Expected Utility & Risk Measures

Question 1

Consumer Choice Theory

Answer 1

Preference of investor (ranking alternatives of PF based on theory of preferences) - formulat
budget constraint; decision process (utility maximisation - rational behaviour to choose PF which maximises own utility) –> utility function

Question 2

Theorem for choice under uncertainty

Answer 2

Expected Utility Theorem (choice about RV- decision based on max- expected value of utility under the belief of investor about P. of different outcomes)
According to this - U.F is only unique up to a stricly increasing affine transformation - will show same preference

Question 3

Portfolio Choice

Answer 3

Optimal Portfolio Choice - choose trading strategy (M,Z) which makes E(U(W)) achieve max. value for initial invested wealth w
(M,Z) optimal strategy if W* gives maximum wealth
Necessary & sufficient condition - 1st oder conditions ofr maximum M*
Necessary - no asset should dominate the other (no arbitrage if Y takes value u & d with positive P.)

Question 4

Risk Measurement

Answer 4

number which captures the risk

Question 5

Risk Measurement

Answer 5

guidance - math.concept for understanding the risk

Question 6

Standard Deviation

Answer 6

One assumes Returns are Normally Distributed
WE use SD to look at the minimal SD
SD is sufficient condition if investor has a mean-var preference, but not enough if Returns aren´t ND

Question 7

Value at Risk

Answer 7

Maximum Loss of an investor over a given period of time with a confidence level a
Probability that Loss of PF (- Return) exceeds VaR will happen with a probability 1-a (VIOLATION)

Question 8

Expected Shortfall

Answer 8

Alternative to VaR (VaR fail to provide any info. about how much loss will exceed VaR)
at CL of a is referred as the expected loss conditioned that VaR has been violated
more sensitive to shape of Loss distribution (R) in the tail of distribution

Question 9

Mean - Var- Analysis

Answer 9

Investor will choose between two asset with same expected return the one with the smallest variance
= Risk-averse assumption (min. risk measured by SD)
= Non-Satiation (everything remaining equal will always seek after more wealth)

Question 10

Mean- Var Analysis

Answer 10

Feasible Set/ Region
Mean Var Portfolio Frontier or Mean-Var Set
Efficient Frontier
Global Minimum Value and Portfolio (weights)

Question 11

Minimum Var = 0

Answer 11

create return Ro by investing in the money market account which is a non-risky investment with 0 VaR

Question 12

One-fund Theorem

Answer 12

there exists a single PF (fund) M of risky assets such that any efficient PF of the mean-var problem can be constructed as a combination of PF M and m.m.a (M;Z)

Question 13

Capital Market Line

Answer 13

line determine by Ro and M (market PF) where all efficient assets must lie

Question 14

Market Price of Risk

Answer 14

Sharp Ratio
Slope of the CML which indicated by how much the expected return of PF increases if SD of that unit increases by one unit

Question 15

CAPM

Answer 15

this holds for any trading strategy and related Return of any risky asset with the efficient PF frontier (CML)
links Risk Premium of any PF with its systematic Risk (whereas E.F links R.P of efficient PF with it´s individual risk)
changes the concept of risk of an asset (SD –< beta of that asset)

Question 16

Expected Excess Return

Answer 16

EER of asset is proportional to EER of Market PF M with B

EER is proportional to Cov (Ri; Rm)

Question 17

CAPM  - corr 1 
Beta is (SD Ri / SD Rm)

Answer 17

PF which is highly risky won´t give high return if no perfectly correlated with Market (corr = 1)
Asset isn´t worth investing (indv. risk of that asset is not worth it as the risk premium is not as high as it can be for an eff. P for which corr = 1)
If Corr= 1 - instead of caring indv. risk we can talk about a PF where it´s idv. risk has been diversified away and hence risk premium is as high as it can get

Question 18

CAPM - B = 0

Answer 18

If B= 0 - this means asset is uncorrelated to PF M and asset only carried individual risk and the ER will be the Return given by the m.m.a but this is inefficient as all it´s idv. risk could be diversified away
the market doesn´t reward an investor investing in such asset by not giving any expected excess return as in this case the Return is equal but not higher as the one invested in the m.m.a

Question 19

Concavity U(W)

Answer 19

The Utility of the expected wealth will always be greater than the the expected utility of a situation giving this wealth
U(E(W)) > E (U(W))
U( p1 * w1 + p2 * w2) > p1 * U(w1) + p2 * U(w2)

Question 20

Risk Premium

Answer 20

EER
EF - ERR of M * individual risk
CAPM - EER pf M* systematic risk
both cases - normalised (divided by sigma (M))

Question 21

Risks parameters

Answer 21

sigma(i) - individual risk
corr * sigma (i) - systematic risk
Beta - systematic risk / sigma(m)