Chapter 11 - Optimal portfolio choice and the Capital Asset Pricing Model

Question 1

Define an optimal portfolio

Answer 1

An optimal portfolio is a portyfolio that retuns the highest expected return given a level of volatility that the investor is “comfortable” with.

Question 2

How do we find an optimal portfolio? (first step)

Answer 2

we first need a method to define a portfolio and analyze its returns.

in order to do this, we use the concept of portfolio weights.

Question 3

elaborate on how we can define a portfolio and analyze its returns

Answer 3

We need the concept of portfolio weights.

xi = Value Of investment i / total portfolio investment value

the sum of xi for all i will therefore add up to 1.

Recall that this is entirely value based. Total value in the denominator, value of single investment at the enumerator. Shares and share price must be accounted for in a way that compute total value of an investment.

The return of the portfolio is the sum of products, where each product is investment portfolio weight multiplied by the specific return of that investment. This is summed together for all the investments in the portfolio.

Since each part is a contribution to percentage, the sum is the total percentage that indicate the return of the portfolio.

Question 4

Elaborate on the portfolio weights

Answer 4

It is important to udnerstand how they vary with time.

firstly, since the total investment value will most likely appreciate or depreciate in value, the denominator (which is the total investment value) will also change. This total investment value should always be equal to the current market value of the investments.

Secondly, since the individual stocks/investments will change over time as well, the enumerators of the weight calculations will also change.

Therefore, the portfolio weights represent dynamic measures.

Question 5

Can we use the portfolio return equaiton to compute expected return?

Answer 5

Yes, we just need to get the formula on the shape of using expected value.

ExpectedReturn = ∑xi E[Ri]

The math is actually:

ExRet = E[Rp] = E[∑xi Ri] = ∑E[xiRi] = ∑xi E[Ri]

The result is extremely intuitive. Expected return of individual stock, multiply it by portfolio weight, sum all investments, get expected portfolio return.

Question 6

Elaborate on variance and its properties

Answer 6

Variance is defined as the expected value of the squared difference between a value and the mean.

In other words: Variance is the expected value of the squared difference between a value X and the expected value of X:

Var(X) = E((X - E(X))^2)

Now we use the rules of expected value etc to get:

= E((X - E(X))(X - E(X)))
= E(X^2 - 2XE(X) + E(X)^2)

= E(X^2) - E(2XE(X)) + E(E(X)^2)

= E(X^2) - 2(E(XE(X))) + E(X)^2

= E(X^2) - 2E(X)E(X) + E(X)^2

= E(X^2) - 2E(X)^2 + E(X)^2

= E(X^2) - E(X)^2

So, we now have the variance result. we can derive all the properties of variance, like Var(aX) = a^2Var(X) from this formula.

Question 7

elaborate on covariance and its rules

Answer 7

Recall that covariance is defined as the expected value of the product between the differences between the means of 2 random variables.

Cov(X,Y) = E( (X - E(X))(Y - E(Y)) )

We use the rules of expected value etc:

= E(XY - XE(Y) - YE(X) + E(X)E(Y))

= E(XY) - E(XE(Y)) - E(YE(X)) + E(E(X)E(Y))

= E(XY) - E(X)E(Y) - E(Y)E(X) + E(X)E(Y)

= E(XY) - E(X)E(Y)

This is the ultimate covariance result. We can use this formula to figure what happens if we have cases like Cov(aX, Y) etc

Question 8

How much risk will remain in a portfolio when we diversify it?

Answer 8

it depends on the level of common risks. Diversifying can in theory eliminate all independent risk, but the common risk (typically market wide risk) remain. It will be up to the individual investment (beta) to determine how much they are impacted by market wide common risks.

Question 9

What does this image show?

Answer 9

It clearly shows how we can reduce volatility (risk) through diverisfication. the portfolios are both less risky than the individual stocks.

This simply happens because of the patterns of returns in the individual stocks. When one stock does poorly, others may do good. The result is something that does not have that much spread.
Look at how the average return is the same for the stocks and the portfolios, but there is just simply less risk involved in the portfolios.

THe image also shows that risk is further reduced if the stocks included in portfolio tend to move in opposite directions (prices move opposite directions instead of together) and the level of common risk is low. This is why the portfolio of OIL + AIR is less risky than AIR + AIR2.

Just as important is the fact that the return is not affected at all. This only means that good diversification reduce risk.

Question 10

How can we know if two stocks tend to move together?

Answer 10

We need the statistics package.

Covariance & Correlation.

Question 11

Elaborate on covariance

Answer 11

Covariance is the expected product of the deviations of two returns from their means:

Cov(Ri, Rj) = E[(Ri - E[Ri])(Rj - E[Rj])]

so if we are interested in covariance over 10 year period (step length 1 year) we have T = 10, and we would sum over 10 different time periods, where Ri,t is return for investment I in year t, while R^(bar)_i is the annual average return etc

Positive covariance indicates that the two stocks are moving together.

Negative covariance indicate that the two stocks move in opposite directions.

Question 12

How can we interpret covariance?

Answer 12

The sign is all we can use. Magnitude has no meaning.

We need to use the correlation to compare results of magnitude.

Question 13

Why is covariance not suitable in terms of maginude?

Answer 13

Covariance will be large either if both stocks are very volatile, or if both are very little volatile

Question 14

What is formula for correlation?

Answer 14

Corr(Ri, Rj) = cov(Ri, Rj) / SD(Rj)SD(Ri)

Question 15

Upper and lower range for correlation?

Answer 15

-1 to +1

Question 16

how do we interpret correlation?

Answer 16

Question 17

how is correlation connected to stock returns and risk?

Answer 17

Correlation closer to 1 represent that there are much common risk.
Close to 0 indicates no correlation, thus no common risk.
Close to -1 indicate that if one stock has common risk, the other will gain from the others downside.

Question 18

What can we say about negative correlation?

Answer 18

Represent a hedging possibility between the picks.

Question 19

What is the covariance of a stock when compared to itself?

Answer 19

Covariance would be equal to variance in such a case.

Question 20

What is the correlation of a stock when compared to itself?

Answer 20

1. Perfectly moving together obviously.

Question 21

When will stock returns be highly correlated with each other?

Answer 21

Stock returns will be highly correlated with each other when they are affected by the same economic events. for instance, stocks within the same industry will usually have strong correlation compared to stocks that are in different industries.

Question 22

In general, will we easily find stocks with negative correlation?

Answer 22

No, it is not that common. We usually find stocks correlated down to close to 0, but it is difficult to be negatively correlated because most firms benefit from strong economic events etc.

Question 23

Elaborate on the variance of a two-stock portfolio

Answer 23

We need the covariance rules from statistics.

The result is shown in the image. We start with var(Rp) = cov(Rp, Rp) and manipulate until we get teh result as shown on the image.

Question 24

How does correlation between stock affect the volatility of the portfolio?

Answer 24

Higher correlation leads to higher volatility of the portfolio.

Question 25

Consider the variance of a portfolio. Can we relate it to covariance somehow?

Answer 25

Yes. we can make it by using the statistics rules for covariance etc.

We end up getting a result that beasically says that the variance of a portfolio is equal to the weighted average of the covariances of each stock with the portfolio itself.

This is extremely important result because it shows that more covariance (or more correlation) means more variance which means more risk.

Question 26

What is the ultimate formula result of multi-stock protfolio variance?

Answer 26

This image shows that the multi-level case is a direct generalization of the two-stock case (should therefore be easy enough to remember it) and it essentially boils down to the fact that the total volatility (variance) in a portfolio depends on the total co-movement of the stocks that partake in the portfolio.

Question 27

What is an equally weighted portfolio?

Answer 27

A portfolio where each stock has the same weight to it.

If the portfolio has n stocks, each stock will have the weight 1/n.

Question 28

what is the variance of an equally weighted portfolio?

What is the takeaway from this?

Answer 28

Key takeaway: as the number of stocks, n, grows large, the variance of the entire portfolio will be more determined by the covariance between the stocks.

What this result basically boils down to, is that if we let n grow towards LARGE, we will tend towards an asymptote which represent the level of NO individual risk, only common risk. This is the level that represent market risk.

The portfolio converge towards the average covariance. Average covariance tells us something about how the stocks move together, and is a represnetative of common risk.

Question 29

Formula for volatility of portfolio with arbitrary weights…?

Answer 29

Question 30

How does the volatility of a portfolio compare against the weighted average volatility of the stocks in it?

Answer 30

The volatility of the portfolio will generally be less than the weighted average volatilities. This is because of the correlation term that will always be smaller than 1 unless the picks move perfectly together.

Question 31

Consider 2 stocks with volatility 25% and 34%. Is it possible to make them lower?

Answer 31

Yes, by diversification, we can lower the volatility of the portfolio that holds both with weights.

Question 32

Define an inefficient portfolio

Answer 32

We say that a portfolio is inefficeint when it is possible to find a portfolio that gives better expected returns and lower volatility. Such a choice would make the “inefficient portfolio” completely useless.

Question 33

Define an efficient portfolio

Answer 33

An efficient portfolio is one that has no portfolio that is better in terms of both volatility AND expected returns. We can probably improve one of them, but that would be at the cost of making the other metric worse.

Question 34

When identifying efficient portfolios, how do we identify the best one?

Answer 34

This is not possible, because “best” is subjective in terms of risk profile. Some will choose portfolio with greatest potential return, other will choose conservative.

Common for all though, is that no one will ever pick an inefficient portfolio whatsoever.

Question 35

How will correlation affect the expected returns of the portfolio?

Answer 35

No effect. Correlation only affects the volatility of the portfolio.

Question 36

Elaborate on the correlation between two stocks if we are to have a case with a portfolio that has no risk at all

Answer 36

The correlation must be perfectly -1. If this is the case, there will exist a set of weights such that the portfolio bears no risk at all.

Question 37

Elaborate on portfolios in regards to including short selling

Answer 37

We still use weights, be short sales require a negative weight. We view the short sale as a negative investment which decrease the total value of the portfolio, but one that might appreciate in value.

the weights still add up to 1, it is just that now some weights are negative and some might be larger than 1.

It is important to understand that when making short sales, the volatility of the portfolio can actually be larger than that of the individual stocks. This is some sort of reverse diversification.

Question 38

How can short sale be leveraged even if you think the stock you shorted will go up?

Answer 38

Say we have a budget of 20 000 usd. If we short stock X worth 10 000 usd, we receive this amount and can use it. This means that we can now buy 30 000 usd worth of stock Y.

We could then end up with a case like this:

weight 1 = 30 000 / 20 000 = 1.5

weight 2 = -10 000 / 20 000 = -0.5

The sum of weights equal to 1.

However, we will quickly see that the addition of short selling greatly increase the risk of the portfolil.

Question 39

What is the efficient frontier?

Answer 39

efficient frontier is the line representing all efficient portfolios.

The efficient frontier will improve when we add more stocks to the portfolio. Improve means it expands and offer same expeted returns at a lower volatility etc

Question 40

Besides from diversificaiton, how can we reduce our risk level?

Answer 40

Keeping some of the money in risk-free treasury bills etc

Question 41

Volatility of risk-free investment like treasury bills?

Answer 41

0

Question 42

Say we allow to invest in risk free bills as well as risky stocks. Elaborate on finding a meaningful representation of the expected returns

Answer 42

We consider investing with a weight “x” in risky stocks and (1-x) in riskfree bills.

ExpectedReturnTotalPortfolio = E[(1-x)rf + xRp]
Using rules of stats:

E[(1-x)rf] + E[xRp]

= (1-x)E[rf] + xE[Rp]

= (1-x)rf + xE[Rp]

= rf - xrf + xE[Rp]

= rf + x(E[Rp] - rf)

This is the end result saying:
Expected return of the portfolio holding both risk free and risky investment is equal to the risk-free rate plus a fraction x multiplied by the risk premium required from holding the risky portfolio. It is the risk premium because we have the expected returns of the risky portfolio LESS the risk free rate. This difference is known as the risk premium.

Question 43

Elaborate on the volatility of a portfolio that allows risky and risk-free invesments

Answer 43

Key thing here is that volatility and variance of the risk-free investment is 0 because it holds zero risk. Same applies to covariance terms that include the risk-free option. Therefore, the variance expression of the portfolio is significantly simplified:

var(Rxp) = (1-x)^2 Var(rf) + x^2Var(Rp) + 2(1-x)xCov(Rp, Rj)

var(Rxp) = x^2 Var(Rp)

SD(Rxp) = sqrt(x^2Var(Rp))

SD(Rxp) = x sqrt(Var(Rp))

SD(Rxp) = x SD(Rp)

This result is extremely important. Notice how x, since being a fraction, will reduce thevolatility of hte portfolio. This makes sense, as including risk-free investments with 0 volatility will drag the overall volatility (risk) of the portfolio down.

Question 44

What happens if we consider the case of investing in a portfolio that is risky and has a risk-free componenent, and the weight we use for the risky-part is greater than 100%?

Answer 44

We are essentially short selling the risk-free option (because we invest proportion x into risky part) which allows us to build a linear relationship between volatility (risk) and returns that extends the line we had from x=0 to x=100%.

In other words: we are borrowing money at the risk free interest rate. Borrowing money to invest in stocks is referred to as buying stocks on margin, or using leverage.

Question 45

name example of a levered portfolio

Answer 45

A portfolio that borrow money at the risk free interest rate to invest in risky stocks is a levered portfolio.

Question 46

No question, just read

Question 47

Elaborate on the sharpe ratio

Answer 47

Sharpe ratio is the slope of the line going through combinations of portoflios that hold riskfree and risky investments.

The idea is that the line with the greatest slope will offer better expected returns compared to volatility.

The sharpe ratio goes like this:

SR = Portfolio excess returns / Portfolio volatility

SR = (E[Rp] - Rf)/ SD(Rp)

Question 48

Elaborate deeply on the sharpe ratio. Quite long card

Answer 48

The Sharpe ratio is something we can use to establish the optimal way of selecting a portfolio that contains a risk-free part and a risky part. The risk-free can be considered for instance us treasury bills, while the risky part is consiereded as a portfolio of stocks.
If the portfolio of stocks has 2 stocks in it, it is easy to graphically illustrate the curve we get and identify the section of efficient portfolios. If we do not include risk-free part, we would lie somewhere along the curve (specifically the section of efficient portfolios called efficient frontier). However, the problem is that it is difficult to select a spefici portfolio, as it will vary widely depending on subjective risk profiles.

However, when introducing the risk-free investment, we can get a better understanding of what portfolio to select.
we use the Sharpe ratio as: (E[Rp] - riskfreeRate)/SD(Rp) as a measure of the risk premium vs volatility reward. If we can identify the point along the efficient frontier that has the highest Sharpe ratio, and draw a line going from the point of “only risk-free investment” to the point on the frontier with this ratio, and further extending the line beyond this point, we have a line that represent portfolios of various combinations of the risk-free investment vs the risky portfolio. Specifically, we “lock” the risky portfolio to be exactly equal to the portfolio in the “Sharpe ratio point” on the efficient frontier (in terms of weights of the individual stocks) and then only adjust “how much” of the risky portfolio we want, and how much risk-free we want.
why?
Because it gives us a line that relates the each level of volatility with its best possible expected return. if we want more risk averse approach, we simply decrease the weight “x” of risky portfolio, and instead increase the holding of the risk-free investment, and vice versa. The big point here is that the Sharpe ratio with the greatest value represent the risky portfolio with the best expected return vs volatility reward, so we want to use that portfolio as a basis, and manage volatility by using risk-free investment.

Question 49

Can we go beyond x=100%?

Answer 49

If we select x=100%, we are at the tangent point on the efficient frontier. This represent the point with the greatest expected return we can get from using our own cash. If we want a higher expected return, this is certainly doable, but we would have borrow money at the risk-free interest rate and accept higher volatility. Note however, that the Sharpe ratio would still remain the same, meaning that the expected benefit vs risk would remain the same, but with greater numbers.

Question 50

Define the efficient portfolio

Answer 50

The efficient portfolio is the one that offers the greatest sharpe ratio, which is the portfolio with the best tradeoff between expected returns and risk. This is the tangent portfolio.

By combining the efficient portfolio with the risk-free investment oppurtunity, an investor will earn the highest possible expewcted returns with the risk level he is comfortable with.

Question 51

IMPORTANT: Derive the expression for the risk of a portfolio that has arbitrary weights and have as many stocks in it as we want.

Then explain the import results/takeaways from this result

Answer 51

We start by finding expression for the variance, although it is the volatility we are ultimately after.

Var(Rp) = Cov(Rp, Rp) = Cov(∑xiRi, Rp)

Since it is covariance, we know that any constant that is not random variable can be moved directly out:

= ∑xi COV(Ri, Rp)

This is a very interesting and important result. It says that the variance of a portfolio is equal to the weighted average covariance of each stock, and the portfolio. Meaning, the variance is large if stocks tend to follow with the overall portfolio. If all stocks represent stocks from the same industry, this would be a case where we can expect high variance because the covariance between each individual stock, and the portfolio as a whole, is large for all stocks.

In order to get a nice result, we take a short detour:

= ∑xi COV(Ri, ∑xj Rj)

= ∑∑xixj COV(Ri, Rj)

This result is basically the same as earlier, but we have changed the “overall portfolio” with the sum of all the other stocks. It gives exactly the same result, but it now visualize the concept from another perspective.

We can move further:

Var(Rp) = ∑xi Correlation(Ri, Rp) SD(Ri) SD(Rp)

We can get the volatility by dividing on the standard deviation of Rp:

Volatility(Rp) = ∑xi Correlation(Ri, Rp) SD(Ri)

This result is crucial. It consists of 3 parts, and those represent the following:
1) xi and the sum represent weighted averages.
2) Correlation represent how much a stock “swing” along with the portfolio as a whole.
3) SD(Ri) says something about how that specific stock’s volatility look like.

Together, these 3 elements create a weighted average volatility multiplied by correlation. Correlation can be seen as a measure of common risk. This is basically where “the magic happens”. If a stock is completely following the portfolio, it means that it responds exactly the same to market wide news and common risks are at a maximum. If all stocks were like this, the volatility of the portfolio would be equal to the weighted average, which would represent a case where no individual risk has been diversified away.
However, this is unlikely. Usually we will have correlation much lower than 1, which will contribute by making the volaitlity of the portfolio being less than the weighted average volatility. This is a crucial point because it shows WHY we reduce risk through large portfolios. It is not the number of stocks that does it, but rather the fact that UNCORRELATED STOCKS WILL DIVERSIFY AWAY IDIOSYNCRATIC RISK.

Question 52

consider a portfolio with equal weighting. Elaborate on the variance of this portfolio, and how it will be affected by the number of stocks

Answer 52

We can get a formula that tells us an important relationship that is also very intuitive.

If we weigh each stock equally, the more stocks we have, the less their individual variance will affect the portfolio variance, and the more the covariances (average covar) between them will determine the overall portfolio variance.

In such a case, if we let number of stocks grow LARGE, the portfolio variance will approach the limit of the average covariance of the portfolio.

So, even with infinite amount of stocks, we still have risk, as the variance is equal to average covariance. This risk is the market risk. Represent all the common risk in the portfolio.

Question 53

name the effect that correlations has on expected return

Answer 53

NO effect

Question 54

Elaborate on the volatility vs expected returns curves for various level of correlation between two stocks

Answer 54

If we have two perfectly correlated stocks, it means that they move together. Therefore, the variance of the portfolio will be determined by the weighted average between them. This will create a straight line between them, representing the fact that if we inlcude more of the more volatile stock, the overall portfolio will also be more volatile.

If we reduce correlation from 1 and downward, we will see that some combinations of portfolios offer synergy effects because they will work together and diversify away some risk due to a degree of opposite movement. This creates a “bending” curve.

If we have perfectly opposite correlated (-1) stocks, there will be a single portfolio that offers a certain expected return for absolutely zero risk. This happens at the point where the weighted average volatility is 0. This is possible because of the negative correlation.

Question 55

elaborate on why a combination of risk-free investment and a portfolio investment gives a straight line graphically

Answer 55

It is because the portfolio that is risky, is fixed in terms of internal weights. The only weights that are open for change, are the weights that determine how much we invest in risk-free part and how much we invest in risky part.

if we choose to increase x (the proportion we invest in risky portfolio), and thereby reduce (1-x), we will include more volatility from the risky part. This volatility is fixed, so we only choose how much of the total portfolio that should include the fixed volatility. Since the risk-free option has no volatility, the overall volatility is given by a proportion of risky portfolio, multiplied by the volaitlity of the risky portfolio.

Therefore, the restul is a straight line.

Question 56

When we extend “x” beyond 100%, we are short selling the risk free investment. BUt what are we actually doing here?

Answer 56

We have to pay the expected return that would follow from the risk free investment. If x=1.25, we would have to pay for the 25% portfolio size that corresponds to the negative amount of risk-free investment we have. This is the same as borrowing money, and paying the risk free interest rate.

Question 57

A portfolio that consists of a short position in the risk-free interest rate is known as …

Answer 57

a levered portfolio

Question 58

elaborate on figuring out whether we can increase the Sharpe ratio by adding more of a certain investment

Answer 58

Say we consider investment in security “i”. Since the current risky portfolio is chosen to be the greatest Sharpe ratio (tangent portfolio), we will not immeditly change it. Instead, we consider giving up some of our risk-free assets. Could be giving them up, could be shorting them (I believe).

Investing in investment i carry expected return E(Ri). Since we give up the risk free returns, we would increase our expected returns by:

E[Ri] - rf

At the same time, we are adding risk. The independent risk is diversified away, so we dont consider it. However, we need to account for the common risk involved with “i” in regards to the tangent portfolio.

What is the common risk?
It is given by the individual volatility of the security multiplied by how much it is correlated with the tangent portfolio.

So, we have: volatility = SD(Ri)Corr(Ri, Rp)

Then we consider the current Sharpe ratio. The ratio tells us how much additional expected return we will get from increasing the risk (volatility) by a unit. Recall Sharpe ratio is given as “(E[Rp]-rf)/SD(Rp)”.

Since the Sharpe ratio offers the expected return GAIN from increasing volatility, we naturally want to use this ratio to compare with what happens when adding the investment “i” into the mix.
Adding investment “i” increase volatility with ‘SD(Ri)Corr(Ri,Rp)’. We multiply this by the Sharpe ratio to get the expected returns we WOULD get from INCREASING THE CURRENT SELECTION WITHOUT INVESTMENT ‘i’ BY THE ADDITIONAL RISK OF ‘i’. This gives us the expected returns from our current selection if we simply choose to increase our risk profile.
This result makes ONE of the comparison parts.

To get the other part, we need to use the expected return GAIN from adding the other investment. We already know that this is given as E(Ri)-rf.
Therefore, we compare the following:

E(Ri)-rf VS SD(Ri)Corr(Ri,Rp) [(E(Rp)-rf)/SD(Rp)]

Now, if the expected returns of investment i (additional increase in expected return) is larger than that of the current tangent portfolio utilization, we can conclude that adding “i” into the portfolio is beneficial, because of the fact that the Sharpe ratio will be improved. If however, current tangent portfolio utilization is greater, then we conclude that adding “i” is not beneficial.

Question 59

What is Beta for investment “i” in regards to portfolio P?

Answer 59

Beta = SD(Ri)Corr(Ri,Rp) / SD(Rp)

Beta measure how much the investment “i” change from a unit change in Portfolio P

Question 60

What is **required return **

Answer 60

Required return is the return necessary for investment i in order to compensate for its additional risk and contribute to the portfolio.

ri = rf + beta (E(Rp) - rf)

Question 61

how do we typically solve a task related to “checking investment i against current portfolio”?

Answer 61

The goal is to compute the required return, because this will tell us whether the new investment is good or not.

To compute the required return, we start by finding the beta - the sensitivity of the investment compared to changes in the current portfolio.

Beta = SD(Ri)Corr(Ri, Rp) / SD(Rp)

Interpretation: If Ri has individual volatility of 20%, and Rp has 10%, and they are fully correlated, we get a beta of 2, indicating that if the portfolio happens to move 1 percent up, the investment Ri will move 2 percent up.

With the beta, we then compute the risk premium movement:

beta x (E(Rp) - rf)

Then we add the risk free return, and get:

rf + beta x (E(Rp) - rf)

This part tells us the required return we need from investment i.

Question 62

What are the assumptions of CAPM?

Answer 62

1) Assume investors can buy and sell securtities with no fees at compatitive market prices

2) Investors hold only efficient portfolios - portfolios that yield maximum return for a given level of risk

3) Investors have homoegenuous expectations regarding volatilities, correlations and expected returna

Question 63

elaborate on the CAPM result

Answer 63

The assumptions of CAPM lead to the following result:

Since each investor knows the same shit, they will all find the same portfolio as the one with the highest sharpe ratio in the economy. This means that all investors will demand the very same portfolio of risky investments, and simply adjust their involvement in it to suit their individual risk level. However: Every investor will hold the same proportion of stocks, because this specific proportion gives the shit with highest sharpe ratio, and is therefore the best.

Since every investor hold the same proportion of the same portfolio, when we sum together all of their portfolios, the result is also equal to the tangent portfolio.

Now, since every stock investment is owned by someone, the sum of portoflios will equal the entire market portfolio. Therefore, the market portfolio equals the tangent portfolio.

NOTE: This model also tells us what happens to stock prices.
If a stock is not good in terms of price and expected return, no one will hold it, and it will not be a part of the tangent portfolio.
If the demand for a stock is not equal to its supply, its price will fall. When a price fall, the expected return of the stock rise. Therefore, eventually the stock price reach a level where the expected return is so good that it becomes a part of the tangent portfolio.

This essentially happens for all stocks, until they are all part of the market portfolio.

Question 64

What is CML?

Answer 64

Capital Market Line.

When the tangent line goes through the market portfolio it is called the capital market line.

According to the CAPM all investors will choose investment combinations that place them along the capital market line.