Portfolio Selection Flashcards
what do investors consider when they are choosing a portfolio?
- portfolio of assets that maximise expected utility subject to wealth constraint
what do these mean?
wk
ukj
rkj
A
pj
rp
wk = terminal wealth in state k
ukj = gross payoff of asset j in state k
rkj = net payoff of asset j in state k
A = investment
pj = price of asset j
rp = net rate of return on the portfolio
making a new portfolio
A > 0
investing
- what is alphaj
- gross rate of return
- net return of portfolio
- alpha j = proportion of portfolio invested in asset j
- gross rate of return of portfolio = wk/A
- net rate of return of portfolio = wk-A/A
reallocation of an existing portfolio
A = 0
moving share of assets around - not investing more
- you cant buy unless you sell
- if wealth stays the same - buying £500 of stock means I sold £500 of bonds
what is FVR?
how do we find a portfolio that maximises expected utility given uncertainty
- 3 states, 2 risky securities and 1 risk free security
- gross payoff of risk free security is constant across states
- how do you choose the number of shares to invest in each security = x0,x1,x2? you want to maximise subject to wealth constraint = A
- take the weighted averages of utility in each final wealth state subject to constraint
explain risk aversion in returns
tradeoff between high returns that are only in a few states and lower returns but more equitably distributed
- portfolios with assets that are negatively correlated dominate portfolios with assets that are positively correlated
- risk averse = prefers riskless assets with same expected return
explain risk neutral in returns
marginal utility of risk neutral investor is constant = utility is linear
- dont care about risk
- only choose the asset with the highest expected return
- will only have a portfolio with multiple assets if they all have the same expected return
for a risk neutral investor explain :
- if E[rkj] = r0 = expected return of risky asset = risk free rate of return
- if E[rkj] > r0
- if E[rkj] < r0
- if E[rkj] = r0
- indifferent
- if E[rkj] > r0
- borrow at r0 and invest in j
- if E[rkj] < r0
- short sell j and invest in risk free asset
what is the mean variance model of portfolio selection?
investors only care about expected return of the risk
- return = weighted average of the returns of assets in different states
- risk = variance
why is covariance important?
need to know assets returns move relative to portfolio
- negative covariance = move in opposite directions - eliminates risk = insurance
what if covariance = 0
what if covariance = negative
what if covariance = positive
0 covariance = independent random variables
negative = cancel each other out
positive = big effect
- huge amounts of returns and states where very low returns
intuition behind risky assets
assets that increase overall risk of the portfolio should compensate risk averse investors by giving higher expected returns than the portfolio expected return
what are the only 2 things that our utility depends on to make a decision
- risk
- risk of portfolio depends on covariance of all of the stuff
- good to have negative covariance - return
what if the covariance of the asset and portfolio is greater than the variance of the portfolio
- LHS denominator > RHS denominator
putting this asset in your portfolio = risk of your portfolio increases
so asset must offer high risk premium than the risk premium of the portfolio
what is the difference between binding and non binding constraints
binding = income is not sufficient to achieve maximum possible utility
I < pxX-
non binding = income is sufficient to achieve maximum possible utility
I >= pxX-
what is the interpretation of the langrian?
equal to the change in the utility of the consumer after a small change in income
non binding constraint
optimum solution of x = x-
FOC = 0
SOC < 0 (maximum)
- taking partial derivative of langrangian WRT income = lambda = 0
- cant benefit from increase in income
what do investors care about with quadratic preferences?
only care about
mean
and
variance
what is the 2 step procedure
- efficient portfolio frontier
- choose portfolio that maximises preferences
how do you construct an efficient portfolio frontier of 2 risky assets
what is an efficient portfolio frontier?
represents the set of optimal portfolios that provide the highest expected return for a given level of risk
Portfolios below the efficient frontier are considered suboptimal because they offer lower returns for the same level of risk
what is the intuition behind risk of portfolio turned into an equaition that depends on variance of each risky asset, correlation coefficient
the variance of the return of the portfolio depends not only on the variances of the returns of the 2 assets but also on their covariance
- the lower the value of the correlation coefficient -1 the lower the variance of the portfolio
because variance in individual returns is cancelled out when the 2 assets are combined
what happens to the variance of the portfolio if the coefficient correlation = +1
p12 = +1
then the risk of the portfolio (standard deviation) is a weighted average of the risks of the 2 assets
- so if one asset goes up so does the other one
assets move in same direction
what happens to the variance of the portfolio if the coefficient correlation = +1
p12 = -1
there exists some alpha that makes the standard deviation of the portfolio equal to 0
- riskless portfolio
- standard deviation is the difference between the 2 of the weighted averages
