Portfolio Risk and Return: Part I

CFA Level I · CFA · 6 min read

Welcome to the World of Portfolio Management!

Hello there! You are about to dive into one of the most exciting and practical parts of the CFA Level I curriculum: Portfolio Risk and Return. Think of this as the "instruction manual" for building a successful investment portfolio. Instead of just looking at one stock in isolation, we are going to look at how different investments play together as a team.

By the end of these notes, you will understand how to measure your "winnings" (returns), how to measure the "scary parts" (risk), and how to pick the best combination of assets to help you sleep at night while still making money. Let’s get started!

1. Measuring Returns: How Well Did We Do?

Before we can manage a portfolio, we need to know how to calculate how much money we actually made. There are several ways to do this, and the CFA exam loves to test which one you should use in different situations.

Holding Period Return (HPR)

This is the most basic return. It’s simply the percentage increase in your wealth over a specific period.
The Formula: $HPR = \frac{P_{end} - P_{beginning} + CashFlow}{P_{beginning}}$
Example: You buy a stock for $100, get a $2 dividend, and sell it for $105. Your HPR is $(105 - 100 + 2) / 100 = 7%$.

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Arithmetic vs. Geometric Returns

Arithmetic Mean: This is your "simple average." You add up the returns and divide by the number of years. It’s great for estimating the return for next year (the next single period).
\nGeometric Mean: This is the "compound average." It accounts for the fact that money grows on top of money.
\nFormula: $RG = [(1+R_1) \times (1+R_2) \times ... \times (1+R_n)]^{1/n} - 1$

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Did you know? The Geometric Mean will always be less than or equal to the Arithmetic Mean unless all returns are exactly the same. The more volatile (bumpy) the returns are, the bigger the gap between the two!

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Money-Weighted vs. Time-Weighted Returns

This is a classic "Common Mistake" zone for students. Let's make it simple:

1. Time-Weighted Return (TWR): This measures the growth of $1 invested. It is NOT affected by when an investor adds or removes money. Because of this, it is the standard way to measure the performance of Investment Managers (who usually can't control when clients give them money).

2. Money-Weighted Return (MWR): This is essentially the Internal Rate of Return (IRR). It IS affected by the timing and size of cash flows. If you add a lot of money right before a stock booms, your MWR will look better than your TWR.

Quick Review:
- Use Time-Weighted to evaluate the Manager.
- Use Money-Weighted to evaluate the Investor's actual wallet experience.

2. Measuring Risk: The "Bumpy Ride"

In finance, we define Risk as uncertainty. Specifically, it’s the chance that the actual return will be different from what we expected.

Variance and Standard Deviation

We use Variance ($\sigma^2$) and Standard Deviation ($\sigma$) to measure risk. The higher the standard deviation, the more "spread out" the possible returns are, and the riskier the asset is.

Don't worry if the math looks scary: On the exam, you'll often be given these numbers, or you'll use your calculator's statistical functions. Just remember that Standard Deviation is the "square root of variance."

Risk Aversion

Most investors are Risk Averse. This doesn't mean they won't take risks; it means they need to be paid (via higher expected returns) to take those risks.
- Risk Neutral: Only cares about return, ignores risk.
- Risk Seeking: Actually enjoys the gamble (not common in professional portfolio theory!).

3. Utility Theory: Picking What’s Best for YOU

How do we decide if a portfolio is "good enough"? We use a Utility Function. It’s a way to give a "happiness score" to an investment.

The Formula: $U = E(R) - \frac{1}{2} \times A \times \sigma^2$
Where:
- $E(R)$ = Expected Return (The "Good" stuff)
- $A$ = The Risk Aversion Coefficient (How much you hate risk. Higher A = More hate)
- $\sigma^2$ = Variance (The "Bad" stuff)

Key Takeaway: An investor will choose the portfolio that gives them the highest Utility (U). If you hate risk (High A), the "penalty" for variance will be very high, and you'll prefer safer assets.

Indifference Curves

Imagine a graph where the Y-axis is Return and the X-axis is Risk. An Indifference Curve connects all portfolios that give you the same amount of happiness (Utility).
- For a risk-averse investor, these curves slope upward (to take more risk, I need more return).
- The steeper the curve, the more risk-averse the investor is.

4. Portfolio Risk: The Power of Diversification

This is the "magic" of portfolio management. When you combine assets, the risk of the portfolio is usually less than the weighted average of the individual risks. Why? Because assets don't move perfectly together!

Covariance and Correlation

Covariance: Measures how two assets move together. If positive, they move together. If negative, they move opposite.
Correlation ($\rho$): This is a cleaner version of covariance. It always stays between -1.0 and +1.0.

- Correlation = +1.0: Perfect positive movement. No diversification benefits.
- Correlation = 0: No relationship. Good diversification.
- Correlation = -1.0: Perfect opposite movement. Maximum diversification (you can actually eliminate risk!).

Analogy: Think of an umbrella shop and a sunscreen shop. On a rainy day, one does well and the other does poorly. On a sunny day, it flips. If you own both, your total income stays relatively steady regardless of the weather. That is diversification!

5. The Capital Allocation Line (CAL)

What happens when we mix a "Risk-Free Asset" (like a Government T-Bill) with a "Risky Portfolio" (like a basket of stocks)? We get the Capital Allocation Line (CAL).

The Formula for Expected Return of this mix: $E(R_p) = R_f + \sigma_p \times \frac{E(R_i) - R_f}{\sigma_i}$

Wait! Look closer at that formula: The part $\frac{E(R_i) - R_f}{\sigma_i}$ is actually the Sharpe Ratio!
The Sharpe Ratio measures "Excess return per unit of risk." In the CAL, the Sharpe Ratio is the slope of the line.

Key Points of the CAL:
1. The Y-intercept: This is the Risk-Free Rate ($R_f$). If you take zero risk, this is what you earn.
2. Moving along the line: As you move to the right, you are putting more money into the risky stocks and less into the risk-free asset.
3. Lending vs. Borrowing: If you are between the Y-axis and the risky portfolio, you are "lending" (buying T-bills). If you move beyond the risky portfolio, you are "borrowing" money at the risk-free rate to buy even more stocks (using leverage).

Summary and Tips for Success

- Remember the goal: Portfolio management is about maximizing return for a given level of risk.
- Watch out for TWR vs MWR: If the question mentions an "Investment Manager," look for Time-Weighted Return.
- Correlation is key: Diversification only works if correlation is less than +1.0.
- The A coefficient: A higher "A" in the utility formula means a more "scared" investor who wants a steeper indifference curve.

You've got this! This chapter sets the stage for everything else in Portfolio Management. Take a moment to review the formulas, and then try some practice problems to see these concepts in action.

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

How is the Holding Period Return (HPR) calculated?

The HPR is calculated as: $HPR = \frac{P_{end} - P_{beginning} + CashFlow}{P_{beginning}}$.

What is the primary difference between the Arithmetic Mean and the Geometric Mean in return calculations?

The arithmetic mean is a simple average used for estimating next-period returns, while the geometric mean accounts for compounding and represents the actual growth rate of wealth over time.

Why is Time-Weighted Return (TWR) the preferred method for evaluating investment managers?

TWR is preferred because it is not affected by the timing or size of cash flows into or out of the account, which are usually controlled by the client rather than the manager.

How are Variance and Standard Deviation used to define investment risk?

Variance ($\sigma^2$) and Standard Deviation ($\sigma$) measure the dispersion of possible returns around the expected mean; a higher value indicates greater uncertainty and risk.

What is the difference between risk-averse, risk-neutral, and risk-seeking investors?

Risk-averse investors require higher returns to accept more risk, risk-neutral investors ignore risk entirely, and risk-seeking investors prefer gambling for higher potential gains despite the risk.

What does the Utility Function formula represent for an investor?

It calculates a "happiness score" for a portfolio: $U = E(R) - \frac{1}{2} A \sigma^2$, where $E(R)$ is expected return and $A$ is the risk aversion coefficient.

What characterizes an indifference curve for a risk-averse investor?

It is an upward-sloping curve connecting portfolios that provide the same utility; a steeper slope indicates higher risk aversion.

How does the correlation coefficient affect the benefits of diversification?

Diversification benefits exist as long as the correlation ($\rho$) is less than +1.0, with maximum benefit occurring at a correlation of -1.0.

What is the Capital Allocation Line (CAL)?

The CAL represents all possible combinations of a risk-free asset and a risky portfolio, showing the trade-off between risk and return.

What does the slope of the Capital Allocation Line (CAL) represent?

The slope of the CAL represents the Sharpe Ratio: $\frac{E(R_i) - R_f}{\sigma_i}$.

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