Lecture 5B - Capital Asset Pricing Model (CAPM)

How is systematic risk defined?

Risk factors that affect a large number of assets Also known as non-diversifiable risk or market risk Includes such things as changes in GDP, inflation, interest rates, etc.

How is unsystematic risk defined?

Risk factors that affect a limited number of assets Also known as unique risk and asset-specific risk Includes such things as labor strikes, part shortages, technical failures etc.

How does diversification affect portfolio risk?

- Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. - This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another. - However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion

How is total risk defined? How does unsystematic risk may affect this definition?

- Total risk = systematic risk + unsystematic risk The standard deviation of returns is a measure of total risk. - For well-diversified portfolios, unsystematic risk is very small. Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.

How does an investor's tolerance towards risk define their investment choice?

All investors have the same CML, but their investment choice depends on their tolerance towards risk: 1. If you are risk averse, you will reduce risk by lending money to the bank at the risk free rate, and choose your portfolio somewhere between rf and M. 2. A risk neutral will choose portfolio M. 3. A risk lover will increase risk by borrowing money at the risk free rate rf and purchase more of the portfolio M, and thus increase expected return but also risk along CML.

How do we calculate the expected return?

By combining a risky asset (Statoil) with the market portfolio M (e.g. NYSE), we get the expected return.

What does the Beta - symbol for the market risk indicates? (most important formula. It must be memorized).

By defining beta we can rewrite the CAPM equation. A low (high) beta indicates low (high) risk. - Example #1: An asset with a beta = 2, has twice as much risk compared to the market portfolio. If the market rise by 1%, the asset will rise by 2%. If the market falls by 2%, the asset will fall by 4%. - Example #2: An asset with a beta = 0.5 has half as much risk compared to the market portfolio.

How do we define the slope of the opportunity set in market equilibrium (M)?

By definition the market portfolio hold the equilibrium weight of each shares. Any "α" holding of asset I (Statoil) would create excess demand for asset I (Statoil). Therefore holding zero extra of asset I (Statoil) must be the market portfolio. By evaluatingthederivatives in α= 0, we find the slope of the opportunity set in market equilibrium (M).

What is the function of Capital Asset Pricing Model (CAPM)?

CAPM tells us the relationship between the risk of one single asset compared to the risk in the market portfolio. E.g. is asset A more or less risky than the market portfolio? And how will asset A affect the return and risk of my portfolio by combining it with the market portfolio?

What is the final step to obtain CAPM?

Combining slope of CML and slope of IMI' will give us CAPM. The result will be: Return on asset i = risk free rate + [risk premium] * market risk

How do we identify the efficient frontier?

Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios. Some portfolios are more desirable than others, providing the same expected return at lower risk. The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

What is Diversifiable risk?

Diversifiable risk is risk that can be eliminated by combining assets into a portfolio. It is often considered the same as unsystematic, unique, or asset-specific risk. If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away.

Reducing risk using a hedge - S&P 500 vs. gold Is gold a perfect hedge? Other candidates for hedging against changes in the stock market?

Gold is not a perfect hedge for a market portfolio (based on yearly return since 967). Over the period, both have had a substantial increase in value. Still, even though the correlation is close to 0 (-0.0695), there is some diversification bonus by combining the gold with a market portfolio.

What is Capital Market Line (CML)?

Now investors can allocate their money across the T-bills and any market portfolio. With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope just touching the efficient frontier. This line is called the Capital Market Line (CML).

How do we calculate the the slope of the Capital Market Line (CML)?

Since we know the risk free rate rf, the expected market return E(rM) and the risk in the market σM, we can calculate the slope of the Capital Market Line (CML).

How can investors do Riskless Borrowing and Lending?

So far we have only considered risky assets, meaning that the expected return cannot be guaranteed (as indicated by standard deviation > 0). But, an investor can always choose to borrow money to increase investment, or put money in the bank (effectively lending) to reduce risk. In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills(US government bonds).

How do we define the Minimum Variance Opportunity Set (MVOS) or the Efficient Set?

Some portfolios are "better" than others. They have higher returns for the same level of risk (or less). The dots circled are called the Minimum Variance Opportunity Set (MVOS) or the Efficient Set. The Minimum Variance Portfolio is defined by the portfolio with the lowest risk (green dot).

What is The Efficient Set?

The efficient set, or efficient frontier, is a graphical representation of a set of possible portfolios that: Minimize risk at specific return levels; and, Maximize returns at specific risk levels.

How does the efficient set relates with correlation?

The shape of this figure is dependent on the correlation between the assets: a) ρ = -1.0, a perfect negative correlation, like the bond/stock-example, provides a perfect diversification and there is one combination that will reduce the risk to 0. b) ρ = 1.0, a perfect positive correlation, eliminates all diversification advantages. c) ρ otherwise will give a curve somewhere in between the two extremes, providing opportunities to diversify, but not perfectly as some risk is still remaining.

Reducing risk using a hedge#2 - S&P 500 vs VIX

VIX is clearly a better hedge as the index has moved opposite of S&P 500. And with very high negative correlation (-0.756), there is a high diversification bonus by combining a VIX index with the market portfolio.

How can be Beta used in the statoil example?

We can now calculate the expected return on Statoil by using its beta (qouted on NYSE), combined with the expected return on the market portfolio and the risk free interest rate.

How do we define the slope of IMI'?

We find the change in expected return E(rp) and return st.dev. σ(rp) of the portfolio pwith respect to the share a that is invested in asset iby taking the derivative of E(rp) and σ(rp) with regards to α.

How do we find the opportunity set for the investor?

When we combine our risky asset (Statoil) with the market portfolio M, we get IMI', which is the opportunity set for the investor. The slope will change according to the share α in Statoil.

How do we find a Market Equilibrium?

With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors.