1: Portfolio Theory Flashcards
Gamble function G(A,B; α)
What does it mean?
A occurs with prob α
B occurs w prob 1 - α
Utility function models
Utility wrt wealth
What are the 6 utility axioms?
- Complete/Comparative
- Transitive
- Independent
- Measurability
- Ranking
- Certainty equivalent
Principle of Non-satiation
Individuals prefer moer wealth to less.
So u’(w) >0
3 types of investors
Risk…
Averse
Neutral
Loving
Risk averse investors
Prefer expectation of risk to the risk itself
E (W) ≻ W.
U fnc concave
Risk neutral investors
W ∼ E (W )
U fnc linear
Risk loving investors
W ≻ E (W )
They prefer the gamble
U fnc convex
Risk premium definition
Max amount a RA investor will pay to avoid uncertainty.
Absolute risk aversion eqn
A(w) = -u’‘(w) / u(w)
Decreasing absolute risk aversion eqn means
We take more risky investments as wealth increases
Relative risk aversion shows
Change in prop of risky assets held, as wealth changes
Relative risk aversion eqn
R(w) = w*A(w)
Mean-Variance Portfolio theory helps us
choose the proportion of each asset in our portfolio
What we want with a portfolio
Max expected returns and min variance.
Certainty equivalent weath equation
c(W) = invU( E[U(W)] )
What is an investment risk measure?
it puts a number of the risk of an asset
Most important investment risk measure
Value at risk
Value at risk (alpha) meaning
The loss value where there’s only a 1-alpha chance of a bigger loss
What is a shortfall measure
int from −∞ to L
g(L−x) f(x) dx
semi-var(X)=
int from −∞ to μ
(x−μ)^2 f(x) dx
How to find the relationship between risk measures and utility functions
Expand out the E() and use taylor series to make it look familiar, (eg having some terms from Variance)
For a portfolio what happens when ρ=-1
basically a risk free asset
For a ptflo what happens when ρ=0
The risk/rtn will lie on the sideways curve
For a ptflo what happens when ρ=1
risk/rtn will lie on a straight line
What does vector Z have
expected rtn for each asset
Vector W shows
weights of each asset
Σ matrix is
symmetrical covariance matrix
What does the two fund theorem make us realise.
An efficient portfolio can be found by combining two or more other efficient portfolios
λ=
(C−μB) / Delta
γ=
(μA−B)/Delta
What does one fund theorum introduce
a risk free asset into our portfolio
what does one fund theorem change
the value of mean and var, and therefore the solution to the least var portfolio eqn.
what is the tangency portfolio
where the efficient frontier meets the one fund line.
to the left of the tangency portfolio, we are still
having some rf asset in the P
