Private Wealth Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 3
Topics: 13
Hard: 30

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006Two assets each have a volatility of 20% a year, and their returns are uncorrelated. What is the volatility of a portfolio split 50/50 between them?Probability and risk of lossCorePrivate banking
Try it first
Pick the volatility of the 50/50 mix.
Show the worked solution
About 14.1%. Each asset at a 50% weight contributes 10% of volatility. With zero correlation, variances add rather than volatilities, so portfolio variance is 0.1 squared plus 0.1 squared, 0.02, and volatility is its square root, 14.14%. The mix is less risky than either asset alone, with no loss of expected return if both assets expect the same.
Why is the answer not simply 20%?
Walk ten steps east and then ten steps north. You have walked twenty steps, but you are only about fourteen steps from where you started, because the two legs point in different directions. Uncorrelated risks are like steps at right angles: they add by Pythagoras, not in a straight line. Only if the two assets always moved together, correlation of one, would the steps line up and the risks add to the full 20%.
Each asset at half weight carries 10% of risk, and because the two are uncorrelated the risks combine at right angles to 14.1%, against 20% if they moved together and zero if they moved exactly opposite.
How do you write the formula without getting lost?
Work in varianceVolatility squared. Variances of independent returns add, which is why risk calculations square first and take the root last. and take the square root at the end. With zero correlation the cross term vanishes, so portfolio variance is just each weight squared times each variance, summed. That is 0.25 times 0.04, twice, which is 0.02, and the square root of 0.02 is 0.1414. A general rule falls out: n equal, uncorrelated assets of the same volatility give that volatility divided by the square root of n.
The relationship
$\sigma_p = \sqrt{w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\,\sigma_1\sigma_2} = \sqrt{0.01 + 0.01 + 0} \approx 14.1\%$
w each asset's weight, 0.5
sigma each asset's volatility, 20%
rho the correlation between the two, here zero
What it says in wordsSquare the weighted risks, add them, add the correlation term, and take the square root.
Say the limitation, because a private banking interviewer will push on it. Correlations measured in calm markets tend to rise in a crisis, when many assets fall together. The 14.1% is the answer to the question as set; a client portfolio needs a stress view as well as the average one.
Where candidates lose it
The fast wrong answer is 20%, averaging the volatilities. It silently assumes perfect correlation and throws away the whole point of diversification, which is the one idea a wealth desk expects you to own.
The second trap is overshooting to zero. Uncorrelated is not opposite: the risks partly cancel, they do not vanish. Only a correlation of minus one takes the mix to zero.
What the interviewer asks next
What is the volatility of the 50/50 mix if the correlation is 0.5?
With four such uncorrelated assets in equal weights, what is the portfolio volatility?
Why do correlations tend to rise in a market crash, and what does that do to this answer?
044A client holds 10 unrelated stocks. Each has a 5% chance of going to zero this year, independently of the others. What is the chance that at least one of them goes to zero?Probability and risk of lossCoreWealth management
Try it first
Pick the closest answer.
Show the worked solution
About 40%. The chance a single stock survives is 95%. The chance all ten survive, if they are independent, is 0.95 to the power 10, which is 0.599. So the chance that at least one goes to zero is 1 minus 0.599, or 40.1%. Small risks that each look negligible add up quickly across a portfolio.
Why work with the chance that nothing goes wrong?
If each of ten wedding vendors has a 5% chance of letting you down, the day goes perfectly only if every one of them turns up. The chance that at least one thing fails is one minus the chance that everything works, and the chance that everything works is the product of each piece working. Adding the 5%s does not work, because it counts the unlucky years where two vendors fail twice.
Each stock added multiplies the chance of no zero by 0.95, so after ten stocks it has fallen to 0.599. The red share above the last bar shows a 40.1% chance that at least one stock goes to zero this year.
The relationship
$P(\text{at least one}) = 1 - (1-p)^{n} = 1 - 0.95^{10} = 0.401$
p the chance any one stock goes to zero, 5%
n the number of independent stocks, 10
What it says in wordsThe chance of at least one failure is one minus the chance that every stock survives.
Does this mean the portfolio is riskier with more stocks?
No, and the distinction is the adviser's real point. The chance of seeing at least one zero rises with the number of stocks, but the damage from each zero shrinks, because each stock is a smaller slice. Ten stocks at 10% each expect 0.5 zeros a year, costing about 5% of the portfolio on average, and the chance of two or more zeros is only 8.6%. With one stock the same 5% chance means a 5% chance of losing everything.
The limit is the word unrelated. Real stocks fail together in a crisis, which makes a year with several zeros more likely than independence suggests, and a year with none also more likely. With 30 stocks at the same odds, the chance of at least one zero rises to 79%, which is why a diversified client should expect to see a disaster in the statement sometimes.
Where candidates lose it
The two fast wrong answers are 5%, ignoring that there are ten chances, and 50%, adding ten 5%s. The second one fails loudly if the interviewer asks about 25 stocks, where addition gives 125%.
Go straight to the complement, give 40%, and then say what it means for the client: expect to see a loser, but diversification caps how much any one loser costs.
What the interviewer asks next
How many such stocks before the chance of at least one zero passes 90%?
What is the chance of exactly one zero among the ten?
How does correlation between the stocks change the answer?
070A client's Rs 5 crore portfolio has an expected return of 8% a year and volatility of 15%. What loss should he be ready for in a one-in-twenty bad year?Probability and risk of lossCorePrivate banking
Try it first
Roughly how much could a one-in-twenty year cost on Rs 5 crore?
Show the worked solution
A loss of about Rs 84 lakh, or -16.75%, and in some years more. If returns are roughly normal, one year in twenty falls more than 1.65 volatilities below the average. That is 8% minus 1.65 x 15%, or -16.75%, which on Rs 5 crore is about Rs 83.75 lakh. It is a threshold, not a worst case: the average year beyond it loses about 23%.
Why translate volatility into rupees?
A weather forecast that says the standard deviation of rainfall is 40 mm helps nobody pack; a forecast that says one monsoon in twenty floods the ground floor tells a family what to prepare for. A client cannot feel 15% volatility, but he can feel a Rs 84 lakh loss in one bad year, and he can decide whether he could live with it. The job is to turn the statistic into a rupee figure and a frequency.
With an 8% expected return and 15% volatility, the worst one year in twenty starts at -16.75%, a loss of about Rs 84 lakh on Rs 5 crore, and the average year inside that tail loses about 23%, near Rs 1.15 crore.
The relationship
$r_{5\%} = \mu - 1.65\,\sigma = 8\% - 1.65 \times 15\% = -16.75\%$
mu expected return, 8%
sigma volatility, the standard deviation of yearly returns, 15%
1.65 standard deviations below the mean that cut off the worst 5% of a normal distribution
What it says in wordsThe one-in-twenty bad year starts 1.65 volatilities below the expected return.
What should you say about the limits of this number?
Three things. It is a cut-off, not a floor: when a bad year comes, the average loss beyond the cut-off is about 23%, near Rs 1.15 crore. Real market returns have fatter tails than the normal curve, so the true one-in-twenty loss is usually worse than the formula says. And one in twenty does not mean once every twenty years on a schedule; two such years can arrive back to back.
For a sterner test, one year in a hundred sits about 2.33 volatilities down: 8% minus 34.95% is -26.95%, about Rs 1.35 crore. Giving the client both lines, one in twenty and one in a hundred, is more honest than a single number.
Where candidates lose it
The common error is using one standard deviation, 8% minus 15%, and calling a Rs 35 lakh loss the bad year. One volatility down happens about one year in six, far more often than one in twenty.
The second miss is presenting the figure as the maximum loss. Say it is the edge of the bad tail, then give the average loss inside it.
What the interviewer asks next
What volatility would keep the one-in-twenty loss under Rs 50 lakh, with the same 8% expected return?
Why might a normal curve understate the chance of a large loss?
How would you explain the one-in-a-hundred year to a client who has only lived through good markets?
082A client's portfolio has an expected annual return of 10% and a volatility of 20%. Assuming returns are normally distributed, what is the chance of a negative year?Probability and risk of lossCoreWealth management
Try it first
Roughly how often does this portfolio lose money in a year?
Show the worked solution
About 31%, roughly one year in three. A loss means a return below zero, which is 10 points under the 10% mean. With volatility of 20, that is half a standard deviation below the mean, a z-score of minus 0.5, and the normal table puts 30.9% of outcomes below that point.
How do you get from 10% and 20% to a chance of loss?
Picture a bus that arrives on average at 9:10 but varies by about 20 minutes either way. Arriving before 9:00 is only 10 minutes early, half of its usual wobble, so it happens often. The chance of a losing year depends on how many standard deviations separate the mean return from zero, not on the mean alone. Here zero is 10 points below the mean and a standard deviation is 20, so zero sits half a standard deviation down.
The relationship
$z = \frac{0 - 10\%}{20\%} = -0.5 \qquad P(R < 0) = \Phi(-0.5) \approx 30.9\%$
z how many standard deviations zero sits from the mean
\Phi the share of a normal distribution below a given z
R one year's return
What it says in wordsConvert zero into standard deviations from the mean, then read the share of outcomes below it.
With a mean of 10% and volatility of 20%, zero sits half a standard deviation below the centre, and the red area below it holds 31% of years, so a healthy expected return still loses about one year in three.
What do you tell the client, and what is the catch?
Tell him in years, not in z-scores: in a typical decade he should expect about three losing years. Saying the chance of a losing year before he invests is what keeps a client invested when one arrives. A client told only the 10% expected return reads a losing year as a broken promise.
Then give the limitation. Real returns have fatter tails than the normal curve, so large losses turn up more often than the bell suggests, and years are not fully independent. The 31% is a clean first estimate, not a forecast.
Where candidates lose it
The weak answer is "rarely, because the expected return is positive". It confuses the centre of the distribution with its spread, and it is exactly the thinking that makes clients sell in the first bad year.
The second slip is getting the z-score wrong way round or using the whole standard deviation: zero is 10 points away, which is half of 20, not one full standard deviation.
What the interviewer asks next
What is the chance of losing more than 10% in a year?
Volatility falls to 10% with the same expected return. What is the chance of a losing year now?
Why might the real chance of a large loss be higher than the normal curve says?

w	each asset's weight, 0.5
sigma	each asset's volatility, 20%
rho	the correlation between the two, here zero

p	the chance any one stock goes to zero, 5%
n	the number of independent stocks, 10

mu	expected return, 8%
sigma	volatility, the standard deviation of yearly returns, 15%
1.65	standard deviations below the mean that cut off the worst 5% of a normal distribution

z	how many standard deviations zero sits from the mean
\Phi	the share of a normal distribution below a given z
R	one year's return

Try it first

Why is the answer not simply 20%?

How do you write the formula without getting lost?

Where candidates lose it

What the interviewer asks next

Try it first

Why work with the chance that nothing goes wrong?

Does this mean the portfolio is riskier with more stocks?

Where candidates lose it

What the interviewer asks next

Try it first

Why translate volatility into rupees?

What should you say about the limits of this number?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get from 10% and 20% to a chance of loss?

What do you tell the client, and what is the catch?

Where candidates lose it

What the interviewer asks next