Chapter 11 - Optimal portfolio choice and CAPM Flashcards
How can we define a portfolio/describe a portfolio
we do so by using weights. The sum of all the weights equals 1, and eahc wiehgt holds teh relative value of the total portfolio that is invested in a specific ivnesment.
How do we find the return of a portfolio
We use the portfolio weights, and take the sum-product of weight and return, which yield the total overall return of the portfolio.
How do we find the expected return of a portfolio?
we can find the expected return of a portfolio by taking expected value of it and using the rules from statistics.
E[Rp] = E[sum(xiRi)] = sum(xiE(Ri))
in general terms, what is the expected return of a portfolio?
The expected return of a portfolio is equal to the weighted average of the expected return of the individual stocks. This is the basic result from the equaiton ealrier, and the weights are of course the same as the portfolio weights.
Elaborate on the “diversification rule”
my own words (there is no real rule): By creating a portfolio of individual stocks that indivudally have the same expected return and volatility, will keep the expected return the same but will decrease the volatility. This creates a scenario where we basically get reduced risk/can more easily predict the returns.
How do we find the risk of a portfolio, in general terms?
We know that the risk of the portfolio will depend on how much the individual stocks swing together with each other. In other words, the correlation between them.
We are interested in the correlation. To find the correlation, we first need the covariance.
Define covariance
covariance is the expected value of the product of the deviations of two returns from their means.
in other words: covariance of two “values” is equal to the expected value of the product of deviations that these two values represent relative to their expected values.
So, if we have two stock returns, we need their returns and their mean/expected return. Then we take the difference between the returns and expected return, and multiply these two differences together. Then we take the expected value of this.
This was the probability distribution definition of it. In real life, we have to estiamte the expected value using an estimator, typically the mean. Therefore, the definition changes slightly in the way that we know basically compute average as expected value.
Cov(Ri, Rj) = 1/(t-1) ∑(Ri,t - Ri^avg)(Rj,t - Rj^avg)
Crucial point: the sum is over time periods.
Define correlation
Mathematically it is defined as the covariance divided by the product of the individual standard deviations.
Correlation is defined on the interval between -1 and 1
Generally speaking, how can we use inuition to say anything about two stocks correlation?
Same industry means more exposure to common risks, which will likely result in correlated stock mvoement.
how does the correlation between two stocks affect the volatility of the portfolio?
Recall the volatility of a 2-stock portfolio:
var(R1, R2) = x1^2var(R1) + x2^2var(R2) + 2x1x2 Cov(R1, R2)
easily interpreted that if the covariance is negative, the variance of the portfolio is smaller as well. Therefore, we want stocks that move in opposite directions (if diversification is the goal).
Find a nice expression for the variance of a portfolio
var(Rp) = cov(Rp, Rp)
= cov(∑xi Ri, Rp)
= ∑xi Cov(Ri, Rp)
where Ri is the return of stock i, while Rp is the return of the portfolio.
Therefore, we see that the variance of the portfolio (and therefore the volaitlity as well) can be written as/interpreted as the weighted sum of covariances of each stock relative to the overall portfolio.
We can do the same with Rp but use Rj:
= ∑∑xixj Cov(Ri, Rj)
This basically tells us that the variance of a portfolio is determined by the weighted sum of covariances in the portfolio.
How can we find the variance of a portfolio with equal weights?
What kind of information does this result tell us?
The book dont bother with the derivation, here is the result:
Var(Rp) = 1/n (Average variance of the individual stocks) + (1-1/n)(Average covariance between the stocks)
The key insight lies in what happens as n grows large. Basically, the larger n is, the less focus it is on the individual stocks, and more focus on the average covariance. This is the key result, and it generalize to cases with unequal weights as well: More stocks in the portfolio generally leads to more focus on covariance as a determinant of portfolio variance.
Say we have a portfolio with equally weighted stocks, and we let n grow towards infinity. What happens?
We eliminate all the risk that could be diverisified. We are left with only market risk.
how can we find the general result of volatility in a portfolio with unequal weights?
What can we use this result for
Start with variance of the portfolio:
var(Rp) = ∑xi Cov(Ri, Rp)
Then we covnert cov to corr
var(Rp) = ∑xi Corr(Ri, Rp)(SD(Ri)SD(Rp))
Now we make use of the fact that var(Rp) = SD(Rp)SD(Rp)
SD(Rp) = ∑xi SD(Ri) Corr(Ri, Rp)
This result is nice because it highlights the dynamics of the volatility of the portfolio.
The overall volatility is a function of stock weight, volatility of the individual stocks, and how correlated the stock is to the rest of the portfolio.
How does the volatility of a portfolio compare to the weighted average volatility of the stocks within it?
Volatility of the portfolio will be less than the weighted average. This is obviously due to the correlation term that basically force the result to be equal to, or lower.
Recall the goal of this chapter?
How an investor can create an efficient portfolio
Recall the formula of varaince of a two stock portfolio
var(Rp) = x1^2 var(R1) + x2^2 Var(R2) + 2x1x2 cov(R1, R2)
Define the optimal portfolio
optimal is subjective to some degree. however, we define optimal portfolio as the portfolio that has the highest expected return for a given level of volatility.
This is an objective principle. Regardless of how risk seeking or risk averse an investor is, he will naturally want to achieve the highest returns for the level of risk he accepts.
Consider the case of a two stock portfolio. What is the core point that the book try to teach here?
We can adjust the portfolio weights and achieve different levels of expected return and volatility. However, due to diversification, some portfolios will actually have better expected return AND volatility than other portfolios.
The point is that due to how correlation works, statistically there is an advantage in having uncorrelated investments together. This creates the diversification. And the outcome of this is that we need to figure out: Which portfolios are superior in both dimensions to others? Or perhaps more importantly, which portfolios are simply inferior, and never even needs to be considered?
I suppose there will be portfolios that will not be better on both dimensions, but can for instance offer a greater expected return at the price of greater volatility. This is still an efficient portfolio, as it is not inferior.
Define an inefficient portfolio
A portfolio is inefficient if there exist weights that make the portfolio better in terms of both volatility and expected return (better on all dimensions).
Elaborate on the effect of correlation in regards to efficient portfolios
First of all, correlation has no effect on expected return. This means that only volatility of the portfolio is affected by correlation.
We can consider the edge cases (correlation -1 and 1) for a two stock portfolio.
if correlation is -1, it means that the stocks move in literal opposite directions of each other. Therefore, there exist a selection of portfolio weights that make the volatility of the portfolio be equal to 0. Since the volatility of the individual stocks are differnet (likely) as well as the expected returns, these weights are not immediately given, but can definitely be found.
If correlation is 1, the volatility vs expected reutrn line is linear.
I think the core idea here is that when two stocks move opposite of each other, will always have less risk involved than those who dont.
What is going on here?
Rperesemnt how a short sale affects return and volatility. The key here is that whether the short sale is an efficient or inefficient portfolio depends on which of the two stocks we short. In general, shorting the stock with the lower expected return will allow you to invest more in the hihg-return stock, which will boost your return per buck. Therefore, shorting low-expected return stock is efficient, while shorting the hihg-return stock is majorly inefficient.
Is portfolio P optimal?
Nope. As we can see, there are portfolios that offer a higher expected return for the same volatility.
if we have a portfolio of a risky and risk free component, what happens if we take osme risk free asset, either by usign whatever we have in it, or by borrowing, and ivnesting the proceeds in investment i?
We give up the risk free rate, but we gain the expected return from investment i. this means that the net result is that we gain the excess return from investment i.
However, we also gain the additional volatility. However, we will only gain the common risk volaitlity, because the independent risk volatility will be diversified away.
The incremental risk is measured as SD(Ri)x Corr(Ri, Rp)
Then the question is: Is the gain we get (incremental return) good enough for us to trade the incremental risk for it?
We’ll take a look at the bang for the buck of this new investment i:
We have a portfolio P with sharpe ratio (E[Rp] - rf) / SD(Rp)
The new investment offer return E[Ri]-rf.
The new investment has incremental volatility SD(Ri)xCorr(Ri, Rp).
This is actually a simple case of “metric per unit” multiplied by “units” to get total metrics.
Sharpe ratio gives return per volatility level. We need to multiply this by the incremental volatility of investment i to get the additional return WE WOULD GET FROM INCREASING RISK VIA CURRENT PORTFOLIO.
SD(Ri)xCorr(Ri, Rp) x SharpeRatio gives us how much expected return we would get from our current portfolio if we had invested in such a way that we level out the volatility at the same level as the new investment would. This is a return that is used for reference.
Then we use the excess return from investment ‘i’ to see if it is better or worse than the expected return of the portfolio. So, we get a comparison situation:
E[Ri] - rf VS SD(Ri)xCorr(Ri,Rp)x(E[Rp]-rf)/SD(Rp)
IF the LHS is greatest, it would mean that the benefit of adding investment i is better than the additional benefit of the current portfolio.
Notice how SD(Ri) x Corr(Ri, Rp) / SD(Rp) is actually a measure of the beta.
The relationship between an investments required and expected return give a condition for how much of the ivnestment we want in our portfolio. How does this generalize?
It generalize by saying that our portfolio is optimal when eahcsecurity has required return equal to the expected return.
In fact, it generalize to all investments. Our portfolio is efficient, optimal, if and only if the expected return of every available investment is equal to its required return.
And as we know, required return will change as a function of beta.
Why does all the securities lie on the line, but not on the CML?
There is no real correlation between a security’s expected return and volatility. The only thing that matters is the volatility that is due to market sensitivity.
