Learning outcomes. After watching this video,

you will be able to define the Capital Allocation Line, that is,

CAL, describe what it means to have asset weight greater than 100% or less than 0%,

calculate the Sharpe Ratio.

Capital allocation. In this video,

we will see how to allocate your money across two investments to maximize your utility.

Last time, we saw that a 60:40 portfolio of

the risky and riskless assets yielded a utility of 0.0885.

But is 0.0885 the highest utility you can achieve? Not necessarily.

To identify the portfolio with the highest utility,

we will first generate all feasible combinations of the two assets.

From among these feasible combinations,

we will select the one with the highest utility.

Going back to the portfolio math for just a bit,

remember that sigma_sub_p equals w times sigma.

We can solve for w as sigma_sub_p divided by sigma.

Also recollect that the expected return on the portfolio is given by

E(r_sub_p) which equals w times E(r) plus one minus w times r_sub_f.

Rearranging, we get E(r_sub_p) equals r_sub_f plus w times E(r) minus r_sub_f.

Substituting for W in this equation,

we now have E(r_sub_p) equals r_sub_f plus

E(r) minus r_sub_f over sigma times sigma_sub_p.

We know that E(r) minus r_f is 0.22 minus 0.05,

which equals 0.17, and that sigma is 0.3429.

So now we have E(r_sub_p) equals r_sub_f plus

0.17 over 0.3429 times sigma_sub_p.

This is a straight line that represents all feasible combinations of the two investments.

This line is called the capital allocation line, CAL,

as it represents the feasible ways

in which you can allocate your capital across the two investments.

The slope of the CAL is E(r) minus r_sub_f over sigma,

which is the risk-reward ratio.

It is the expected excess return per unit of risk.

It is also commonly known as the Sharpe ratio.

We can see the different feasible combinations

of the two investments along the CAL in this figure.

First, we have the point where w is zero,

that is all your capital is in the riskless asset.

Then as you move along the line,

you can see points with different weights in

the two investments until you reach the point where w equals 100.

This is where all your wealth is in

the risky asset and none of it is in the riskless asset.

What about the points on the CAL beyond where w equals 100 percent?

At these points, w is clearly greater than 100 percent.

What does it mean to have a weight greater than 100 percent?

If you have $100,

then you are investing more than $100 in the risky asset.

If we invest, say,

$120 in the risky asset,

then we have w equals 120 percent.

But how is that possible?

Remember that weights must add up to 100 percent.

So if the weight in the risky asset is 120 percent,

then the weight in the riskless asset must equal -20 percent.

The negative sign means that you borrow at the riskless rate of return.

But how much must you borrow?

You have $100 but you want to invest $120 in the risky asset,

which means that you must borrow the additional $20.

Denoting a negative sign for borrowing,

the weight in the riskless asset is -$20 over your initial wealth of $100,

which gives us a -20 percent.

This is the exact weight in the riskless asset that we

needed to make the weights add to 100 percent.

Going back to our initial problem of trying to maximize utility,

which of the feasible portfolios on the CAL maximizes

the utility for an investor with a risk aversion coefficient of 3?

With some math, we can show that the optimal allocation in the risky investment,

may be written as w^star,

equals E(r) minus r_sub_f over A time sigma-squared in the denominator.

If we plug in the values we know,

we have w^star equals 0.22 minus 0.05 over three times 0.3429^2,

which works out to be 0.4819.

As an investor with a coefficient of risk aversion of three,

you will maximize your utility by putting 48.19 percent of your money in

the risky asset and the balance 51.81 percent in the riskless asset.

You can calculate the utility from this portfolio to be

0.09096 using the quadratic utility function.

To understand this in terms of indifference curves, see the figure.

The picture shows that an indifference curve with the utility of 0.05 is not optimal.

You can do better by moving towards the top left.

Remember, non-satiation. But then,

an indifference curve of 0.12 does not include any feasible portfolios.

So the optimal investment is a point where the CAL is a tangent to an indifferent curve.

In our example, that is the indifference curve with a utility of 0.09096.

In the real world, we have a large number of risky assets in which we can invest.

Next time, we will see how to extend this analysis to more number of risky assets.