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
how does the tradeoff between risk and return vary across portfolios
- as alpha changes
- solve standard deviation of portfolio for alpha
substitute this solution into the expected returns of the portfolio
can create a graph
will tell you how changing the alpha - changing the risk of the portfolio will change the expected return of the portfolio
what is the equation of the graph
expected rate of return of portfolio = alpha(intercept) + beta(slope)*risk of the portfolio
why does the graph always have to have a positive slope?
you wouldnt invest in both assets if the slope was negative
- would be one pointless asset with lower return and higher risk
analyse the graph
shape
if you put all of your money into asset 1 = expected high return buy risk is high
if you put all money into asset 2 = expected low return and risk is lower
bold line = the efficient frontier
why would you never invest on the thin line
no point
for the same level of risk you could get a higher expected return
what does the efficient portfolio frontier look like with 2 risky assets p12= extremes
if correlation coefficient = 1
- EPF = weighted average of the returns of the 2 assets - draw a straight line between
if correlation coefficient = -1
- there is always an alpha where the risk of the portfolio is equal to 0
why is it good to have not perfectly correlated diversification
level of risk that is below the weighted average risk of the 2 assets
because they cancel out
in what circumstance can a risk free portfolio be achieved?
perfectly negative correlated
what is a risk free asset
asset with a constant payoff across all states of nature
what is the expected return of the portfolio with 1 risky and 1 riskless asset
weighted average of the return of the risk free asset and the expected return of the risky asset
what is a risk premium
the difference between expected return of risky asset and return on riskless asset
what shape is the efficient portfolio frontier of risky and riskless asset
linear
relationship between return of the portfolio and risk is a straight line
describe the optimal portfolio selection of risky and riskless graph combines
you choose amount of risky and riskless to invest in according to indifference cruve - you like northwest
- everyone has different risks - different preferences
what does allocation on left on Z indicate?
portfolios with lending at the risk free rate = investment in risk free rate
what does allocation to the right of Z correspond to
portfolios with borrowing at the risk free rate
- borrowing at r0 and investing in the risky asset
- increases the portfolios expected return increases and risk
what does imperfect capital markets imply for borrowing and lending rate
borrowing rate is higher than lending rate
so after Z slope will be shallower
can risk averse people be at point C
yes as long as they are compensated enough for their risk aversion
what to we take prices as
investor that takes asset prices in a competitive financial market as given
