Measurement of Risk 📐📈

Knowing that an investment is "risky" isn't enough for a portfolio manager. We need to know how much risk is involved. To do this, we use statistical measures that look at how much actual returns deviate from the expected return. In this chapter, we will cover the foundational measures: Range, MAD, and Variance.

---

1. Range

The Range is the simplest measure of risk. It is the difference between the highest possible outcome and the lowest possible outcome.

Range = Maximum Return - Minimum Return

• Pros: Very easy to calculate.
• Cons: It ignores all information in between the extremes and doesn't tell us how likely those extremes are.

---

2. Mean Absolute Deviation (MAD)

MAD measures the average distance between each actual return and the expected (mean) return, ignoring whether the difference is positive or negative.

Formula:

MAD = Σ |Ri - E(R)| / n

Where:

• |Ri - E(R)|: The absolute difference (magnitude).
• n: Number of observations.

---

3. Variance

Variance (sigma^2) is the most powerful measure in modern portfolio theory. It calculates the average of the squared deviations from the mean.

Formula for Historical Data:

Variance = Σ (Ri - Mean)^2 / n

Formula for Probabilistic Data:

Variance = Σ [ Pi * (Ri - E(R))^2 ]

• Why Square the deviations?
  1. It removes negative signs (just like absolute value).
  2. It penalizes large deviations more heavily, which reflects the psychological pain of big losses.

---

---

4. Comparison of Risk Measures

---

Summary

• Range gives a quick "worst-to-best" picture.
• MAD gives an "average error" picture.
• Variance is the standard mathematical tool for risk (though it results in squared units, e.g., 25% squared).
• In the next chapter, we will learn how to turn Variance back into a usable percentage through Standard Deviation.

---

Quiz Time! 🎯