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# P1.T1.60. Sharpe-Lintner-Mossin capital asset pricing model derivation (CAPM; Elton & Gruber)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
AIMs: Understand the derivation and components of the CAPM. Describe the assumptions underlying the CAPM.

Questions:

60.1. Roger is an analyst employing the Sharpe-Lintner-Mossin capital asset pricing model (CAPM) to estimate the return of a portfolio. However, his colleague Sally makes four observations. If true, each observation violates an assumption of this standard version of the CAPM. Which of the following observations, if true, does NOT violate an assumption such that, by itself, is consistent with the CAPM; i.e., each of the following is a standard CAPM assumptional violation EXCEPT?

a. Sally proves that the portfolio securities' returns are heavy-tailed (leptokurtic)
b. Sally observes that other investors have different views about the expected returns and variances of the portfolio securities
c. If Roger makes a purchase of a security in the portfolio, his purchase will NOT increase (or impact) the price of the security
d. Sally points out that Roger incurs transaction costs, cannot short sell, and cannot borrow at the risk-rate

60.2. In regard to the derivation of the Sharpe-Lintner-Mossin capital asset pricing model (CAPM), each of the following is true EXCEPT for:

a. All investors will hold combinations of only two portfolios: the Market portfolio ( M) and a Riskless security. This is called the "two mutual fund theorem" because all investors would be satisfied with a market fund, plus the ability to lend or borrow a riskless security
b. All portfolios and risky assets must lie along (on) the capital market line (CML)
c. In equilibrium, the Market portfolio lies at the tangency point between the original efficient frontier of risky assets and a straight line ("ray") passing through the riskless return
d. The security market line (SML) implies: the relationship between the expected return on any two assets can be related simply to their difference in Beta; the higher Beta is for any security, the higher must be its equilibrium return; and the relationship between Beta and expected return is linear

60.3. Which of the following is strictly true about the standard (Sharpe-Lintner-Mossin) version of the capital asset pricing model (CAPM)?

a. The security market line (SML) states that the expected return on any security is the riskless rate of interest plus the market price of risk times the amount of risk in the security or portfolio
b. If CAPM is valid, then the return of a high-beta should be higher than the return of a low-beta stock over the next calendar year, or for that matter, any given calendar year
c. All other things being equal, the security market line (SML) implies that higher non-systematic (aka, idiosyncratic) risk will produce higher expected returns
d. While CAPM characterizes equilibrium in terms of rate of return, it cannot be similarly extended to prices

#### Miss FRM

##### New Member
Q1. C
Q2. B
Q3. B

Are the above correct?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Miss FRM Almost, I agree with you re Q1 and Q2, but Q3 is not (B):
(B) is FALSE: "Invariably, when a group of investors is first exposed to the CAPM, one or more investors will find a high- Beta stock that last year produced a smaller return than low- Beta stocks. The CAPM is an equilibrium relationship. High- Beta stocks are expected to give a higher return than low- Beta stocks because they are more risky. This does not mean that they will give a higher return over all intervals of time. In fact, if they always gave a higher return, they would be less risky, not more risky, than low- Beta stocks. Rather, because they are more risky, they will sometimes produce lower returns. However, over long periods of time, they should on the average produce higher returns."

#### Miss FRM

##### New Member
@David Harper CFA FRM CIPM

I agree with your argument, but among all choices it seemed like the best available answer.
For Q3 - The question seeks a true statement.
Answer A is false, because SML states that expected return on an asset is the riskless interest rates plus market risk premium times systematic risk. The CML is the riskless rate of interest plus market price of risk times amount of risk.
Answer C is false, SML does not reward non-systematic risk. Beta (systematic risk) is the only priced risk.
Answer D is false, CAPM can be extended to prices. Reference reading: Mordern Portfolio Theory, Chap 13, Standard CAPM, pg 292. Section Prices and the CAPM
"CAPM could be used to describe equilibrium in terms of either return or prices"

So where am I going wrong ??

#### Heidi

##### Member
Subscriber
@Miss FRM, the correct answer to no 3 is A.

From your definition above, the market risk premium is the price of risk while the systematic risk is the quantity of risk.

This is same as option A of the daily question.

@David, I hope am correct

#### superpocoyo

##### Member
Hi David,

I do not fully understand for Q3, why A is true? "The security market line (SML) states that the expected return on any security is the riskless rate of interest plus the market price of risk times the amount of risk in the security or portfolio"

Looking at the SML formula, I can not see "plus the market price of risk times the amount of risk in the security or portfolio".

E(R) = Rf + beta*(E(Rm) - Rf)

Can you explain please. Thank you.

Melody

#### Alex_1

##### Active Member
Hi @superpocoyo , answer A is indeed true.

Market price of risk is the return in excess of the risk-free rate that the market wants as compensation for taking risk, therefore it is "E(Rm) - Rf"
The amount of risk in the security or portfolio is "beta", which is the nondiversifiable or systematic risk of a security or portfolio.

Which means "plus the market price of risk times the amount of risk in the security or portfolio" is exactly "+ beta*(E(Rm) - Rf)".

I hope this helps.

#### Miss FRM

##### New Member
@Alex_1 and @Heidi

Thank you for the feedback. But I just want to point out the following:
[E(Rm) - Rf] is called market risk premium
[E(Rm)-Rf]/std. dev of market is called market price of risk.
The two concepts are not the same.
Ref: Standard Capital Asset pricing model, Chapter 13 Modern Portfolio Theory, Section 'Deriving the CAPM a simple approach', page 285. "The term (Rm - Rf)/std dev M, can be though of as the market price of risk for all efficient portfolios".

One way to think about this is, when we talk about 'Price', it is always in terms of a base eg, $2/1 kg for potatoes, or$1000/1 ounce for gold.
Similarly, for markets, the price is 5% excess return/1 unit risk.

This stills leaves us to find the correct answer.

@David Harper CFA FRM CIPM - Help?

#### superpocoyo

##### Member
Thanks @Alex_1 & @Miss FRM

I am confused by the meaning of "Market Price of risk" here. I always know [E(Rm) - Rf] is called market risk premium, never come across "market price of risk".

@Alex_1, can I treat "beta" as a factor sensitivity in the APT model? If [E(Rm) - Rf] is the "market price of risk", then "beta" could be the sensitivity of a security or portfolio to "market price of risk"?

CAPM model is a single factor APT model?

#### Alex_1

##### Active Member
Hi @superpocoyo, well, yes, my understanding (at least until now - I have passed part 1 in May, but maybe I have forgotten some things from then) was that CAPM might be considered a "special case" of APT in that the SML represents a single-factor model of the asset price and beta is the sensitivity of the asset with regard to changes in value of the market portfolio. @Miss FRM might be right in her description about market price of risk, I am currently not 100% sure about it.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@Miss FRM

I pulled (A) from later in Elton, page 297 - 298 (although these pages are 9th edition in anticipation of 2015), where he says (emphasis mine):"Because σ(iM)/σ(M) is a definition of the risk of any security, or portfolio, we would see that the security market line, like the capital market line, states that the expected return on any security is the riskless rate of interest plus the market price of risk times the amount of risk in the security or portfolio." and he's referring to:
$${{\overline{R}}_{i}}={{R}_{F}}+\left( \frac{{{\overline{R}}_{M}}-{{R}_{F}}}{{{\sigma }_{M}}} \right)\frac{{{\sigma }_{iM}}}{{{\sigma }_{M}}}$$

But I guess the denominator moves around because he gives this alternative and says "many authors write the CAPM equation as" (note only market volatility shifts):
$${{\overline{R}}_{i}}={{R}_{F}}+\left( \frac{{{\overline{R}}_{M}}-{{R}_{F}}}{\sigma _{M}^{2}} \right){{\sigma }_{iM}}$$

And I admit that when I wrote (A) I was thinking really only of the familiar version:
$${{\bar{R}}_{i}}={{R}_{F}}+\left( {{{\bar{R}}}_{M}}-{{R}_{F}} \right)\frac{{{\sigma }_{iM}}}{\sigma _{M}^{2}}={{R}_{F}}+\left( {{{\bar{R}}}_{M}}-{{R}_{F}} \right){{\beta }_{iM}}$$
... because there was a historical tendency to "English describe" CAPM (which is still in Amenc, I *think*) as: E[return] = Rf + beta*MRP, where beta is "quantity of risk" and market risk premium is "price of risk" but I'm sort of with you in realizing that's imprecise.

I think your point about [R(m) - R(f)] not really being "adjusted" into a price of risk has a lot of merit. I think I could both agree with you but still find (A) true simply in the refuge that the product (multiplication) in SML seems to allow for us to re-map from beta to a different flavor of "quantity of risk" (i.e., beta multiplied by market volatility) and from MRP ("market risk premium) to a different flavor of "market price of risk" (i.e., MRP divided by market volatility). I hope that is helpful, but I am grateful for the precise observations, thank you!

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