Risk Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 17
Topics: 13
Hard: 30

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002A credit card balance carries interest of 3.5% a month, compounded monthly. What is the effective annual rate?Compounding and drawdownsWarm upNBFC credit riskBank credit risk
Try it first
Answer inside ten seconds: roughly what is the effective annual rate?
Show the worked solution
About 51.1% a year. Rs 100 left unpaid grows by 3.5% each month on a balance that already includes last month's interest, so after twelve months it is 100 times 1.035 to the 12th, which is Rs 151.1. The simple rate of 12 times 3.5% is 42%, so compounding adds about 9.1 points.
Why is 42% the wrong answer?
Picture a jar of rice where a helper adds 3.5% of whatever is in the jar every month. In month two the helper adds 3.5% of a bigger jar than in month one. Monthly compounding charges interest on the interest already added, so the annual rate is always above twelve times the monthly rate. The 42% figure is what you would pay only if the lender added interest to a separate pile that never itself earned interest.
Rs 100 left on a card at 3.5% a month climbs in monthly steps to Rs 151.1 after a year, while adding a flat Rs 3.50 a month reaches only Rs 142, so compounding adds about 9.1 points to the annual rate.
The relationship
$\text{EAR} = (1 + m)^{12} - 1 = 1.035^{12} - 1 \approx 51.1\%$
m the monthly rate, 3.5%
12 the number of compounding periods in a year
What it says in wordsGrow one rupee for twelve months at the monthly rate and subtract the rupee you started with.
How do you get 1.035 to the 12th without a calculator?
Square it in steps. 1.035 squared is about 1.0712. Square again for four months, about 1.1475. Cube that for twelve months: 1.1475 squared is about 1.3168, and times 1.1475 again is about 1.511. Three multiplications you can do out loud get you to within a tenth of a point. A faster check is the log approximation: twelve times 3.44%, the log of 1.035, is 41.3%, and e to the 0.413 is about 1.51.
Then say why a credit risk team cares. The effective rate is what a borrower who rolls the balance actually pays, and a borrower paying above 50% a year is a borrower whose debt can outgrow their income quickly. The stated monthly figure is how the product is sold; the effective annual rate is the number that belongs in a comparison with other loans.
Where candidates lose it
The trap is answering 42% because the question sounds like a multiplication. It misses that the lender adds interest to the balance every month, and it understates the cost by about 9 points.
The second loss is freezing on the arithmetic. Say the formula, then square in steps: 1.035 squared, squared again, then cubed. Reaching 1.51 out loud is worth more than a silent calculator answer.
What the interviewer asks next
What monthly rate gives an effective annual rate of exactly 36%?
If the card compounds daily at the same annual simple rate, is the effective rate higher or lower, and by how much?
A borrower pays only the minimum of 5% of the balance each month. How long until the balance halves?
014A stock goes from 100 to 150 and back to 100. Compare the average of the simple returns with the average of the log returns.Compounding and drawdownsCoreAsset manager riskQuant risk
Try it first
What is the average simple return over the two periods?
Show the worked solution
The simple returns average +8.3%, the log returns average 0%. The simple returns are +50% and -33.3%, and their average suggests a gain although the price is back at 100. The log returns are +40.5% and -40.5%, which add to zero and match what happened. Log returns add over time; simple returns do not.
Why does the simple average show a gain that never happened?
Imagine a shop that raises a price from Rs 100 to Rs 150 and then cuts it back to Rs 100. The rise is 50 on a base of 100, and the cut is 50 on a base of 150. Simple returns are each measured against a different starting price, so averaging them mixes percentages of different bases and overstates growth whenever prices bounce around. The arithmetic mean of +50% and -33.3% is +8.3%, but the investor has exactly the money they started with.
For a price that goes 100, 150, 100, simple returns of +50% and -33.3% average +8.3% although the price made nothing, while log returns of +40.5% and -40.5% add to zero and average zero.
The relationship
$r_t = \ln\frac{P_t}{P_{t-1}}, \qquad r_1 + r_2 = \ln\frac{150}{100} + \ln\frac{100}{150} = \ln\frac{100}{100} = 0$
r_t the log return in period t
P_t the price at the end of period t
What it says in wordsLog returns add up across periods to the log of the total change, so a round trip sums to zero.
Which one should a risk manager use?
It depends on what you are adding up. Log returns add across time, so they are the natural choice for compounding a return over many days and for most statistical models of a single asset. Simple returns add across assets, so a portfolio's return for one period is the weighted average of its holdings' simple returns, which log returns do not give you. The gap between the two averages is the volatility dragThe shortfall of compound growth below the arithmetic average return, which grows with the variance of returns.: roughly half the variance, and here the swings are wild enough to make it 8 points.
Close with the practical warning. A fund that reports the arithmetic average of its yearly returns will look better than the growth its investors actually got, and the more volatile the fund, the larger the flattering gap. The honest single number for growth over time is the geometric average, which here is zero.
Where candidates lose it
The trap is quoting +8.3% as the average return and calling it performance. It is a correct average of the wrong thing: percentages taken on different bases.
The second miss is saying log returns are simply better. They add over time but not across assets, so a risk manager needs both and should say when each applies.
What the interviewer asks next
What is the geometric average return here, and how does it relate to the log returns?
A fund returns +20% and -20% in alternate years. What is its compound growth rate?
Why do most VaR models use log returns for single assets but simple returns to aggregate a portfolio?
027A stock has 2% daily volatility and no drift. What is the probability that it touches a level 10% below today's price at some point in the next 20 trading days, and how does that compare with the probability of ending below that level?Compounding and drawdownsHardBank market riskQuant risk
Try it first
Compared with the chance of finishing below the level, the chance of touching it at some point is:
Show the worked solution
About 24% to touch, against about 12% to finish below: roughly double. Twenty days at 2% a day is 8.94% of volatility. A 10% fall is ln 0.9, or 10.5% in log terms, which is 1.18 standard deviations, so the chance of ending below is 11.9%. The reflection principle doubles it for a touch: 23.9%.
Why is touching twice as likely as finishing below?
Picture a drunk walker on a pavement with a kerb one step to the left. By the end of the evening he may be well to the right, yet at some point he almost certainly stepped off the kerb. Paths wander. Once a driftless path has touched the level, it is equally likely to finish above it or below it, so for every path that finishes below there is a twin that touched and came back. This is the reflection principleFor a random walk with no drift, the path after it first hits a level is equally likely to be its mirror image about that level.: flip the path after the first touch and you swap an above-finisher for a below-finisher.
A path that touches the 10% stop level on day 9 and recovers has a mirror twin, drawn dashed, that finishes below the line; because the two are equally likely, the chance of touching is twice the chance of finishing below, 23.9% against 11.9%.
How do you get the numbers in the room?
Scale volatility by the square root of time: 2% times the square root of 20 is 8.94%. Work in log returns, because a 10% fall is a log move of 10.5%, not 10%. That is 1.18 standard deviations, and the normal table gives 11.9% below it. Double for the touch. If you skip the log step and use 10% flat you get 26.4%, close enough to earn credit if you say it is an approximation.
The relationship
$P(\text{touch}) = 2\,\Phi\!\left(\frac{\ln 0.9}{0.02\sqrt{20}}\right) = 2\,\Phi(-1.18) \approx 2 \times 0.119 = 0.239$
\Phi the standard normal cumulative probability
0.02\sqrt{20} 20-day volatility, 8.94%
\ln 0.9 the log return of a 10% fall, minus 0.105
What it says in wordsFind the chance of ending beyond the level, then double it because every finisher beyond has a twin that touched and came back.
Where does the doubling rule stop being exact?
Two places. With drift the mirror twins are no longer equally likely, and a positive drift pulls the touch probability below double. And if the stop only triggers on daily closes, a path can dip through during the day and recover by the close, so fewer touches count; a standard correction for daily monitoring gives about 19%. The practical reading is that a stop-loss level is hit far more often than a finishing-price distribution suggests, which is why stops set from end-of-period VaR fire more than the desk expects.
Where candidates lose it
Most candidates compute the finishing probability, 11.9%, and offer that as the answer to the touching question. The interviewer asked about touching precisely to see whether you know the two differ, and by how much.
The other loss is knowing the doubling rule but not why. Draw the mirror path out loud; it takes ten seconds and proves you are reasoning, not reciting.
What the interviewer asks next
What is the probability of touching a level 10% above instead?
How does a positive drift change the answer?
A barrier option knocks out at this level. Why is it cheaper than the vanilla option by more than the finishing probability suggests?
039A retiree holds Rs 1 crore and withdraws Rs 10 lakh at the end of each year. Compare two years of minus 20% then plus 20% with plus 20% then minus 20%. Where does each order leave the retiree?Compounding and drawdownsCoreAsset manager risk
Try it first
Without withdrawals, both orders end at Rs 96 lakh. With the Rs 10 lakh withdrawals, which order ends better?
Show the worked solution
Up first ends at Rs 78 lakh; down first ends at Rs 74 lakh. Without withdrawals both orders end at Rs 96 lakh, because returns multiply. With withdrawals, the Rs 10 lakh taken out after year one sits out year two: it escapes a 20% fall in one order and misses a 20% gain in the other, a gap of 10 x 0.4, Rs 4 lakh.
Why does order matter only once money is withdrawn?
Multiplication does not care about order: 0.8 x 1.2 is the same as 1.2 x 0.8, 0.96 either way. A withdrawal breaks the chain, because rupees taken out before a return do not experience it. Think of a farmer who sells part of the harvest each year. A good year followed by a bad one lets him sell from a big crop first; a bad year first forces him to sell from a small one, and the portion sold never gets the chance to recover. This is sequence riskThe risk that the order of returns, not just their average, changes the outcome, which happens whenever money is being added or withdrawn..
With Rs 10 lakh withdrawn each year, a fall of 20% followed by a rise of 20% leaves Rs 74 lakh, while the same returns in the opposite order leave Rs 78 lakh; without withdrawals both orders would end at Rs 96 lakh.
Where exactly does the Rs 4 lakh gap come from?
Track the first withdrawal. In both orders the retiree takes Rs 10 lakh at the end of year one. That Rs 10 lakh then skips year two: in the up-first order it skips a 20% fall and saves Rs 2 lakh, in the down-first order it skips a 20% rise and loses Rs 2 lakh. The difference is 10 x (1.2 minus 0.8), Rs 4 lakh. The second withdrawal is taken at the very end and does not depend on the order at all.
The relationship
$W_2 = (W_0(1+r_1) - D)(1+r_2) - D \;\Rightarrow\; W_2^{\text{up first}} - W_2^{\text{down first}} = D\,(r_{\text{up}} - r_{\text{down}}) = 10 \times 0.4 = 4$
W_0 starting wealth, Rs 100 lakh
D the yearly withdrawal, Rs 10 lakh
r_1, r_2 the two years' returns
What it says in wordsThe starting wealth sees both returns in any order; only the withdrawn money depends on which return it misses.
Scale it up and the effect grows. Over twenty years of withdrawals, a bad run in the first few years forces sales at low prices from a pot that then has less left to recover, and two retirees with the same average return can end decades apart. The limit of this puzzle is its size: two years and one gap of Rs 4 lakh undersell how much the order of returns matters over a long retirement.
Where candidates lose it
The instinct is to say order cannot matter because multiplication is commutative. That is true only without cash flows, and the question gave you a withdrawal precisely to break it.
The second miss is getting 74 and 78 but being unable to say why. The single sentence about the first withdrawal skipping year two is what the interviewer wants to hear.
What the interviewer asks next
What if the retiree added Rs 10 lakh each year instead of withdrawing it?
How would you reduce sequence risk for a new retiree without changing the expected return?
Over thirty years, why does a bad first five years matter more than a bad last five?
052An asset has an arithmetic average return of 10% a year and a volatility of 30%. Roughly what is its compound annual growth rate?Compounding and drawdownsCoreAsset manager risk
Try it first
Pick the closest before you calculate.
Show the worked solution
About 5.5% a year. Compound growth is roughly the arithmetic mean minus half the variance. The variance is 0.30 squared, 0.09, and half of it is 4.5 points, so 10% becomes about 5.5%. A quick check: +40% then -20% averages 10% with a 30% spread, and Rs 100 ends at Rs 112, which is 5.8% a year.
Why is the average return not the growth rate?
A shopkeeper who marks a shirt up 40% and later cuts it 20% has not made 20% on the pair of moves. He has made 12%, because the cut is taken from the higher price. Returns work the same way. A loss is always taken from whatever the gain left you, so ups and downs of the same average size leave you with less than steady growth at that average. The more the returns swing, the bigger the shortfall.
Rs 100 rising 40% and then falling 20% ends at Rs 112, a compound 5.8% a year, even though the two returns average 10%. The rule of thumb puts compound growth at the 10% average less half the variance of 0.09, which is about 5.5%.
Where does half the variance come from, and how good is it?
Take logs. The log of one plus a return is roughly the return minus half its square, so averaging the logs knocks off about half the variance. The drag grows with the square of volatility: at 15% volatility it is about 1.1 points, at 30% it is 4.5. The rule is an approximation that works best for small returns. The two-year example lands at 5.8%, and other return patterns with the same average and spread give slightly different answers.
The relationship
$g \approx \mu - \frac{\sigma^2}{2} = 0.10 - \frac{0.30^2}{2} = 0.055$
g the compound annual growth rate
\mu the arithmetic average annual return, 10%
\sigma the annual volatility, 30%
What it says in wordsCompound growth is the average return less half the variance.
For a risk manager this is why two funds with the same average return are not the same fund. The one with double the volatility has four times the drag, and its investors end up with less money even though the average looks identical.
Where candidates lose it
The trap is answering 10% because the question gave you 10%. The interviewer is checking whether you know that averages of returns overstate what an investor compounds, and that the gap is driven by volatility.
The second slip is subtracting the whole variance, or subtracting half the volatility, and landing at 1% or -5%. Square first, then halve: 0.09 over 2 is 4.5 points.
What the interviewer asks next
What volatility would make the compound growth zero with a 10% average?
If you lever this asset 2x, what happens to the average and to the compound growth?
Which number should a fund report to investors, and why?
065You can bet repeatedly at even odds with a 60% chance of winning each bet. Compare the long-run growth of your capital if you bet 20% of it each time with betting 50% each time.Compounding and drawdownsHardQuant riskAsset manager risk
Try it first
Betting 50% a time on a 60% edge: what happens over many bets?
Show the worked solution
Betting 20% grows capital about 2.0% a bet; betting 50% shrinks it about 3.3% a bet. Growth per bet is the average log return: 0.6 ln(1 + f) + 0.4 ln(1 - f). At 20%, the Kelly fraction 2p - 1, that is +2.01%. At 50% it is -3.40%. After 100 bets the typical 20% bettor has about 7.5 times the start; the 50% bettor about 0.03 times.
How can a bet with a positive edge lose money?
A shopkeeper who stakes half his stock on every festival season, and wins more seasons than he loses, can still go under: one bad season halves him, and a good one only adds half back. Losses on a shrunken base are expensive to recover. The typical outcome of repeated betting is set by the average log return, not the average return, and at 50% the losses dominate the logs.
With a 60% chance of winning at even odds, growth per bet peaks at the 20% Kelly fraction, about +2.0% a bet, crosses zero near 39% and is -3.4% a bet at 50%, so oversizing a winning bet turns steady growth into steady decline.
Where does the 20% come from?
Set the slope of the growth curve to zero. For even odds the Kelly fractionThe fraction of capital to bet each time that maximises the long-run compound growth rate of the capital. is the probability of winning minus the probability of losing: 0.6 minus 0.4, or 20%. Betting less than Kelly gives up some growth for a much smoother ride; betting more than about twice Kelly gives up growth altogether. Here the curve crosses zero near 39%, just under twice Kelly.
The relationship
$g(f) = p\ln(1+f) + (1-p)\ln(1-f) \qquad f^* = 2p - 1 = 0.2$
f the fraction of capital bet each time
p the probability of winning a bet, 0.6
g(f) the expected log growth of capital per bet
What it says in wordsLong-run growth is the probability-weighted average of the log of what each outcome does to your capital.
For a risk manager the lesson is position sizing. A desk with a genuine edge can still destroy capital by running too much leverage, and the average P&L will look fine right up until the drawdowns compound. Real edges are also uncertain, which is why practitioners bet a fraction of Kelly.
Where candidates lose it
The trap is answering that 50% grows faster because each bet has a positive expected value and bigger bets earn more of it. That is true of the arithmetic average and false for the capital you actually end up with, which compounds.
The second slip is knowing Kelly but not being able to show why 50% loses. Have the one-line check ready: six wins and four losses at 50% multiply capital by about 0.71.
What the interviewer asks next
What is the Kelly fraction if you win 55% of the time at even odds?
What if the payoff is 2 to 1 with a 40% chance of winning?
Why do traders often bet half Kelly rather than full Kelly?
077An index rises 10% one day and falls 9.09% the next, ending flat. A fund promises twice the index's daily return. Where does the fund end after the two days, and what happens if the pattern repeats?Compounding and drawdownsCoreAsset manager riskBank market risk
Try it first
After the two days, where is the 2x fund?
Show the worked solution
The fund ends at about 98.18, down 1.82% while the index is flat. Day one is plus 20%, taking 100 to 120. Day two is twice minus 9.09%, minus 18.18%, and 18.18% of 120 is 21.82, leaving 98.18. Each repeat of the up and down pair multiplies the fund by 0.9818, so after five pairs it sits at 91.23.
Why does doubling each day not double the two-day result?
Think of walking up an escalator that runs down. If you climb 10 steps and slip back 10, you are where you started. Now imagine every climb is measured as a share of your height above the ground, and so is every slip: slipping 18% from a higher point loses more steps than climbing 20% from a lower one gained. A daily-leveraged fund resets its exposure every day, so each day's percentage move is applied to a new base, and the losses land on the larger base.
The relationship
$(1 + 2 \times 0.10)(1 - 2 \times 0.0909) = 1.20 \times 0.8182 = 0.9818$
0.10 the index's up day
0.0909 the index's down day, 1 minus 1/1.1
2 the fund's daily leverage
What it says in wordsCompound each day's leveraged return, and the product is below one even though the index's two days multiply to exactly one.
Over ten days of alternating plus 10% and minus 9.09%, the index returns to 100 every second day while the 2x daily fund falls to 98.18 after the first pair and 91.23 after five, losing ground on every round trip.
How big is the drag, and what makes it worse?
Expand the product: for a leverage of L and an index that goes up r and then back to where it started, the fund loses about L times (L minus 1) times r squared on each pair, divided by 1 plus r. The drag grows with the square of the daily move and roughly the square of the leverage, so it is small in calm markets and fierce in choppy ones. With 2x and 10% moves that is 2 times 1 times 0.01 over 1.1, and each pair costs 1.82%; at 3x the same pair would leave 94.55. In a steady trending market the effect can run the other way and the fund beats twice the index.
Say where a risk manager meets this. A client who holds a daily-leveraged product for months is not holding twice the index; the product's prospectus usually says as much, and the gap is a suitability question as well as a maths one.
Where candidates lose it
The fast wrong answer is 100: the index is flat, so twice flat must be flat. It forgets that the fund compounds daily and that the down day is applied to 120, not 100.
The second loss is getting 98.18 and stopping. The interviewer wants the pattern: the drag scales with leverage squared and volatility squared, and it repeats every round trip.
What the interviewer asks next
Would a 2x fund beat twice the index if the index rose 1% every day for a month?
What is the fund's value after the same two days if it is 3x leveraged?
How would you explain this decay to a client who bought the fund for a year?
090A fund's NAV over six observations is 100, 120, 90, 130, 100 and 140. What is its maximum drawdown?Compounding and drawdownsWarm upAsset manager risk
Try it first
What is the maximum drawdown?
Show the worked solution
25%. Drawdown is measured from the highest value reached so far. The fund peaks at 120 and falls to 90, a 25% drop. It then peaks at 130 and falls to 100, a 23.1% drop. The larger of the two, 25%, is the maximum drawdown. Measured from the start the worst point looks like only 10%, which understates the pain.
Why measure from the running peak?
If your savings climbed to Rs 1.2 lakh and then fell to Rs 90,000, you would not console yourself that you started with Rs 1 lakh. You lost Rs 30,000 of money you had. Drawdown measures the fall from the highest value an investor has held so far, because that is the loss an investor who bought at the top actually suffers. The running peak resets upward each time the fund makes a new high, and each drawdown is measured against it.
The fund's NAV falls 25.0% from its peak of 120 to 90 and 23.1% from its later peak of 130 to 100, so the maximum drawdown is 25%, although the worst point measured from the start of 100 is only 10% down.
The relationship
$DD_t = \frac{V_t}{\max_{s \le t} V_s} - 1 \qquad MDD = \min_t DD_t = \frac{90}{120} - 1 = -25\%$
V_t NAV at time t
max V_s the running peak up to time t
MDD maximum drawdown, the deepest fall from a running peak
What it says in wordsEach point's drawdown is how far it sits below the best value seen so far; the maximum drawdown is the deepest of them.
What does the number not tell you?
Two things worth saying. Maximum drawdown is a single worst episode, so it depends heavily on the sample: a longer history can only make it larger, never smaller. It also ignores time. The fall from 120 to 90 took one period and the recovery took one more; a fund that takes three years to climb out of a 25% hole is a different experience from one that recovers in a quarter. A risk team reports duration of the drawdown and time to recovery alongside the depth, and remembers that a 25% fall needs a 33.3% gain to get back.
Where candidates lose it
The fast wrong answer is 10%, measuring the lowest point, 90, against the start, 100. It ignores that investors held the fund at 120.
The other slip is picking the most recent fall, 23.1%, because it is fresh, or measuring 130 to 90, which mixes a later peak with an earlier trough. The peak must come before the trough.
What the interviewer asks next
What gain does the fund need to recover from its maximum drawdown?
Why is maximum drawdown hard to compare across funds with different track record lengths?
How would you combine drawdown with volatility in one risk-adjusted measure?

m	the monthly rate, 3.5%
12	the number of compounding periods in a year

r_t	the log return in period t
P_t	the price at the end of period t

\Phi	the standard normal cumulative probability
0.02\sqrt{20}	20-day volatility, 8.94%
\ln 0.9	the log return of a 10% fall, minus 0.105

W_0	starting wealth, Rs 100 lakh
D	the yearly withdrawal, Rs 10 lakh
r_1, r_2	the two years' returns

Try it first

Why is 42% the wrong answer?

How do you get 1.035 to the 12th without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the simple average show a gain that never happened?

Which one should a risk manager use?

Where candidates lose it

What the interviewer asks next

Try it first

Why is touching twice as likely as finishing below?

How do you get the numbers in the room?

Where does the doubling rule stop being exact?

Where candidates lose it

What the interviewer asks next

Try it first

Why does order matter only once money is withdrawn?

Where exactly does the Rs 4 lakh gap come from?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the average return not the growth rate?

Where does half the variance come from, and how good is it?

Where candidates lose it

What the interviewer asks next

Try it first

How can a bet with a positive edge lose money?

Where does the 20% come from?

Where candidates lose it

What the interviewer asks next

Try it first

Why does doubling each day not double the two-day result?

How big is the drag, and what makes it worse?

Where candidates lose it

What the interviewer asks next

Try it first

Why measure from the running peak?

What does the number not tell you?

Where candidates lose it

What the interviewer asks next