Finance-Theory · Unit 15 · Video 3 · Interactive Practice

Why Hold a Portfolio? Diversification, Mean and Variance

IKey Formulas

Formula	Name	What it measures
$\mu_p \equiv \text{E}[R_p]$	Mean (reward)	Central tendency of the return distribution
$\text{Prob}(R_p \le q_{5\%}) = 5\%$	5th percentile	How bad the losses get, one year in twenty
$\sigma_p^2 \equiv \text{E}[(R_p - \mu_p)^2]$	Variance (risk)	Symmetric dispersion around the mean
$\sigma_p = \text{SD}[R_p] = \sqrt{\sigma_p^2}$	Standard deviation	Dispersion in the units of the return

Key Insight: The mean-variance investor likes a higher $\text{E}[R_p]$ , dislikes a higher $\sigma_p^2$ , cares about nothing else, and judges only the overall portfolio: a stock earns a place only if it raises $\text{E}[R_p]$ or lowers $\text{SD}[R_p]$ .

IIWhich Number Summarizes Reward?

The mean, the 75th percentile and the 5th percentile need not rank two funds the same way.

Summary

IIIVariance: Spread Around the Mean

With four equally likely scenarios, $\mu_p$ is their balance point and $\sigma_p$ their root-mean-square distance from it, both probability-weighted ( $p = \tfrac{1}{4}$ each): a distribution of outcomes, not a sample, so the variance divides by $4$ , not $T - 1$ .

💡 Challenge: keep $\mu_p = 7\%$ but double $\sigma_p$ to $24\%$ ; then find a different arrangement with the same mean and the same $\sigma_p$ .

IVWhen a Positive Return Is the Loss

For a short seller the stock's positive return is the loss, yet the standard deviation does not change.

VQuiz Questions

Problem 1 · The Mean of a Return Distribution

Given: An illustrative fund's return next year is $-10\%$ with probability $\tfrac{1}{4}$ , $14\%$ with probability $\tfrac{2}{4}$ , and $22\%$ with probability $\tfrac{1}{4}$ . Find its expected return $\mu_p = \text{E}[R_p]$ .

✅ Correct! Each scenario counts in proportion to its probability:

-2.5 + 7 + 5.5 = 10

❌ Not quite.

14\%

is the single most likely outcome, not the probability-weighted average; the

-10\%

and

22\%

scenarios pull the mean away from it.

❌ Not quite.

(-10 + 14 + 22)/3 \approx 8.7

treats the three scenarios as equally likely, but

14\%

has probability

\tfrac{2}{4}

❌ Check the probabilities. The weight

\tfrac{2}{4}

belongs to the

14\%

scenario, not the

22\%

one.

❌ Not quite. Multiply each return by its probability and add.

Show solution

The mean is the probability-weighted average of the scenarios:

\mu_p = \tfrac{1}{4}(-10\%) + \tfrac{2}{4}(14\%) + \tfrac{1}{4}(22\%) = -2.5\% + 7\% + 5.5\% = 10\%

The probabilities add to one ( $\tfrac{1}{4} + \tfrac{2}{4} + \tfrac{1}{4} = 1$ ), so this is a complete distribution, and its central tendency is $\mu_p = 10\%$ .

Problem 2 · Risk for a Short Seller

Given: Rowan's return next year is modelled (illustrative) as four equally likely scenarios: $-18\%$ , $6\%$ , $14\%$ , $38\%$ , so $\text{Prob}(R < 0) = \tfrac{1}{4}$ and $\sigma = 20\%$ . You have sold Rowan short, so your return in each scenario is the negative of Rowan's.

Probability that your position loses money

Standard deviation of your short position's return

✅ Correct! The "probability of a loss" depends on which side you hold; the standard deviation does not.

❌ That is Rowan's own probability of a negative return. You are short: Rowan's three positive scenarios are your losses.

❌ Not quite. Count the scenarios in which Rowan's return is positive: each one is a loss for a short seller.

❌ A standard deviation is never negative. It is the square root of an average of squared deviations; flipping every return flips every deviation, and squaring removes the sign.

❌ Not quite. Negating every return negates every deviation from the mean, and the squares are unchanged.

Show solution

Losses. Your returns are $18\%$ , $-6\%$ , $-14\%$ , $-38\%$ . Three of the four equally likely scenarios are losses, so the probability is $\tfrac{3}{4}$ , although Rowan itself falls only with probability $\tfrac{1}{4}$ . A risk measure built on "a negative stock return" would get this investor exactly backwards.

Spread. Your probability-weighted mean is $\tfrac{1}{4}(18 - 6 - 14 - 38) = -10\%$ , and your deviations are $28$ , $4$ , $-4$ , $-28$ ; each squared deviation is weighted by its probability $\tfrac{1}{4}$ (a distribution, not a sample, so no $T - 1$ ):

\sigma^2 = \tfrac{1}{4}\left(28^2 + 4^2 + 4^2 + 28^2\right) = \tfrac{1}{4}(1600) = 400, \qquad \sigma = \sqrt{400} = 20\%

Rowan's own deviations are $-28$ , $-4$ , $4$ , $28$ , with the same squares, so both sides of the trade have $\sigma = 20\%$ : the symmetric measure never has to decide which direction is bad.

Problem 3 · Same Center, Wider Spread

Given: An illustrative fund's return next year is $-13\%$ , $-13\%$ , $27\%$ or $27\%$ , each scenario with probability $\tfrac{1}{4}$ . Quill (illustrative) has $\mu_p = 7\%$ and $\sigma_p = 12\%$ . Find the fund's standard deviation (probability-weighted), then compare it with Quill for a mean-variance investor.

The fund's standard deviation

Compared with Quill, the fund is

✅ Correct! Two distributions centred at the same point: the wider one is riskier, and a mean-variance investor prefers the narrower one.

❌ Wrong divisor.

\sqrt{1600/3} \approx 23.1

divides by

T - 1 = 3

, the sample-variance divisor for estimating from past returns. These four outcomes are a probability distribution, so each squared deviation is weighted by its probability

\tfrac{1}{4}

❌ Not quite.

40

is the range,

27 - (-13)

. The standard deviation measures spread around the mean: average the squared deviations, then take the square root.

❌ That is the variance.

\sigma_p^2 = 400

(in squared percent); take the square root to get back to percent.

❌ Not quite. Find the mean first, then average the squared deviations (each with weight

\tfrac{1}{4}

) and take the square root.

❌ The upside comes with an equal downside. The fund's spread is symmetric: its

-13\%

scenarios sit as far below the mean as its

27\%

scenarios sit above it.

❌ Not quite. The mean-variance investor looks at exactly two numbers: equal means, so the standard deviations decide.

Show solution

Mean: $\mu_p = \tfrac{1}{4}(-13 - 13 + 27 + 27) = \tfrac{28}{4} = 7\%$ , the same as Quill.

Deviations from the mean: $-20$ , $-20$ , $20$ , $20$ .

\sigma_p^2 = \tfrac{1}{4}\left(20^2 + 20^2 + 20^2 + 20^2\right) = \tfrac{1600}{4} = 400, \qquad \sigma_p = \sqrt{400} = 20\%

These are probability-weighted (population) figures for a distribution of possible outcomes, so each squared deviation gets its probability $\tfrac{1}{4}$ ; the $T - 1$ divisor belongs to the sample variance of past returns.

Comparison: both have $\mu_p = 7\%$ , but $20\%$ is larger than $12\%$ . The investor likes a higher mean, dislikes a higher variance and cares about nothing else, so with the means tied the fund with the larger spread is worse.

Problem 4 · Who Gets Into the Club?

Given: A mean-variance investor holds a well-diversified portfolio and considers four candidate stocks, one at a time. Which candidate has no reason to join the portfolio?

✅ Correct! It neither raises the portfolio's expected return nor lowers its risk, so its impressive standalone mean is irrelevant.

❌ Not quite. The investor cares only about the overall portfolio: this stock lowers

\text{SD}[R_p]

at no cost in expected return, whatever its risk held alone.

❌ Not quite. It raises

\text{E}[R_p]

, so there is a reason to consider it; whether the extra risk is worth it is a trade-off, not a clear "no".

❌ Not quite. Ask of each candidate: does it raise the portfolio's expected return, or lower its risk?

Show solution

Adding a stock is like inviting someone into a club: the question is what the newcomer does to the portfolio's two numbers, $\text{E}[R_p]$ and $\text{SD}[R_p]$ .

(a) lowers risk at no cost in expected return: it has a reason to join.
(b) raises the expected return at the price of more risk: a trade-off the investor may accept.
(c) raises the expected return at no extra risk: it has a reason to join.
(d) lowers the expected return and raises the risk: the answer to both questions is no, so it does nothing for the portfolio.

How a stock looks on its own (the riskiest, the highest mean) does not decide anything; only its effect on the overall portfolio does.

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