Risk and Rates of Return: Stand-Alone Risk, Portfolio Risk, and the Capital Asset Pricing Model (CAPM)
1. What is investment risk?
Investment risk means the chance that you might make less money than expected or actually lose money.
Imagine three places you could put your $100.
Option A: A very safe investment
You put in $100 and expect to get $103 back.
Low risk
You know approximately what you're going to get.
Option B: A risky investment
You put in $100. You might get $120, but you could also end up with $70.
High risk
You could make more money, but you could lose more too.
Remember: More risk does not guarantee more money. It just means the outcome is less certain.
2. The two types of investment risk
Your notes describe two ways to think about risk.
Stand-alone risk: How risky is one investment by itself?
Portfolio risk: How risky are all your investments when you put them together?
Imagine you have only one basketball player on your team. If he has a bad game, your whole team struggles. That's like stand-alone risk.
Now imagine you have a whole team of players with different strengths. If one has a bad game, the others might help. That's like portfolio risk.
A portfolio is simply a collection of investments.
3. Probability: What are the chances?
Probability means the chance that something will happen.
Imagine a company could experience five different economic situations:
Economic situation | Chance it happens |
|---|---|
Recession | 10% |
Below average | 20% |
Average | 40% |
Above average | 20% |
Boom | 10% |
Total | 100% |
Think of it like a spinner with 100 spaces. Ten spaces represent a recession, 20 represent below-average conditions, and so on.
The chances must add up to 100% because these five situations cover all the possible outcomes in this example.
What is a probability distribution?
It's just a chart or list showing:
What could happen.
How likely each outcome is.
Your notes also mention that a narrow group of outcomes means lower risk, while outcomes spread far apart mean higher risk.
For example, earning between $2 and $4 is more predictable than possibly earning $30 or losing $25.
4. Expected rate of return: How much money do you expect to make?
The expected rate of return is the average return you expect from an investment after considering all the possible outcomes and their chances.
Imagine you are choosing an investment called High Tech.
Economy | Chance | High Tech return |
|---|---|---|
Recession | 10% | −29.5% |
Below average | 20% | −9.5% |
Average | 40% | 12.5% |
Above average | 20% | 27.5% |
Boom | 10% | 42.5% |
You can't just add these five returns and divide by five because some situations are more likely than others.
For example, an average economy has a 40% chance of happening, so it matters more than a recession with only a 10% chance.
The formula is:
\[ \hat r=\sum P_i r_i \]
Don't let the symbols scare you!
\(P_i\) = the probability, or chance, of something happening.
\(r_i\) = the return you would earn in that situation.
\(\hat r\) = the expected return.
Let's calculate it together
Multiply each return by its probability, then add everything together.
\[ \begin{aligned} \hat r={}&0.1(-29.5)+0.2(-9.5)\\ &+0.4(12.5)+0.2(27.5)\\ &+0.1(42.5) \end{aligned} \]
\[ =-2.95-1.90+5.00+5.50+4.25 \]
\[ \boxed{\hat r=9.9\%} \]
That means High Tech has an expected return of 9.9%.
If you invested $100, that would mean an expected gain of $9.90, or 100 for sure.
Choice B: You might receive $150, but you might receive only $50.
Some people might choose B. Others would rather take the guaranteed 100.
Risk aversion means preferring less risk and requiring compensation to accept additional risk.
That extra compensation is called the risk premium.
\[ \text{Risk premium}= \text{Risky investment return} -\text{Risk-free return} \]
If a safe investment earns 3% and a risky investment is expected to earn 8%, the risk premium is:
\[ 8\%-3\%=\boxed{5\%} \]
The extra 5 percentage points are the reward investors expect for accepting risk.
9. Diversification: Don't put all your eggs in one basket!
Diversification means spreading your money across different investments instead of putting everything into one.
Imagine you invest 100:
$50 in High Tech.
$50 in Collections.
High Tech tends to do well when the economy is booming but poorly during recessions. Collections behaves in the opposite way in the example from your notes.
That means one investment may help offset losses in the other.
Your notes calculate the portfolio's expected return like this:
\[ \begin{aligned} \hat r_p&=0.5(9.9\%)+0.5(1.2\%)\\ &=\boxed{5.5\%} \end{aligned} \]
The portfolio's standard deviation is only 4.6%, compared with 20.0% for High Tech and 11.2% for Collections individually.
The big lesson: Combining investments can reduce risk without necessarily reducing expected return by the same amount.
This works especially well when the investments don't move together.
What is correlation?
Correlation tells you how two investments move in relation to each other.
Perfect positive correlation: +1
Both investments move together. Diversification doesn't reduce risk in this case.
Perfect negative correlation: −1
They move in opposite directions. With the right proportions, their movements can cancel out completely.
Partial correlation: between −1 and +1
They sometimes move together and sometimes differently. Diversification can reduce risk, but usually cannot eliminate it.
Two kinds of risk
Your notes divide total stand-alone risk into two parts:
\[ \text{Total risk}= \text{Market risk}+\text{Diversifiable risk} \]
Market risk (systematic risk): Risk that affects the overall market, such as recessions, inflation, and interest-rate changes. Diversification cannot eliminate it.
Diversifiable risk (company-specific risk): Risk affecting a particular company, such as a lawsuit, strike, or failed product. Diversification can reduce it.
This is why owning several different investments is generally better than owning just one stock.
10. Beta: How risky is a stock compared with the overall market?
Beta measures how much a stock tends to move when the overall market moves.
Think of the market as a roller coaster.
Beta = 1.0: The stock has average market risk.
Beta greater than 1.0: The stock tends to move more than the market.
Beta less than 1.0: The stock tends to move less than the market.
Negative beta: The stock tends to move in the opposite direction of the market.
Examples from your notes:
Investment | Beta | Simple meaning |
|---|---|---|
High Tech | 1.31 | More market-sensitive |
Market Portfolio | 1.00 | Average market risk |
U.S. Rubber | 0.88 | Less market-sensitive |
T-Bills | 0.00 | No market risk in this model |
Collections | −0.50 | Tends to move opposite the market |
Important: Beta measures market risk, not every kind of risk. A stock can have a low beta but still have company-specific problems.
11. CAPM: Calculating the return investors require
CAPM stands for the Capital Asset Pricing Model.
The name sounds complicated, but the idea is simple:
If an investment is riskier compared with the market, investors usually demand a higher return.
The formula is:
\[ \boxed{r_i=r_{RF}+(r_M-r_{RF})b_i} \]
Here is what each part means:
\(r_i\) = required return on the investment.
\(r_{RF}\) = risk-free rate.
\(r_M\) = required return on the overall market.
\(b_i\) = the investment's beta.
\(r_M-r_{RF}\) = market risk premium.
Your notes give these values:
Risk-free rate = 3%.
Market return = 8%.
Market risk premium = \(8\%-3\%=5\%\).
Example: High Tech
High Tech has a beta of 1.31.
Step 1: Write the formula.
\[ r_i=r_{RF}+(r_M-r_{RF})b_i \]
Step 2: Put in the numbers.
\[ r_i=3\%+(5\%)(1.31) \]
Step 3: Multiply and add.
\[ r_i=3\%+6.55\% \]
\[ \boxed{r_i=9.55\%} \]
This means investors would require a return of 9.55% to compensate them for High Tech's market risk, using the assumptions in your notes.
12. The Security Market Line (SML)
The Security Market Line is a graph showing the relationship between beta and required return.
Think of it as a line that tells you how much return investors demand for different levels of market risk.
Higher beta → higher required return.
Lower beta → lower required return.
Your notes also explain two things that can change the line:
If expected inflation increases: The risk-free rate increases, shifting the whole line upward if the market risk premium stays constant.
If investors become more afraid of risk: The market risk premium increases, making the line steeper. High-beta investments experience a larger increase in required return.
13. Undervalued vs. overvalued stocks
This is one of the most important things to understand for exam questions.
Compare two numbers:
Expected return: What you think the investment will earn.
Required return: What it needs to earn to compensate you for its risk.
Comparison | Meaning | Decision in your notes |
|---|---|---|
Expected return > Required return | The investment offers more return than required | Undervalued — attractive to buy |
Expected return = Required return | The investment offers exactly the required return | Fairly valued |
Expected return < Required return | The investment offers less return than required | Overvalued — unattractive to buy |
Example
Suppose a stock is expected to earn 12%, but investors require only 9%.
The stock offers more return than required.
\[ 12\%>9\% \]
Therefore, it is undervalued under the model in your notes.
Now suppose another stock is expected to earn 6%, but investors require 9%.
\[ 6\%<9\% \]
That stock is overvalued under the model.
The logic is that an investment offering too little return for its risk is not attractive at its current price.