Learning Outcomes. After watching this video, you will be able to define the Capital Allocation Line, that is C-A-L. 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. By this point 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 b divided by sigma. Also recollect, that the expected return on the portfolio is given by E r sub b which equals wE(r) + 1 (1- w)r sub f. Rearranging, we get E(r sub p) = r sub f + w[E(r)- r sub f]. Substituting for w in this equation, we now have E(r sub p) = r sub f + [E(r)- r sub f] / sigma times sigma sub p. We know that minus rf is 0.22 minus .05, which equals 0.17. And that sigma is 0.3429. So now we have 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, C-A-L, 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- 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 C-A-L in this figure. First, we have the point where w is 0. 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 C-A-L beyond where W equals 100%? At these points, W is clearly greater than a 100%. What does it mean to have a weight greater than a 100%? 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 a 120%. But how was that possible? Remember, that weights must add up to 100%. So, if the weight in the risky asset is 120%, then the weight in the riskless asset must equal negative 20%. 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 the weight in the riskless asset is negative $20 over initial wealth of $100, which gives us a negative 20%. This is the exact weight in the riskless asset that we needed to make the weights add to 100%. Going back to our initial problem of trying to maximize utility, which of the feasible portfolios on the CL maximizes the utility for an investor with the risk aversion coefficient of three? With some math, we can show that the optimal allocation in the risky investment maybe returned as w* = E(r)- r sub f / A times sigma squared in the denominator. If you plug in the values we know, we have w star equals 0.22- 0.05 over 3 times 0.3429 squared, which works out to be 0.4819. As an investor, with the coefficient of risk aversion of 3, you will maximize your utility by putting 48.19% of your money in the risky asset. And the balance 51.81% 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 a utility of .05 is not optimal. You can do better by moving towards the top left. Remember, non cessation. But then, an indifference curve of 0.12 does not include any feasible portfolios. So the optimal investment, is a point where the C-A-L is a tangent to an indifference curve. In our example, that is the indifference curve with a utility of 0.09096. In the real world, we have 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.