Portfolio Selection Flashcards

1
Q

what do investors consider when they are choosing a portfolio?

A
  • portfolio of assets that maximise expected utility subject to wealth constraint
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2
Q

what do these mean?

wk
ukj
rkj
A
pj
rp

A

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

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3
Q

making a new portfolio
A > 0
investing

  • what is alphaj
  • gross rate of return
  • net return of portfolio
A
  • 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
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4
Q

reallocation of an existing portfolio
A = 0
moving share of assets around - not investing more

A
  • you cant buy unless you sell
  • if wealth stays the same - buying £500 of stock means I sold £500 of bonds
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5
Q

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
A
  • 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
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6
Q

explain risk aversion in returns

A

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
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7
Q

explain risk neutral in returns

A

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
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8
Q

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
A
  • 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
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9
Q

what is the mean variance model of portfolio selection?

A

investors only care about expected return of the risk

  • return = weighted average of the returns of assets in different states
  • risk = variance
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10
Q

why is covariance important?

A

need to know assets returns move relative to portfolio
- negative covariance = move in opposite directions - eliminates risk = insurance

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11
Q

what if covariance = 0
what if covariance = negative
what if covariance = positive

A

0 covariance = independent random variables

negative = cancel each other out

positive = big effect
- huge amounts of returns and states where very low returns

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12
Q

intuition behind risky assets

A

assets that increase overall risk of the portfolio should compensate risk averse investors by giving higher expected returns than the portfolio expected return

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13
Q

what are the only 2 things that our utility depends on to make a decision

A
  1. risk
    - risk of portfolio depends on covariance of all of the stuff
    - good to have negative covariance
  2. return
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14
Q

what if the covariance of the asset and portfolio is greater than the variance of the portfolio
- LHS denominator > RHS denominator

A

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

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15
Q

what is the difference between binding and non binding constraints

A

binding = income is not sufficient to achieve maximum possible utility
I < pxX-
non binding = income is sufficient to achieve maximum possible utility
I >= pxX-

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16
Q

what is the interpretation of the langrian?

A

equal to the change in the utility of the consumer after a small change in income

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17
Q

non binding constraint

A

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
18
Q

what do investors care about with quadratic preferences?

A

only care about
mean
and
variance

19
Q

what is the 2 step procedure

A
  1. efficient portfolio frontier
  2. choose portfolio that maximises preferences
20
Q

how do you construct an efficient portfolio frontier of 2 risky assets

21
Q

what is an efficient portfolio frontier?

A

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

22
Q

what is the intuition behind risk of portfolio turned into an equaition that depends on variance of each risky asset, correlation coefficient

A

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
23
Q

what happens to the variance of the portfolio if the coefficient correlation = +1

p12 = +1

A

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
24
Q

what happens to the variance of the portfolio if the coefficient correlation = +1

p12 = -1

A

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
25
how does the tradeoff between risk and return vary across portfolios - as alpha changes
1. 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
26
what is the equation of the graph
expected rate of return of portfolio = alpha(intercept) + beta(slope)*risk of the portfolio
27
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
28
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
29
why would you never invest on the thin line
no point for the same level of risk you could get a higher expected return
30
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
31
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
32
in what circumstance can a risk free portfolio be achieved?
perfectly negative correlated
33
what is a risk free asset
asset with a constant payoff across all states of nature
34
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
35
what is a risk premium
the difference between expected return of risky asset and return on riskless asset
36
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
37
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
38
what does allocation on left on Z indicate?
portfolios with lending at the risk free rate = investment in risk free rate
39
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
40
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
41
can risk averse people be at point C
yes as long as they are compensated enough for their risk aversion
42
what to we take prices as
investor that takes asset prices in a competitive financial market as given