Risk Management puzzles, solved step by step

Puzzles
100
Traced to a firm
17
Topics
13
Hard
30
Topic
All topicsCapital and leverage6Compounding and drawdowns8Correlation and diversification8Counterparty exposure and collateral7Credit risk arithmetic10Duration and rates7Liquidity and balance sheet7Logic, estimation and brainteasers7Operational loss and fraud7Options and Greeks7Probability and base rates8Statistics and estimation10VaR and expected shortfall8
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AnyWarm upCoreHard
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Showing 1–2 of 2 · filtered from 100Clear filters
  1. 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 unpaid: monthly steps against a straight line100110120130140150036912Months since the balance was left unpaid151.1142.0Compounded: 1.035 to the 12thSimple: 12 x 3.5% = 42%Interest on interest+9.1 points a year
    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
    EAR=(1+m)12−1=1.03512−1≈51.1%\text{EAR} = (1 + m)^{12} - 1 = 1.035^{12} - 1 \approx 51.1\%
    mthe monthly rate, 3.5%
    12the 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?
  2. 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.

    Drawdown is measured from the running peak, not from the start8010012014010012090130100140120 to 90-25.0%130 to 100-23.1%from the start: only -10%running peak (dashed)t0t1t2t3t4t5ObservationNAV
    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
    DDt=Vtmax⁡s≤tVs−1MDD=min⁡tDDt=90120−1=−25%DD_t = \frac{V_t}{\max_{s \le t} V_s} - 1 \qquad MDD = \min_t DD_t = \frac{90}{120} - 1 = -25\%
    V_tNAV at time t
    max V_sthe running peak up to time t
    MDDmaximum 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?