6B.6 SIPs and rupee-cost averaging · M06B Funds and ETFs

Motivation

The SIP is how most new Indian investors buy funds: more than ten crore SIP accounts contributed in September 2026 (6B.5.4). It is often sold with the claim that averaging lowers your cost. The claim is true in a narrow sense and false in the sense that matters. Both halves can be proved.

Average cost as a harmonic mean#

The mechanism A SIP is a standing instruction: on a fixed date each month the bank account is debited by a fixed amount $P$ , and units are allotted at that day's applicable NAV (6B.2). Units bought in month $i$ are $P/p_i$ . M00 (0.2) gives the future value at a constant return.

Proposition 6B.6.1average cost is the harmonic mean

If $P$ is invested at each of the prices $p_1, \dots, p_n > 0$ , the average cost per unit is the harmonic mean of the prices, and it does not exceed their arithmetic mean:

\bar c = \frac{nP}{\sum_{i}P/p_i} = \frac{n}{\sum_i 1/p_i} = H(p) \;\le\; \frac1n\sum_i p_i = A(p), \tag{6B.14}

with equality if and only if all prices are equal. For prices $p_i = A(1 + \delta_i)$ with small dispersion, $\sum_i \delta_i = 0$ and $\sigma_\delta^2 = \frac1n\sum_i\delta_i^2$ ,

H(p) \approx A(p)\,\big(1 - \sigma_\delta^2\big). \tag{6B.15}

Proof

The first equalities are definitions: $nP$ rupees bought $\sum_i P/p_i$ units. By the Cauchy–Schwarz inequality, $\big(\sum_i \sqrt{p_i}\cdot 1/\sqrt{p_i}\big)^2 \le \sum_i p_i \sum_i 1/p_i$ , that is $n^2 \le nA \sum_i 1/p_i$ , which rearranges to $H \le A$ ; equality holds iff $\sqrt{p_i}$ is proportional to $1/\sqrt{p_i}$ , that is iff all $p_i$ are equal. For (6B.15), $1/(1 + \delta) = 1 - \delta + \delta^2 + O(\delta^3)$ , so $\frac1n\sum_i 1/p_i = \frac1A(1 + \sigma_\delta^2 + \dots)$ and $H = A/(1 + \sigma_\delta^2 + \dots) = A(1 - \sigma_\delta^2) + \dots$ ∎

Worked example 6B.8

₹10,000 a month at NAVs of 100, 80, 125 and 100 buys $100 + 125 + 80 + 100 = 405$ units for ₹40,000: an average cost of ₹98.77, below the average NAV of ₹101.25. More units are bought when the price is low.

Averaging versus a lump sum#

What the proposition does not say $H \le A$ compares a SIP with a strategy that buys the same number of units each month. It does not compare a SIP with investing the money at once, which is the decision an investor with a lump sum actually faces.

Proposition 6B.6.2averaging is not an edge over a lump sum

An investor has $W$ at time 0. The risky fund has expected gross return $1 + \mu$ per period, cash earns $1 + r_f$ , and $\mu > r_f$ . Strategy L invests $W$ at once. Strategy S keeps the money in cash and invests $W/n$ at times $0, 1, \dots, n-1$ . With returns independent across periods, expected wealth at time $n$ is

\mathbb E[W_L] - \mathbb E[W_S] = \frac{W}{n}\sum_{j=0}^{n-1}\Big[(1+\mu)^{n} - (1 + r_f)^{j}(1+\mu)^{n-j}\Big] > 0. \tag{6B.16}

Proof

Under L, $\mathbb E[W_L] = W(1+\mu)^n$ because the expectation of a product of independent returns is the product of expectations. Under S, the tranche invested at time $j$ sat in cash for $j$ periods and in the fund for $n - j$ , so its expected value is $(W/n)(1+r_f)^j(1+\mu)^{n-j}$ . Each bracket is non-negative because $(1+r_f)^j \le (1+\mu)^j$ , and it is strictly positive for $j \ge 1$ .∎

S has lower variance than L, but only because it holds less equity on average: for most of the averaging period much of the money is in cash. Constantinides (1979) made the stronger point. A rule that fixes the purchase schedule in advance ignores information that arrives along the way (the realised prices and wealth). For any such schedule there is a policy that conditions on the current state and does at least as well for any risk-averse investor, so dollar-cost averaging as a decision rule for a lump sum is dominated. Sequential averaging buys a lower risk exposure, not a lower cost.

Worked example 6B.9

$W = \text{₹}12$ lakh, $\mu = 1\%$ a month, $r_f = 0.5\%$ a month, $n = 12$ . Lump sum: $\mathbb E[W_L] = \text{₹}13{,}52{,}190$ . Averaging over the year: ₹13,15,974. The expected cost of averaging is ₹36,216, the price of holding cash for an average of 5.5 months.

The behavioural case for SIPs#

What SIPs really buy Most salaried investors do not have a lump sum: they have a monthly surplus. For them a SIP is not an alternative to a lump sum; it is investing each rupee when it is earned, which is what Proposition 6B.6.2 recommends. The real benefits are behavioural and operational: automation removes the monthly decision (and with it the temptation to time the market), and the instruction keeps running through a bear market, when stopping is most expensive. Measure a SIP's result with XIRR, M00 (0.3), never with "absolute return".

Practitioner notes

Received a bonus, an ESOP sale or an inheritance? The default from Proposition 6B.6.2 is to invest it according to your target allocation now. If you spread it over a few months to manage regret, call the cost what it is: a temporary underweight in equity.
SIP dates do not matter in expectation. Spreading one SIP over four weekly dates changes nothing but the paperwork.
A step-up SIP (raising the amount each year) is an income-matching device, not a return enhancer.