Risk Management puzzles, solved step by step

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

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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?
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?
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?

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

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

g	the compound annual growth rate
\mu	the arithmetic average annual return, 10%
\sigma	the annual volatility, 30%

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

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 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

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