Finance-Theory · Unit 14 · Video 5 · Interactive Practice

Unpredictable Over Time, Connected Across Stocks: What Stock Returns Look Like

IKey Formulas

Formula	Name	What it says
$R_t = \dfrac{D_t + P_t - P_{t-1}}{P_{t-1}}$	Total return (one month)	Everything the asset pays counts: dividends plus price change for a stock, coupons plus price change for a bond
$\text{Corr}[R_{m,t},\, R_{m,t-1}] \approx 0$	Autocorrelation (time series)	The same security in two periods: last period's return says very little about this period's
$\text{Corr}[R_{GM,t},\, R_{m,t}] > 0$	Cross-sectional correlation	Two securities in the same period: strongly positive, well below 1
$\sigma_T^2 \approx T\sigma_1^2 \;\Rightarrow\; \sigma_T \approx \sigma_1\sqrt{T}$	Square-root-of-time scaling	$\sigma_{\text{month}} \approx \sigma_{\text{day}}\sqrt{21}$ and $\sigma_{\text{year}} \approx \sigma_{\text{month}}\sqrt{12}$ ; valid because the autocorrelation is close to zero

Key Insight: A strong correlation across stocks in the same month gives no forecast of the market over time. And volatility itself moves: a quiet month at $1\%$ a day is about $15.9\%$ as an annual rate, while a crisis month at $5\%$ a day is about $79.4\%$ .

IISame Security Over Time vs Two Securities at Once

Across months the market's return is nearly uncorrelated with itself, yet within a month a stock moves with the market.

Month t 15

💡 With only 24 illustrative months, chance alone pushes the lag correlation beyond $\pm 0.3$ in roughly one fresh sample in six; the long market histories in the video put it close to zero. The market's months are drawn independently here, so any lag correlation you see is sampling noise.

IIIVolatility Comes and Goes

Each point is one month's standard deviation of its 21 daily returns, and those points do not stay level.

Month 36

IVWhy Volatility Grows With √T

Nearly uncorrelated returns make variances add over time, so the standard deviation grows with the square root of the horizon.

Horizon T (trading days, σ₁ = 1.3% a day) 21

💡 A year is taken as $12 \times 21 = 252$ trading days, so $\sqrt{21}\cdot\sqrt{12} = \sqrt{252} \approx 15.87$ turns a daily $\sigma$ into an annual rate in one step.

VQuiz Questions

Problem 1 · Annualize a Monthly Volatility

Given: an illustrative stock index has a monthly return standard deviation of $5.9\%$ , and its monthly returns are nearly uncorrelated from one month to the next. Using the video's scaling, find its volatility expressed as an annual rate.

✅ Correct! Twelve monthly variances add, so the standard deviation grows by

\sqrt{12} \approx 3.46

5.9\% \times 3.46 \approx 20.4\%

❌ That multiplies by 12. It is the variance that scales with the number of months; the standard deviation scales with

\sqrt{12}

, not

12

❌ Wrong factor.

\sqrt{21}

takes a daily figure to a monthly one; a year holds 12 months, so the monthly figure is multiplied by

\sqrt{12}

❌ Wrong direction. A year is longer than a month, so its standard deviation is larger: multiply by

\sqrt{12}

, don't divide.

Show solution

With monthly returns nearly uncorrelated, the variances of 12 months approximately add:

\sigma_{\text{year}}^2 \approx 12\,\sigma_{\text{month}}^2 \quad\Rightarrow\quad \sigma_{\text{year}} \approx \sigma_{\text{month}}\sqrt{12}

\sigma_{\text{year}} \approx 5.9\% \times 3.464 \approx 20.44\% \approx 20.4\%

Multiplying by 12 instead ( $70.8\%$ ) would scale the standard deviation as if it were a variance.

Problem 2 · Correlated Is Not Predictable

Given: an analyst finds that the monthly return of a fictional stock, Tamsin Rail, has an illustrative correlation of $0.7$ with the market's return in the same month. She concludes that watching Tamsin Rail this month will let her forecast the market next month. What is wrong with the conclusion?

✅ Correct! A strong relation within a month says nothing about the market from one month to the next, and that relation is nearly absent.

❌ Not quite. The 0.7 links two returns in the same month. It says nothing about next month; for that you need the correlation across periods, which is close to zero.

❌ Not quite. Changing the data frequency does not change the question. Daily data were exactly where the market's today-vs-yesterday scatter came out round.

❌ Not quite. The video met that objection: with GM removed from the index the positive relation remained. And even a genuine same-month relation would not forecast next month.

Show solution

Two different questions are being mixed:

Cross section — two securities, same period: $\text{Corr}[R_{TR,t},\, R_{m,t}] = 0.7$ (illustrative). Strong.
Time series — the same security, two periods: $\text{Corr}[R_{m,t},\, R_{m,t-1}] \approx 0$ . Almost nothing.

Forecasting next month's market from this month's information is a time-series question. By the time Tamsin Rail's return for this month is known, so is the market's for this month, and this month's market return says very little about next month's. A strong correlation across stocks gives no way to forecast the market itself.

Problem 3 · From Daily to Monthly to Annual

Given: in a quiet month, an illustrative market's daily returns have a standard deviation of $0.8\%$ . A month holds about 21 trading days and a year 12 months. Find the monthly standard deviation and the annual rate.

Monthly standard deviation?

Annual rate?

✅ Correct!

0.8\% \times \sqrt{21} \approx 3.7\%

a month, and

3.67\% \times \sqrt{12} \approx 12.7\%

as an annual rate — the same as

0.8\% \times \sqrt{252}

❌ That multiplies by 21. The 21 daily variances add; the standard deviation grows only by

\sqrt{21} \approx 4.58

❌ Check the monthly step. Days to a month uses

\sqrt{21}

(not

\sqrt{12}

), and a month's standard deviation is larger than a day's.

❌ That multiplies the monthly figure by 12. Months to a year uses

\sqrt{12} \approx 3.46

, not 12.

❌ That multiplies by 252. Adding the standard deviations of 252 days treats them as variances; the factor is

\sqrt{252} \approx 15.87

❌ Check the annual step. Chain both factors:

\sigma_{\text{year}} \approx \sigma_{\text{day}} \sqrt{21}\sqrt{12} = \sigma_{\text{day}}\sqrt{252}

Show solution

Step 1 — day to month. With daily returns nearly uncorrelated, 21 daily variances add:

\sigma_{\text{month}} \approx 0.8\% \times \sqrt{21} = 0.8\% \times 4.583 \approx 3.67\% \approx 3.7\%

Step 2 — month to year.

\sigma_{\text{year}} \approx 3.666\% \times \sqrt{12} = 3.666\% \times 3.464 \approx 12.70\% \approx 12.7\%

Check in one step: $0.8\% \times \sqrt{252} = 0.8\% \times 15.87 \approx 12.70\%$ .

Problem 4 · Is It Just Correlated With Itself?

Given: a fictional stock makes up about $3\%$ of an illustrative index, and over 120 months its return has correlation $0.6$ with the index's return in the same month. A critic says the correlation mostly reflects the stock being correlated with itself, since it sits inside the index. Which check answers the critic, and what did the video find for GM and the S&P 500?

✅ Correct! Removing the stock from the index removes any self-correlation; GM still moved with the other stocks.

❌ Not quite. Autocorrelation is the time-series question (one security, two periods). The critic's claim is about the same-month relation with the index.

❌ Not quite. Higher volatility says how much a stock fluctuates, not whether its co-movement with the index comes from its own weight in it.

❌ Not quite. That round scatter answers a different question — the whole market, last period against this period — not one stock against the index in the same month.

Show solution

The critic's worry is mechanical: if the stock is part of the index, part of the index's return is the stock's return. The direct test removes that part.

Build the index without the stock and recompute the same-month correlation.
If the relation came only from self-correlation, it would vanish. For GM against the S&P 500 it did not: the same positive pattern remained, because GM moves with the other stocks.

The time-series scatters are a separate matter: they were whole-market portfolios, one portfolio in two periods, and there the cloud is round.

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