Ex-post return
Also written Historical return · Realised return
A return calculated from historical data — what an investment actually earned — as distinct from the forward-looking, probability-weighted ex-ante return.
In plain language
"Ex post" means after the event. An ex-post return is not an estimate; it is the record.
The workbook draws the line in one sentence in Chapter 3: the probability-weighted expected return is ex-ante, while returns which are calculated based on historical data are called ex-post returns.
The distinction matters because the two are used for different jobs. Portfolio construction in Chapter 3 runs on forecasts — expected returns, standard deviations and correlations for the future. Fund evaluation in Chapter 7 runs on history — what a fund returned in each past year, how widely those returns varied, and whether the manager beat the benchmark.
How it works
Mean of historical returns (Chapter 7, Box 7.3). The average of the returns achieved each year:
Mean (R̄) = ΣRᵢ ÷ n
Historical risk (section 7.6.1). Standard deviation measures the volatility of the historical returns around that mean, and is used to ascertain the historical risk of the fund:
σ = √[ Σ(Rᵢ − R̄)² ÷ n ]
The workbook's Example 7.7:
| Yr 1 | Yr 2 | Yr 3 | Yr 4 | Yr 5 | |
|---|---|---|---|---|---|
| Fund A | 12.50% | 11.25% | 13.65% | 17.30% | 14.65% |
| Fund B | 21.50% | −0.50% | 16.75% | 20.25% | 21.90% |
Mean A = 69.35 ÷ 5 = 13.87% σ_A = 2.06%
Mean B = 79.90 ÷ 5 = 15.98% σ_B = 8.44%
Fund B returned more on average, but with about four times the dispersion. The workbook concludes that a prudent investor will choose Fund A.
Chapter 7 also warns of the limit: certain risks — macro-economic, legal and regulatory, operational, country-specific, cyber security — cannot be measured from past performance at all.
A worked example
The Rs 1 crore investment is illustrative; the yearly returns are the workbook's Example 7.7.
An investor put Rs 1 crore into each fund five years ago. Grow the money year by year at the actual returns:
Fund A: 1 × 1.1250 × 1.1125 × 1.1365 × 1.1730 × 1.1465 ≈ Rs 1.91 crore
Fund B: 1 × 1.2150 × 0.9950 × 1.1675 × 1.2025 × 1.2190 ≈ Rs 2.07 crore
Fund B ended about Rs 16 lakh ahead. So why does the workbook prefer Fund A?
Because the ex-post record also shows how the money got there. Fund B's investor sat through a year of −0.50% while Fund A never returned less than 11.25%. Per unit of historical risk, A delivered more: roughly 13.87 ÷ 2.06 ≈ 6.7 percentage points of mean return per point of standard deviation, against 15.98 ÷ 8.44 ≈ 1.9 for B.
A second lesson sits in the arithmetic. The mean of Fund A's yearly returns is 13.87%, but Rs 1 crore growing to Rs 1.91 crore over five years is a compound rate of about 13.85%; for Fund B the gap is wider — mean 15.98%, compound about 15.65% — because B's returns vary more.
Why NISM asks about it
Chapter 3, section 3.3, defines ex-post return in contrast with ex-ante return. Chapter 7, section 7.6 (Box 7.3 and Example 7.7), computes the mean and standard deviation of historical returns to judge a fund's risk, and Chapter 8 benchmarks realised fund returns against indices. Expect the ex-ante/ex-post definitional pair and a mean-and-standard-deviation computation from a table of past returns.
Common exam traps
- Ex-post uses historical data; ex-ante uses probabilities. A question that hands you five years of returns wants ex-post measures.
- Chapter 7 divides by n (the number of years) in its standard deviation formula. Use the workbook's formula.
- A higher historical mean is not automatically better. The workbook picks the lower-mean fund because its standard deviation is far lower.
- The arithmetic mean of yearly returns is not the compound growth rate — the more volatile the returns, the bigger the gap.
- Past returns do not capture every risk. The workbook lists macro-economic, legal and regulatory, operational, country-specific and cyber risks as not measurable from past performance.
Check yourself
1.A stock is forecast to return 15 per cent in a boom (probability 0.3), 10 per cent in normal conditions (probability 0.5) and 2 per cent in a recession (probability 0.2). Its expected return is:
- a)9.00 per cent
- b)9.90 per cent
- c)10.00 per cent
- d)27.00 per cent
Show the answer
Answer: (b) 9.90 per cent
Expected rate of return of an individual investment opportunity is the SUM OF THE EXPECTED RETURNS MULTIPLIED WITH THEIR CORRESPONDING PROBABILITIES.
R = 0.3(15%) + 0.5(10%) + 0.2(2%) = 4.5% + 5.0% + 0.4% = 9.9 per cent
This is the workbook's own Stock A from Table 3.1.
Check the probabilities first, always: 0.3 + 0.5 + 0.2 = 1.0. Any distribution failing this test is wrong before the arithmetic starts.
Option 3 is the trap of taking a simple average of 15, 10 and 2, which ignores that the three states are not equally likely. Option 4 is the sum of the returns with no weighting at all.
And note what the answer is not. The stock will return 15, 10 or 2 per cent - never 9.9. The expected return is the centre of the probability distribution, not a prediction of any single outcome. It is also an EX-ANTE figure, built from forecasts; returns which are calculated based on historical data are called EX-POST returns.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Holding period returnThe total of coupons, income earned on reinvesting them and any capital gain, expressed as a percentage of the purchase price — a crude return for the period actually held, with no compounding in it.
- RiskThe possibility that actual returns turn out different from what was expected — measured as the dispersion of returns around their own average, and not the same thing as uncertainty.
- Standard deviationA measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
- BenchmarkThe independently published index a scheme's performance is measured against, chosen to match its investment objective, asset allocation and strategy, and disclosed in the Scheme Information Document.
- Ex-ante returnThe forward-looking expected rate of return: the sum of the possible returns multiplied by their probabilities. The opposite of ex-post return, which is computed from historical data.