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Product notation

Read finite products, indices and bounds; compound successive returns using gross factors; distinguish an empty product from zero and products from sums.

Product notation writes a finite multiplication compactly using the capital Greek letter pi, ∏\prod. For a scalar sequence,
∏i=mnai=amam+1⋯an,m≤n.\prod_{i=m}^{n}a_i=a_m a_{m+1}\cdots a_n, \qquad m\le n.
The integer index ii moves upward in steps of one; the lower and upper bounds are inclusive, so there are n−m+1n-m+1 factors. The expression aia_i tells you which factor to use. This is the multiplication counterpart of summation notation, which uses ∑\sum to add terms.
The index is a dummy variable: replacing every bound occurrence of ii with jj changes no value. Changing the bounds does change which factors enter. For example,
∏t=23(1+rt)=(1+r2)(1+r3).\prod_{t=2}^{3}(1+r_t)=(1+r_2)(1+r_3).
There are two factors here, not three. Expand a short product before calculating whenever the indexing is unfamiliar. The University of Western Australia's notation chapter develops this finite-product convention.

Three returns, three changing bases

An investment starts with USD 100 and earns simple returns of +10%, −10% and +5% over three consecutive periods. Assume no deposits or withdrawals, no distributions, and no costs. Each return acts on the value at the start of its own period:
PeriodSimple return rtr_tGross factor 1+rt1+r_tEnding wealth, USD
1+10%1.10110.00
2−10%0.9099.00
3+5%1.05103.95

The wealth recursion Vt=Vt−1(1+rt)V_t=V_{t-1}(1+r_t) gives

V3=V0∏t=13(1+rt)=100(1.10)(0.90)(1.05)=103.95.V_3=V_0\prod_{t=1}^{3}(1+r_t) =100(1.10)(0.90)(1.05)=103.95.
The product is 1.0395, a dimensionless growth multiple. Subtract one to obtain the cumulative simple return: 3.95%. Multiplying the percentages themselves would give a completely different quantity.
Synthetic wealth path in USD: 100 at the start, 110 after plus 10%, 99 after minus 10%, and 103.95 after plus 5%. The respective dollar changes are plus 10, minus 11 and plus 4.95. Adding the simple returns instead implies 105.
Synthetic wealth path in USD: 100 at the start, 110 after plus 10%, 99 after minus 10%, and 103.95 after plus 5%. The respective dollar changes are plus 10, minus 11 and plus 4.95. Adding the simple returns instead implies 105.
Original synthetic example. The gross factors multiply to 1.0395; adding the simple returns gives 5% and overstates cumulative return by 1.05 percentage points.
Why don't +10% and −10% cancel? The loss is 10% of USD 110, or USD 11, after a gain of USD 10. Their two-period product is 1.10×0.90=0.991.10\times0.90=0.99. Compound interest uses the same mechanism: successive percentage changes act on a changing base. The university notes by Heng Peng at HKBU derive multiperiod returns from products of one-period gross returns.
This is aggregation through time for one investment. For different assets held over the same period, fixed holdings and beginning-of-period value weights give a weighted sum of simple total returns. Multiplying those assets' gross returns does not produce the portfolio return.

Distributions require an explicit total-return convention, including reinvestment timing, before linking periods. External deposits and withdrawals must be separated from investment performance; multiplying returns alone will not account for those added or removed dollars.

The product before the first period

An empty product equals one:
∏t=10(1+rt)=1.\prod_{t=1}^{0}(1+r_t)=1.
With no periods elapsed, wealth should remain V0×1=V0V_0\times1=V_0. Zero would erase the starting capital. Empty means no factors, rather than a factor whose value is zero: any product containing a zero factor is zero. Equal bounds select one factor, so ∏i=mmai=am\prod_{i=m}^{m}a_i=a_m.

For a positive integer nn, two useful algebra checks are

∏i=1n(cai)=cn∏i=1nai,∏i=1n(aibi)=(∏i=1nai)(∏i=1nbi).\prod_{i=1}^{n}(ca_i)=c^n\prod_{i=1}^{n}a_i, \qquad \prod_{i=1}^{n}(a_i b_i) =\left(\prod_{i=1}^{n}a_i\right)\left(\prod_{i=1}^{n}b_i\right).
A constant inside every factor appears nn times. These are finite scalar products; matrix multiplication needs an ordering convention and does not generally allow reordering.

When multiplication becomes addition

For strictly positive factors, the logarithm gives
ln⁡(∏t=1n(1+rt))=∑t=1nln⁡(1+rt).\ln\left(\prod_{t=1}^{n}(1+r_t)\right) =\sum_{t=1}^{n}\ln(1+r_t).

Thus log returns add through time, while simple returns compound in gross form. Recover the cumulative simple return by exponentiating the sum and subtracting one. The real-log identity requires every rt>−1r_t>-1. A −100% return makes the gross product zero; it does not give a finite log return. This algebraic identity needs no assumption that returns are independent.

For a guided explanation of the log laws, see Exponentials and Logarithms. For the distinction between time compounding and portfolio aggregation, read Financial Return Characteristics. Both provide public previews; their full lessons require membership.

Sources

  • University of Western Australia, Summation and Product Notation, Chapter 10, product-notation section — finite products, indices and the empty-product convention.
  • Heng Peng, Hong Kong Baptist University, Financial Time Series and Their Characteristics, 17 January 2021, “Multiperiod Simple Return,” “Continuously Compound Return” and “Portfolio Return.”
  • John C. Hull, Options, Futures, and Other Derivatives, 9th Global Edition, §4.2, “Measuring Interest Rates” — reinvestment and compounding conventions.
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  • finite products
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  • logarithms
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