Risk Management puzzles, solved step by step
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037Forecaster A has a bias of 1 point and a forecast error standard deviation of 2. Forecaster B is unbiased with a standard deviation of 2.5. Using the mean squared error decomposition, which forecaster is better?BlackRockNew York · 2026
Try it first
Which forecaster has the lower mean squared error?
Show the worked solution
Forecaster A, with a mean squared error of 5 against 6.25. Mean squared error splits into bias squared plus variance. A pays 1 squared for its bias and 2 squared for its spread, 5 in all. B pays nothing for bias but 2.5 squared, 6.25, for its spread. A's typical error, the square root, is 2.24 against 2.50.
How can a biased forecaster beat an unbiased one?
Two archers. One groups every arrow tightly but slightly left of centre; the other is centred on average but scatters arrows all over the target. Ask which one lands closer to the bullseye on a typical shot, and the tight grouping wins. Mean squared error charges for two things, how far off you are on average and how much you scatter, and a small, steady bias can cost far less than a large scatter. Being unbiased only removes the first charge.
The relationshipbias the average forecast error, forecast minus actual variance the spread of the errors around their own average, the standard deviation squared What it says in wordsSquared error on average equals the squared average error plus the spread of errors around it.Forecaster A's mean squared error is 1 of bias squared plus 4 of variance, 5 in total, while unbiased Forecaster B carries 6.25 of pure variance, so A's small bias buys a larger cut in variance and gives the lower error. When would your answer flip, and what would you do with A?
Solve for the tie: A matches B when bias squared plus 4 equals 6.25, so a bias of 1.50. Below that, A wins. More useful still, a bias that is stable can be measured and subtracted: correct A by one point and its MSE falls to 4, better than either original. That is the practical lesson for a risk team: a model that is consistently off in one direction is fixable, while a noisy model is not. The trade-off is also why risk teams use shrinkagePulling a noisy estimate towards a simpler, steadier target, accepting a little bias in return for much lower variance. on covariance matrices built from short histories.
The limit: MSE punishes large errors heavily because it squares them, and it treats over-forecasts and under-forecasts alike. A risk manager forecasting losses may care more about under-forecasting than over-forecasting, in which case a symmetric score is the wrong yardstick and the ranking could change.
Where candidates lose it
Candidates pick B on reflex because unbiased sounds like correct. The question is built to see whether you know that MSE has two parts and can do the two-line arithmetic.
The quieter miss is stopping at 5 against 6.25. Add that A's bias can be corrected, taking its MSE to 4, and you have turned a statistics answer into a model-risk judgement.
What the interviewer asks next
- What bias would make the two forecasters exactly equal?
- Why might a regulator prefer the unbiased forecaster even with a higher MSE?
- How would you test whether A's bias is stable over time?
Asked at BlackRock, Restructuring, New York, 2026 (Wall Street Oasis):
Which equities have duration ? multiple stocks vs value stocks MSE Forecasting equation
048You estimate a desk's daily P&L variance from five observations, once dividing the sum of squared deviations by 5 and once by 4. Which estimator is unbiased, which is consistent, and how large is the bias?UBSZurich · 2021
Try it first
Which statement is true?
Show the worked solution
Dividing by 4 is unbiased; both estimators are consistent; dividing by 5 is low by one fifth. The sample mean is estimated from the same five points, which uses up one degree of freedom, so the divide-by-n estimator averages (n - 1)/n of the true variance, 80% here. If the true variance is 4, it centres on 3.2, a bias of -0.8. As n grows that factor tends to 1, so both estimators converge on the truth.
Why does dividing by n come out too low?
Measure how spread out five friends' heights are by comparing each to the group's own average, and you will understate the spread, because that average was pulled towards those five people. Deviations measured from the sample mean are smaller on average than deviations from the true mean, so their sum of squares understates the spread by exactly one observation's worth. Dividing by n minus 1, the degrees of freedomThe number of independent pieces of information left after estimating something from the same data; estimating the mean uses up one., corrects it exactly.
With five observations and a true variance of 4, the divide-by-4 estimator is centred on 4 while the divide-by-5 estimator is centred on 3.2, 80% of the truth, yet the biased version is narrower and has a lower mean squared error, 5.76 against 8.00. What is the difference between unbiased and consistent?
Unbiased is about the average over many repeated samples of the same size; consistent is about what happens to one estimate as the sample grows. Divide-by-5 fails the first: repeat the five-day exercise many times and the estimates average 3.2, not 4. It passes the second: with 250 days the bias is only -0.016, and it shrinks to zero with more data. An estimator can be unbiased but inconsistent too, such as using only the first observation to estimate a mean: right on average, never improving.
The relationshipn the number of observations, 5 \bar{x} the sample mean, estimated from the same five points \sigma^2 the true variance, 4 in the illustration What it says in wordsDividing by n recovers only (n minus 1) over n of the true variance on average.Now the twist a model validator should add. For normal data the unbiased estimator has variance 8.00 here, while the divide-by-5 version has 5.12 plus a squared bias of 0.64, a mean squared error of 5.76. The biased estimator is closer to the truth on a typical sample. Which you prefer depends on the use: unbiasedness matters when estimates are averaged across many desks; a smaller typical error matters for a single desk's limit. With five data points, neither is reliable, and that is the more important thing to say.
Where candidates lose it
The usual slip is to treat unbiased and consistent as the same thing, and so to call the divide-by-5 estimator inconsistent. The interviewer asked both words together precisely to hear you separate them.
The second miss is answering from memory without the reason. One sentence on the sample mean using up a degree of freedom shows you know why n minus 1 exists, not just that it does.
What the interviewer asks next
- Give an example of an estimator that is unbiased but not consistent.
- Is the sample standard deviation, the square root of the unbiased variance, itself unbiased?
- With 250 days of P&L, does the choice between n and n - 1 matter for VaR?
Asked at UBS, Risk Management, Zurich, 2021 (Wall Street Oasis):
And several other questions on econometrics - what is an unbiased estimator vs consistent estimator?
062Your prior estimate of a hidden fair price is 10 with variance 4. A noisy measurement comes in at 12 with measurement variance 1. After one Kalman filter update, what is your new estimate and its variance?UBSLondon · 2022
Try it first
Where does the new estimate land?
Show the worked solution
The new estimate is 11.6 with variance 0.8. The Kalman gain is the prior variance over the total, 4 over 5, or 0.8. The estimate moves 80% of the way from 10 towards 12, landing at 11.6. The variance becomes (1 - 0.8) x 4 = 0.8, smaller than either the prior's 4 or the measurement's 1.
How does the filter decide how far to move?
Two friends guess your commute time. One has ridden with you a hundred times, the other once. You would average their guesses, but lean heavily on the first. A Kalman filterA method that updates an estimate of something you cannot see directly each time a noisy measurement arrives, weighting old estimate and new data by how precise each is. does exactly that: it weights the prior and the measurement by their precision, one over variance. Precision 0.25 against 1 gives the measurement 80% of the weight.
The prior centred at 10 is wide, with variance 4, and the measurement at 12 is narrow, with variance 1. The update lands at 11.6, four fifths of the way to the measurement, and its variance of 0.8 makes it narrower than either source. Why is the new variance smaller than both inputs?
Because two independent pieces of evidence together know more than either alone. Precisions add: 1 over 4 plus 1 over 1 is 1.25, and one over 1.25 is a variance of 0.8. That is the part candidates skip. The filter does not just move the estimate; it becomes more confident with every measurement, until new data carry little weight and the estimate settles.
The relationshipP the prior variance, 4 R the measurement variance, 1 K the Kalman gain, the weight on the new measurement \hat{x} the updated estimate What it says in wordsMove from the prior towards the measurement by the gain, and shrink the variance by the same share.Say the limitation: this single step assumes both errors are normal and the hidden price did not move between the prior and the measurement. A full filter adds a prediction step that lets the price drift and widens the variance before each update. Risk teams use the idea to track hidden quantities such as a hedge ratio that changes over time.
Where candidates lose it
The trap is averaging the two numbers and answering 11. That treats the measurement and the prior as equally trustworthy, which the variances say they are not.
The second trap is getting 11.6 and then saying the variance is somewhere between 1 and 4. Combining evidence always reduces uncertainty, so the new variance must be below both: 0.8.
What the interviewer asks next
- A second measurement of 11 arrives with variance 1. What is the estimate now?
- What happens to the gain as the number of measurements grows?
- How would you use a Kalman filter to estimate a time-varying hedge ratio?
Asked at UBS, Risk, London, 2022 (Wall Street Oasis):
Explain what kalman filter is.
