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FINANCE > Portfolio Selection > Flashcards

Portfolio Selection Flashcards

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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

A
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
Q

how does the tradeoff between risk and return vary across portfolios
- as alpha changes

A
  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
Q

what is the equation of the graph

A

expected rate of return of portfolio = alpha(intercept) + beta(slope)*risk of the portfolio

27
Q

why does the graph always have to have a positive slope?

A

you wouldnt invest in both assets if the slope was negative
- would be one pointless asset with lower return and higher risk

28
Q

analyse the graph
shape

A

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
Q

why would you never invest on the thin line

A

no point
for the same level of risk you could get a higher expected return

30
Q

what does the efficient portfolio frontier look like with 2 risky assets p12= extremes

A

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
Q

why is it good to have not perfectly correlated diversification

A

level of risk that is below the weighted average risk of the 2 assets
because they cancel out

32
Q

in what circumstance can a risk free portfolio be achieved?

A

perfectly negative correlated

33
Q

what is a risk free asset

A

asset with a constant payoff across all states of nature

34
Q

what is the expected return of the portfolio with 1 risky and 1 riskless asset

A

weighted average of the return of the risk free asset and the expected return of the risky asset

35
Q

what is a risk premium

A

the difference between expected return of risky asset and return on riskless asset

36
Q

what shape is the efficient portfolio frontier of risky and riskless asset

A

linear
relationship between return of the portfolio and risk is a straight line

37
Q

describe the optimal portfolio selection of risky and riskless graph combines

A

you choose amount of risky and riskless to invest in according to indifference cruve - you like northwest

  • everyone has different risks - different preferences
38
Q

what does allocation on left on Z indicate?

A

portfolios with lending at the risk free rate = investment in risk free rate

39
Q

what does allocation to the right of Z correspond to

A

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
Q

what does imperfect capital markets imply for borrowing and lending rate

A

borrowing rate is higher than lending rate

so after Z slope will be shallower

41
Q

can risk averse people be at point C

A

yes as long as they are compensated enough for their risk aversion

42
Q

what to we take prices as

A

investor that takes asset prices in a competitive financial market as given

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