Learn about SIP vs lumpsum investing

WHY does this choice even matter?

Markets go up and down. The price at which you buy determines your future return. If you could buy only at the bottom, you'd choose lumpsum at that instant. But nobody knows the bottom in advance — this is the problem of timing the market.

• Lumpsum bets that now is a good time (or that time-in-market beats timing).
• SIP refuses to bet on timing; it spreads purchases so no single price dominates.

HOW does rupee-cost averaging actually work? (Derivation)

WHAT we want: the average price you effectively pay per unit under SIP.

Suppose you invest a fixed amount $A$ each period, and the unit price (NAV) in period $i$ is $p_i$.

Step 1 — units bought each period. $u_i = \frac{A}{p_i}$ Why this step? A fixed rupee amount $A$ divided by price $p_i$ gives units. Low $p_i$ ⇒ large $u_i$.

Step 2 — total units after $n$ periods. $U = \sum_{i=1}^{n} \frac{A}{p_i} = A\sum_{i=1}^{n}\frac{1}{p_i}$

Step 3 — total money invested. $M = nA$

Step 4 — average cost per unit = money spent ÷ units received: $\bar{p}_{\text{SIP}} = \frac{M}{U} = \frac{nA}{A\sum \frac{1}{p_i}} = \frac{n}{\sum_{i=1}^{n}\frac{1}{p_i}}$

This is the ==harmonic mean== of the prices!

Step 5 — the inequality (why SIP helps in a volatile market). By the AM–HM inequality, for positive prices: $\text{HM} \le \text{AM}, \quad\text{equality iff all } p_i \text{ equal.}$ So $\bar{p}_{\text{SIP}} \le \bar{p}_{\text{AM}}$. SIP's average cost never exceeds the simple average of prices, and is strictly lower whenever prices vary. This is the mathematical heart of rupee-cost averaging.

Worked Examples

Steel-manned Mistakes

The 80/20 takeaway

Flashcards

#flashcards/stock-market

Connections

• Rupee-Cost Averaging
• Mutual Funds and NAV
• ETFs and Index Funds
• Time in Market vs Timing the Market
• Compounding and Power of Long-Term Investing
• AM-GM-HM Inequalities
• Dollar-Cost Averaging
• Behavioral Finance and Investor Discipline

Concept Map

Hinglish (regional understanding)