Estimation risk
The potential for error that comes from having to estimate the returns, risks and correlations that feed portfolio optimisation — the output is only as accurate as those statistical inputs.
In plain language
Modern Portfolio Theory is exact mathematics wrapped around guesses.
To build an optimum portfolio, the manager has to estimate the expected return of every asset, the standard deviation of each, and the correlation between every pair. None of those numbers can be looked up. They are forecasts.
The workbook's warning (section 3.9) is short: the output of the portfolio allocation depends on the accuracy of the statistical inputs. The potential source of error that arises from these estimations is estimation risk.
It is the risk that the portfolio looks efficient on the spreadsheet and is not efficient in the world — because a correlation of 0.2 was really 0.6, or an expected return of 15% was really 9%.
How it works
The problem grows much faster than the portfolio. For n securities, the manager needs:
- n expected returns,
- n standard deviations, and
- (n² − n) ÷ 2 correlation estimates — one for every pair.
The workbook's figure: a portfolio of 50 securities needs 1,225 correlation estimates.
(50² − 50) ÷ 2 = (2,500 − 50) ÷ 2 = 1,225
| Securities | Correlations to estimate |
|---|---|
| 10 | 45 |
| 20 | 190 |
| 50 | 1,225 |
| 100 | 4,950 |
Double the universe from 50 to 100 and the correlation count roughly quadruples. Every one of those is a chance to be wrong.
For portfolios with more than two securities, the workbook says the calculation needs a variance-covariance matrix, which can be computed on a spreadsheet — but a spreadsheet will faithfully optimise whatever errors it is given.
The formula
Number of correlation estimates = (n² − n) ÷ 2
Total inputs for n securities = n (returns) + n (standard deviations) + (n² − n) ÷ 2 (correlations)
A worked example
Inputs follow the workbook's Table 3.4; the two alternative correlations are illustrative.
A Category III AIF manager is sizing a Rs 100 crore book split equally between two stocks, each with expected return 15% and standard deviation 5%. The analyst estimates their correlation at 0.20. In reality it turns out to be 0.60.
σ²p = 0.25(0.0025) + 0.25(0.0025) + 2(0.5)(0.5)(r)(0.05)(0.05)
= 0.00125 + 0.00125r
Estimated (r = 0.20): σ²p = 0.00150 → σp = 3.87%
Actual (r = 0.60): σp² = 0.00200 → σp = 4.47%
On Rs 100 crore, a one-standard-deviation move is Rs 3.87 crore in the plan and Rs 4.47 crore in reality — about 15% more risk than the manager thought they were running, with no change at all to expected return.
Now scale it. A 50-stock book needs 1,225 such correlations. If a meaningful share are under-estimated in the same direction — which tends to happen in a crisis, when correlations rise together — the optimiser will have loaded up on "diversification" that is not there.
The inputs also include 50 return forecasts and 50 standard deviations. The optimiser treats all 1,325 numbers as facts.
Why NISM asks about it
Chapter 3, section 3.9 (Estimation issues), names estimation risk and gives the 1,225 figure for 50 securities, with the (n² − n)/2 formula in a footnote. It follows section 3.8 on the optimisation inputs. Expect a numeric question ("how many correlation estimates for a portfolio of 50 securities?") and a conceptual one asking what estimation risk is the risk of.
Common exam traps
- Fifty securities means 1,225 correlations, not 2,500 and not 2,450. Each pair counts once and an asset is not paired with itself.
- Estimation risk is not market risk. It is the risk of a wrong input, not of a price move.
- It affects portfolio construction even if every calculation is correct. The arithmetic can be flawless and the output still wrong.
- The inputs are expected returns, standard deviations and correlations — plus any investor constraints. Beta and the risk-free rate belong to the later CAPM sections.
- More securities means more estimation risk, not less. Diversification reduces unsystematic risk but multiplies the estimates needed.
Check yourself
1.For a portfolio drawn from a universe of 50 securities, the number of correlation estimates required is:
- a)50
- b)100
- c)1,225
- d)2,500
Show the answer
Answer: (c) 1,225
For a portfolio of 50 securities, the number of correlation estimates is 1,225, computed from the workbook's formula (n² - n) / 2:
(50² - 50) / 2 = (2,500 - 50) / 2 = 1,225
The formula counts distinct pairs - each of 50 securities pairs with 49 others, giving 2,450 ordered pairs, halved because the correlation of A with B is the same as B with A.
Put this alongside the other two inputs the manager must estimate: the expected return of every asset (50) and the standard deviation of each asset's expected returns (50). Correlations outnumber everything else by more than twelve to one.
Why that matters, in the workbook's own words: "It is important to bear in mind that the OUTPUT of the portfolio allocation DEPEND ON THE ACCURACY OF THE STATISTICAL INPUTS." And the danger has a name: the potential source of error that arises from these estimations is referred to as ESTIMATION RISK.
The computation itself is not the problem. Portfolios with more securities require calculation of a VARIANCE-COVARIANCE MATRIX, for which there are several methods, and which can be easily calculated on a spreadsheet or visual basic application. The difficulty is that every entry is a forecast.
Where this is taught
- Series XXI-B · Chapter 15: Introduction to Modern Portfolio Theoryintroduced here
- Series XIX-C · Chapter 4: Introduction to Modern Portfolio Theoryintroduced here
- Series XIX-E · Chapter 3: Introduction to Modern Portfolio Theory and Capital Market Theoryintroduced here
- Series X-A · Chapter 14: Introduction to Modern Portfolio Theoryintroduced here
Related terms
- CovarianceA measure of the degree to which the returns of two assets move together relative to their own averages — the building block of portfolio risk, standardised into the correlation coefficient.
- Efficient frontierThe umbrella-shaped curve of portfolios offering the highest expected return for each level of risk, or the lowest risk for each level of return — any portfolio below it is dominated.
- Modern Portfolio TheoryMarkowitz's framework for building portfolios on expected return and risk together, in which the co-movement between holdings — not their individual riskiness — decides the risk of the whole.
- Optimum portfolioThe combination of investments, chosen from the feasible portfolios, that has desirable risk-return characteristics within the investor's constraints and meets that investor's objectives.
- Variance-covariance matrixThe grid of every asset's variance and every pair's covariance that is needed to compute the risk of a portfolio holding more than two securities.
- Homogeneous expectationsThe Capital Market Theory assumption that all investors estimate identical probability distributions for future returns — one the workbook says can be relaxed with minor effects if differences are small.