V_0
Give the portfolio value/payoff equation at time 1
V_1 = h_0 + ∑h_k S_1^k [k=1, n]
what happens to our equaitons if there is no risk free asset?
we just let h_0 be 0
give teh equation for initial portfolio cost that is more clean than the one previously given
Also give the payoff version
use portfolio WEIGHTS instead. We define w_k = h_k S_0^k, and get:
V_0 = w_0 + ∑w_k [k=1, n]
V_1 = w_0 1/B_0 + ∑w_k (S_1^k / S_0^k)
V_1 = w_0 1/B_0 + ∑w_k (S_1^k / S_0^k)
This is not really cleaner… why bother with it?
It is basically returns. We can write:
V_1 = w_0 R_0 + ∑w_k R^k
or in matrix form:
V_1 = w_0 R_0 + w^T R
Given the matrix form of the payoff v_1 of the portfolio:
V_1 = w_0 R_ 0 + w^T R
what is the expected value of the portoflio payoff?
E[V_1] = E[w0R0 + w^T R]
= E[w_0 R_0] + E[w^T R]
= w_0 R_0 + w^T E[R]
= w_0 R_0 + w^T mu
What is the variance of the payoff V_1?
Var(V_1) = W^T ∑ W
We have:
Var(V_1) = Var[w_0 R_0 + w^T R]
= Var(w^T R)
the assumption of this chapter is that…´
the investor wants to solve one of the following closely related problems:
NB: This is all without the risk free asset
1) The tradeoff problem
Maximize expected returns, less some constant multiplied by the variance of the returns, subject to the capacity/bdget constraints.
max E[V_1] - C/V_0 Var[V_0]
s.t. ∑h_k S_0^k <= V_0
In other words:
max: E[∑h_k S_1^k] - C/V_0 Var[h_k S_1^k]
s.t.
∑h_k S_0^k <= V_0
2) maximization problem
This problem seeks to maximize the expected value of hte payoff subject to the regular budget constraint AND some specific uncertainty constraint. We specify a level of volatility that we want as a limit, and find the portfolio that offer the greatest expecged payoff given this.
max: E[∑h_k S_1^k]
s.t.
Var[∑h_k S_1^k] <= sigma_0^2 V_0^2
∑h_k S_0^k <= V_0
The variance constrain basically say “we do not want to risk more than X amount”.
3) Minimize variance given some expected return boundaryu.
min: Var[∑h_k S_1^k]
s.t.
E[∑h_k S_1^k] >= mu_0 V_0
∑h_k S_0^k <= V_0
Give the matrix vector notation of the tradeoff problem
max w^T mu - c/(2V_0) w^T ∑ w
s.t. w^T 1 <= V_0
why the 1/2 term?
It produce a nicer looking solution
when can the variance minimizing method be a good one
If we only have historical data for the mean and covariances, we are likely going to select the variance minimizing portfolio. this is because the means are extremely difficult ot measure like that, while the variance tend to be more stable.
what is R_0
The return of the bond:
R_0 = 1/B_0, where B_0 is the price of the bond.
is it optimal to invest all?
yes. we should never see a spolution with not all invested. This is becasue we can always invest in the risk free asset.
elaborate on the sharpe ratiop
Gives the expected excess return divided by the variance. It is therefore a representation of how much bang for hte buck we are getting.
