Slide 12.

Learning outcomes.

After watching this video, you will be able to use the quadratic

utility function to identify the better of two investments.

Calculate the expected return and

risk of a portfolio consisting of a risk and risk free asset.

>> Investment choices and portfolios.

In this video we first answer the question as to which of the two investments we

saw earlier.

Is there better investment choice?

And then talk about calculating expected returns and risk of portfolios.

Using the quandratic utility function,

let's try to answer the question as to which investment is better.

Remember the first investment has an expected return of 5% and

zero risk and the second has an expected return of 22% and the risk of 34.29%.

The answer as to which investment is better depends

on the coefficient of risk aversion.

Let's say that the coefficient of risk aversion a equals three.

Then the utility from the first investment is its expected

return 0.05 minus half times the coefficient of risk

of aversion 3 times its risk 0 squared which is equal to a utility of 0.05.

The utility from the second investment is it's expected return

of 0.22- 1/2 -the same coefficient of risk aversion 3 x it's risk,

which is 0.3429 squared which gives us the utility of 0.0436.

Clearly, the first investment has a higher utility and as investor,

with a coefficient of risk aversion of 3, would prefer the first investment.

What if the investor were risk-neutral?

Then A equals zero.

In which case, the second investment has the higher utility of 0.22,

versus a utility of .05 for the first.

So a risk neutral investor would prefer the second investment.

In reality, investment does not mean investing in only one or the other asset.

We can always form portfolios of assets by investing some of

our wealth in each asset.

Let's start off by looking at some portfolio math.

If you form a portfolio of two assets,

how will we determine its expected return and risk?

For now, let's assume that one investment is risky and the second one is risk-less.

The expected return of a portfolio with these two investments,

denoted as Sub-p, is w times Plus 1 minus w times r sub-f,

where Is the expected return of the risky investment,

r sub-f is the return of the riskless investment, and

w is the proportion of your money invested in the risky investment.

Consequently, 1 minus w is the proportion of your money invested in

the riskless asset.

Note the proportional weight must always add to 1.

The portfolio variance denoted as sigma sub b squared

is w squared times sigma squared.

Where sigma squared is the variance of the risky investments and return.

Consequently, the standard deviation of the portfolio

denoted as sigma sub b is w times sigma.

Recall our earlier investment choice, the first investment was riskless and

had a return of 5% which means r sub f is equal to 5%.

For the risky investment we had E(r) equals 22% and

sigma equals 34.29%.

But what is w?

Let's say that we fix w to be 60%.

That is, 60% of your wealth is put in the risky asset and

the balance 40% in the riskless.

The expected return of the 60/40 portfolio is 0.6

times 22% plus 0.4 times 5% which equals 15.2%.

The standard deviation of this portfolio is 0.6

times 32.9%, which equals 20.57%.

Now, is this 60/40 portfolio better than putting all of

your money in one of the two investments?

To answer that question,

we need to look at the utility you derive from the 60/40 portfolio.

Remember, earlier when we said the coefficient of risk aversion to 3,

the risky investment give a utility of 0.0436 and

the risk class one gave a utility of 0.05.

The utility from this 60 for the portfolio is its expect return

of 0.152 minus half times the coefficient of risk aversion

of 3 times its variance which is 0.2057 squared.

This gives us a utility of 0.885, clearly,

the 60/40 portfolio is far better than putting your

money in only one of the two investments.

But is this the highest utility we can achieve or

will another portfolio with different rates give us an even higher utility?

Next time, we will discuss, as to how to go about identifying

the one portfolio that maximizes your utility.