Chapter 2 Calculating Expected Utility Theory

[Bill SD]

Hi,

The first question on pg 5 (Chap 2 course notes) asks to calculate the expected next-period utility when there's only a proportion, 'a' of wealth, w, invested in Equity A.

Realise its a basic point but dont understand how to calculate the answer. Why is it 0.25*(log(1-.04a)w) + 0.75*log(1+.08a)w rather than:
0.25*(log(.96a)w) + 0.75*log(1.08a)w?

Sim. unclear how to calculate the given answer for Q(iii) at the bottom of that page/top of next

 

[Busy_Bee4422]

Hi Bill

Firstly, congrats on beginning studies so early.

You invest your wealth with proportion a in equity and 1 - a non-interest bearing account.

At the end of the period, you have 1 - a in the non-interest bearing account and either 0.96a or 1.08a for the equity.

This gives total wealth as
either
1 -a + 0.96a = 1 - 0.04a
or of
1 - a +1.08a = 1 + 0.08a

Then you compute the expected utility.

 

[Bill SD]

thanks allot for your helpful answer.
I have two additional but separate questions on this Chapter:

1) Pg 31 of Acted notes (section 7.2 -Finding the maximum premium) quotes the Core Reading that: "the individual's expected utility is:
E[U(a-X)] = int(0,15): sqrt(15-x) dx/ 15"

Where does this formula come from, and presume specific to the uniform distribution?

2) The solution to the Chap 2 practice question 2.1 (on pg 41) states:
"Vp=x_A^2 + x_B^2+ 2*x_A*x_B*sigmaA*sigmaB*rhoAB"

Where does this formula come from?

 

[Steve Hales]

1) Pg 31 of Acted notes (section 7.2 -Finding the maximum premium) quotes the Core Reading that: "the individual's expected utility is:
E[U(a-X)] = int(0,15): sqrt(15-x) dx/ 15"

The "1/15" is the pdf of the uniform distribution U(0,15)

2) The solution to the Chap 2 practice question 2.1 (on pg 41) states:
"Vp=x_A^2 + x_B^2+ 2*x_A*x_B*sigmaA*sigmaB*rhoAB"

This comes from the standard result that the variance of the sum of two random variables is equal to the sum of their individual variances plus twice the covariance between them. There's more on this on pages 8 & 9 of Chapter 6