The assumptions of capital market theory are:
• Alarkowitz investors. All investors use the Markowitz mean-variance framework to select securities. This means they want to select portfolios that lie along the efficient frontier, based on their utility functions.
Unlimited risk-free lending and borrowing. Investors can borrow or lend any amount of money at the risk-free rate.
• Homogeneous expectations. This means that when investors look at a stock, they all see the same risklreturn distribution.
• One-period horizon. All investors have the same one-period time horizon.
• Divisible assets. All investments are infinitely divisible.
• Frictionless markets. There are no taxes or transaction costs.
• No inflation and constant interest. There is no inflation, and interest rates do not change.
• Equilibrium. The capital markets are in equilibrium .
. The Markowitz d'ficient frontier is constructed using only risky assets. Adding a risk- free asset to the Markowitz portfolio construction process extends portfolio theory into capital market theory. Here's the bottom line (no pun intended): The introduction of a
Study Session 12
Cross-Reference to CFA Institute Assigned Reading #51 - An Introduction to Asset Pricing Models
If you invest a portion of your total funds in a risky Portfolio M and the remaining portion in the risk-free asset, the equation for the expected return of the resulting portfolio will be:
where:
RFR E(RM )
the risk-free rate
the expected return on Portfolio M
percentage (weight) of the total portfolio value invested in Portfolio M the percentage (weight) of the total portfolio value invested in the risk-free asset
Professor's Note: rou do not need to memorize the preceding fOrmula! We are
~ going to use it to derive the equation fOr the CAPM, but on the exam, you can , . . " . always determine the E(RpJ fOr a two-stock portfolio (even ifit contains the risk-
free asset) using the equation: E(RpJ =wRFRRFR + wME(RM).
If you combine the risk-free asset with a risky portfolio, the equation for the expected standard deviation of the resulting portfolio will be the same as for a two-risky-asset portfolio:
where:
aRFR aM PRFR,M
=standard deviation of the risk-free asset
=standard deviation of the expected returns on Portfolio M
=correlation between the risk-free asset and Portfolio M
When one of the assets is risk-free, the calculation is much easier! By definition, under the assumptions of portfolio theory and capital market theory, if an asset is risk-free, its return does not vary. Thus, its variance and standard deviation are zero. If an asset has no variance, its expected return doesn't move. If the risk-free rate, RFR, is constant, it can't co-vary with other assets. In other words, the risk-free rate is stationary. Thus, its correlation coefficient with all other assets is zero.
Since aRFR =PRFR,M =0, the equation for portfolio standard deviation simplifies to:
If we put 40% of our portfolio assets in the risky portfolio and the remainder in the risk-free asset, the resulting portfolio has 40% of the standard deviation of the risky portfolio! The risklrerurn relationship is now linear.
Study Session 12 Cross-Reference to CFA Institute Assigned Reading#51 - An Introduction to Asset Pricing Models Combining this with our expected return equation gives us the following linear
equation for the expected portfolio return as a function of portfolio standard deviation:
T.his is the equation for the capital market line (CML). The CML represents aU possible portfolio allocations between the risk-free asset and a risky portfolio. The CML has an intercept of RFR and a slope equal to:
How do we select the optimal risky portfolio when a risk-free asset is also available?
First, let's pick a risky portfolio (like Portfolio X in Figure 1) that's on the Markowitz efficient frontier, since we know that these efficient portfolios dominate everything below them in terms of return offered for risk taken. Now, let's combine the risk-free asset with Portfolio X. Remember, the risk/return relationship resulting from the combination of the risk-free asset and a risky portfolio is a straight line.
Now, choose a risky portfolio that is above Portfolio X on the efficient frontier, such as PortfolioY. Portfolios on the line from RFR to Y wiU be preferred to portfolios on the line from RFR to X because we get more return for a given amount of systematic risk.
Actually, we can keep getting better portfolios by moving up the efficient frontier. At point M, you reach the best possible combination. Portfolio M is at the point where the risk-return tradeoff line is just tangent to the efficient frontier. The line from RFR to M represents portfolios that are preferred to aU the portfolios on the "old" efficient frontier, except M.
Figure 1: Capital Market Line
Capital Market Line
RFR
Efficient Frontier
L - Or
Study Session 12
Cross-Reference to CFA Institute Assigned Reading #51 - An Introduction to Asset Pricing Models
investing the rest in Ponfolio M. To the right of M, invescors hold more than 100% of Ponfolio M. This means they are borrowingfunds to buy more of Ponfolio M. The Levered positionsrepresent a 100% investment in Ponfolio M and borrowing to buy even more of Ponfolio M.
The introduction of a risk-free asset changes the Markowitz efficient frontier into a straight line called the CML.
LOS 51.b: Identify the market portfolio, and describe the role of the market portfolio in the formation of the capital market line (CML).
All investors have to do co get the risk and return combination that suits them is to simply vary the proponion of their investment in the risky Portfolio M and the risk- free asset. So, in the CML world, all investors will hold some combination of the risk- free asset and Portfolio M. Since all investors will want to hold the same risky ponfolio, risky Portfolio M must be the market portfolio.
The market ponfolio is the ponfolio consisting of every risky asset; the weights on each asset are equal co the percentage of the market value of the asset to the market value of the entire market portfolio. For example, if the market value of a stock is $100 million and the market value of the market portfolio is $5 billion, that stock's weight in the market portfolio is 2% ($100 million / $5 billion).
Logic tells us that the market portfolio, which will be held by all invescors, has to contain all the stocks, bonds, and risky assets in existence because all assets have to be held by someone. This market portfolio theoretically includes all risky assets, so it is completely diversified.
LOS Sl.e: Define systematic and unsystematic risk, and explain why an investor should not expect to receive additional return for assuming unsystematic risk.
When you diversify across assets that are not perfectly correlated, the portfolio's risk is less than the weighted sum of the risks of the individual securities in the ponfolio. The risk that disappears in the portfolio construction process is called the asset's
unsystematic risk (also called unique, diversifiabLe, or firm-specific risk). Since the market portfolio containsaLlrisky assets, it must represent the ultimate in
diversification. All the risk that can be diversified away must be gone. The risk that is left cannot be diversified away, since there is nothing left to add to the portfolio. The risk that remains is called the systematic risk (also called nondiversifiabLe riskor market risk) .
The concept of systematic risk applies to individual securities as well as to portfolios.
Some securities are very sensitive to market changes. Typical examples of firms that are very sensitive to market movements are luxury goods manufacturers such as Ferrari automobiles and Harley Davidson motorcycles. Small changes in the market will lead to large changes in the value of luxury goods manufacturers. These firms have high systematic risk (i.e., they are very responsive to market, or systematic, changes). Other
Study Session 12 Cross-Reference to CFA Institute Assigned Reading #51 - An Introduction to Asset Pricing Models These firms have very little systematic risk. Hence, total risk (as measured by standard
deviation) can be broken down into its component parts: unsystematic risk and systematic risk. Mathematically:
total risk=systematic risk + unsystematic risk
e Professor's Note: Know this concept!
Do you actually have to buy all the securities in the market to diversify away
unsystematic risk? No. Academic studies have shown that as you increase the number of stocks in a portfolio, the portfolio's risk falls toward the level of market risk. One study showed that it only took about 12 to 18 stocks in a portfolio to achieve 90% of the maximum diversification possible. Another study indicated it took 30 securities.
Whatever the number, it is significantly less than allthe securities. Figure 2 provides a general representation of this concept. Note, in the figure, that once you get to 30 or so securities in a portfolio, the standard deviation remains constant. The remaining risk is systematic, or nondiversifiable, risk. We will develop this concept later when we discuss beta, a measure of systematic risk.
Figure 2: Risk vs. Number of Portfolio Assets
°" (risk)
Systematic Risk
Total Risk
~ (unsystema~ic:isk) / +systematic fisk Unsystematic Riskr
Market I _ _~t~ -===::::::::::::;;=-_
Risk r- (O"mk,)
Number of securities in the portfolio :::: 30
Systematic Risk is Relevant in Portfolios
One important conclusion of capital market theory is that equilibrium security returns depend on a srock's or a portfolio's systematic risk, not its rotal risk as measured by standard deviation. One of the assumptions of the model is that diversification is free.
The reasoning is that investors will not be compensated for bearing risk that can be eliminated at no cost. If you think about the costs of a no-load index fund compared to buying individual stocks, diversification is actually very low cost if not actually free.
Study Session 12
Cross-Reference to CFA Institute Assigned Reading #51 - An Introduction to Asset Pricing Models
that the drug is effective and safe, stock returns will be quite high. If, on the other hand, the subjects in the clinical trials are killed or otherwise harmed by the drug, the stock will fall to approximately zero and returns will be quite poor. This describes a stock with high standard deviation of returns (i.e., high total risk).
The high risk of our biotech stock, however, is primarily from firm-specific factors, so its unsystematic risk is high. Since market factors such as economic growth rates have little ~o do with the eventual outcome for this stock, systematic risk is a small
proportion of the total risk of the stock. Capital market theory says that the
equilibrium return on this stock may be less than that of a stock with much less firm- specific risk but more sensitivity to the factors that drive the return~fthe overall market. An established manufacturer of machine tools may not be a very risky investment in terms of total risk, but may have a greater sensitivity to market (systematic) risk factors (e.g., GOP growth rates) than our biotech stQck. Given this scenario, the stock with more total risk (the biotech stock) has less systematic risk and will therefore have a lower equilibrium rate of return according to capital market theory.
Note that holding many biotech firms in a portfolio will diversify away the firm- specific risk. Some will have blockbuster products and some will fail, but you can imagine that when 50 or 100 such stocks are combined into a portfolio, the uncertainty about the portfolio return is much less than the uncertainty about the return of a single biotech firm stock.
To sum up, unsystematic risk is not compensated in equilibrium because it can be eliminated for free through diversification. Systematic risk is measured by the contribution of a security to the risk of a well diversified portfolio and the expected equilibrium return (required return) on an individual security will depend on its systematic risk.