Understand systematic vs unsystematic risk — Stock-Market notes

WHAT are we splitting?

HOW to derive the split (from scratch)

Start from the model $R_i = \alpha_i + \beta_i R_m + \varepsilon_i$ .

Step 1 — Take the variance of both sides. $\sigma_i^2 = \text{Var}(\alpha_i + \beta_i R_m + \varepsilon_i)$ Why this step? $\alpha_i$ is a constant, so it adds nothing to variance ( $\text{Var}(c+X)=\text{Var}(X)$ ).

Step 2 — Use independence of $\varepsilon_i$ and $R_m$ . Since $\text{Cov}(\varepsilon_i, R_m)=0$ , variance of the sum = sum of variances: $\sigma_i^2 = \beta_i^2\,\text{Var}(R_m) + \text{Var}(\varepsilon_i)$ Why? $\text{Var}(aX+Y) = a^2\text{Var}(X)+\text{Var}(Y)$ only when the cross-covariance is zero.

HOW diversification kills unsystematic risk

Build an equally weighted portfolio of $N$ stocks, weight $w_i = 1/N$ . Portfolio return: $R_p = \frac{1}{N}\sum_i R_i = \frac{1}{N}\sum_i(\alpha_i+\beta_i R_m+\varepsilon_i)$ .

The firm-specific piece of the portfolio is $\bar\varepsilon = \frac{1}{N}\sum_i \varepsilon_i$ .

Step 1 — Variance of the average noise. Assume each $\varepsilon_i$ has variance $\bar\sigma_\varepsilon^2$ and they're uncorrelated: $\text{Var}(\bar\varepsilon) = \frac{1}{N^2}\sum_{i=1}^{N}\sigma_{\varepsilon_i}^2 = \frac{1}{N^2}\cdot N\bar\sigma_\varepsilon^2 = \frac{\bar\sigma_\varepsilon^2}{N}$ Why this step? Independent terms → variances add; the $1/N^2$ front factor comes from squaring the weight.

Step 2 — Take the limit. $\lim_{N\to\infty} \frac{\bar\sigma_\varepsilon^2}{N} = 0$

Figure — Understand systematic vs unsystematic risk

Worked examples

Common mistakes

Active recall

What are the two components of total risk?

Systematic (market) risk + unsystematic (firm-specific) risk.

Which component can be diversified away?

Unsystematic (idiosyncratic) risk — it's independent across firms and averages to zero.

Give the risk-decomposition formula.

\sigma_i^2 = \beta_i^2\sigma_m^2 + \sigma_{\varepsilon_i}^2

Why does unsystematic risk vanish in a large portfolio?

Independent shocks average out; variance of average noise

=\bar\sigma_\varepsilon^2/N \to 0

N\to\infty

Which risk earns a risk premium and why?

Only systematic risk — you can't avoid it, so the market pays you to bear it. Diversifiable risk earns nothing.

What measures a stock's systematic risk sensitivity?

Its beta,

\beta_i = \text{Cov}(R_i,R_m)/\text{Var}(R_m)

Does diversification remove ALL risk?

No — the systematic floor

\beta_p^2\sigma_m^2

always remains.

Why does adding the 30th stock help less than the 3rd?

Risk reduction scales as

1/N

, so benefits diminish; ~20–30 stocks capture most of it.

Recall Feynman: explain to a 12-year-old

Imagine a class where everyone's grade depends on two things: (1) whether the whole school has a good or bad year (a big storm cancels exams for everyone) and (2) each kid's own luck (you got sick the day of one test). If you average the grades of the whole class, personal bad-luck days cancel out — some kids were sick, some were lucky. But if the storm hit the whole school, averaging doesn't help — everyone was hurt together. Own luck = unsystematic risk (goes away in a group). The storm = systematic risk (stays). And you only get "rewarded" for surviving the storm, because dodging personal bad luck was free (just join the group).

Connections

Beta and CAPM — beta is the measure of systematic risk; CAPM prices it.
Diversification and Correlation — low correlation is what lets specific shocks cancel.
Portfolio Variance and Covariance — the general two-asset variance formula this specializes.
Efficient Frontier — built by diversifying away unsystematic risk.
Security Market Line — plots return vs beta (systematic risk), not total risk.
Sharpe Ratio — rewards return per unit of total risk; contrast with CAPM's beta.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, kisi bhi stock ka total risk do hisson mein banta hai. Ek hota hai systematic risk yaani market risk — jaise interest rate badhna, recession, war, inflation. Ye poori economy ko hilata hai, isliye chahe aap 5 stock lo ya 500, ye risk kabhi khatam nahi hota. Doosra hota hai unsystematic risk — ye kisi ek company ka apna problem hai, jaise CEO ka scandal, factory mein aag, ya kisi product ka fail hona. Ye company-specific hota hai.

Ab magic ye hai: agar aap bahut saare alag-alag companies ke stock ek saath rakho (diversify karo), to har company ki apni bad luck aur good luck aapas mein cancel ho jaati hai. Formula se dekho — average noise ka variance $\bar\sigma_\varepsilon^2 / N$ hota hai, aur jaise-jaise $N$ badhta hai ye zero ki taraf jaata hai. Matlab unsystematic risk diversification se free mein khatam ho jaata hai. Lekin systematic risk (market ka floor) waise ka waise reh jaata hai.

Isliye ek bahut important baat: market sirf systematic risk ka paisa (premium) deta hai. Kyunki jo risk aap free mein hata sakte ho, uske liye koi extra return nahi milega — koi bewakoof nahi jo aisi cheez ke liye pay kare. Yahi se poora CAPM aur beta ka concept nikalta hai: return depend karta hai stock ke beta par, na ki uske total volatility par.

Ek aur cheez yaad rakho — diversification ke fayde $1/N$ ke hisaab se ghatte hain. Pehle 20-30 stocks tak risk bahut kam hota hai, uske baad 40th ya 50th stock add karne se koi khaas farak nahi padta. Toh smartly ~20-30 stocks mein hi kaafi diversification mil jaata hai.

Test yourself — Portfolio Theory

Connections