Hedge Funds puzzles, solved step by step
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014A corporate bond has a spread duration of 6 and convexity of 50. Its credit spread widens by 50 basis points. Roughly what happens to its price?AQR Capital ManagementGreenwich · 2021
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Which is closest?
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The price falls by about 2.94%. Spread duration of 6 says a 0.50 percentage point widening costs 6 x 0.50% = 3.00%. Convexity of 50 adds back one half x 50 x 0.005 squared, about 0.06%, because the price curve bends upwards. On a bond priced at 100 that is a move to about 97.06. At 50 basis points the convexity term is small; at 300 or 500 it is not.
What do duration and convexity each measure?
Picture a playground slide that curves and flattens towards the bottom. Judge the drop from the steepness at the top and you overstate it, because the slide levels off as you go. Spread durationThe percentage change in a bond price for a one percentage point change in its credit spread, holding the risk-free rate fixed. is the steepness at today's spread; convexity is the flattening, so the straight-line estimate always overstates the loss when spreads widen. Duration gives the first-order move, 6 x 0.50% = 3.00% down; convexity corrects it by a term that depends on the square of the move.
For a 50 basis point widening, duration of 6 gives minus 3.00% and convexity of 50 adds back 0.06%, a fall of 2.94%; the convexity cushion grows with the square of the move, to 2.25 points at 300 basis points and 6.25 at 500. The relationshipD_s spread duration, 6 C convexity, 50 \Delta s the change in spread as a decimal, 50 basis points = 0.005 What it says in wordsThe price moves by the duration term plus a smaller correction that grows with the square of the spread change.When does the convexity term start to matter?
It grows with the square of the move. At 50 basis points convexity is worth 0.06% against a 3.00% duration loss; at 300 basis points duration says -18% and convexity adds back 2.25%, which is no longer small. That is why a credit desk can run duration-only risk for everyday moves but needs convexity for stress scenarios. One more distinction marks a strong answer: for a fixed-coupon bond spread duration and rate duration are close, but a floating-rate note has almost no rate duration and still carries several years of spread duration.
Say the limitation plainly. Both numbers are local, measured at today's spread, and a distressed bond stops behaving like this long before default, when its price starts tracking the expected recovery instead. For a bond trading near par, as here, the two-term estimate is good to a few hundredths of a per cent for moves of this size.
Where candidates lose it
Candidates give minus 3% and stop, which is fine as a first line but ignores the second number the question handed you. Worse is using convexity with the wrong sign, making the loss bigger: for a plain bond convexity always cushions a spread widening.
The other slip is units. Fifty basis points is 0.005 in the formula; squaring 0.50 instead turns a 0.06% correction into 6.25% and produces a price that rises when spreads widen.
What the interviewer asks next
- What if the spread tightens by 50 basis points instead?
- Why can a callable bond have negative convexity?
- How would you hedge the spread risk of this bond?
Asked at AQR Capital Management, Investment Research, Greenwich, 2021 (Wall Street Oasis):
Discussion on credit spreads on fixed income products and duration.
039Depreciation rises by Rs 10 and the tax rate is 25%. Walk the change through net income, the cash flow statement and the balance sheet.Millennium ManagementNew York · 2024
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What happens to cash?
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Net income falls Rs 7.5, cash rises Rs 2.5 and the balance sheet shrinks by Rs 7.5 on both sides. Pre-tax profit falls 10, tax falls 2.5, so net income falls 7.5. The cash flow statement starts at -7.5 and adds back the non-cash 10: cash up 2.5. On the balance sheet, cash is up 2.5 and fixed assets are down 10, so assets fall 7.5, matched by retained earnings down 7.5.
Why does cash go up when an expense goes up?
Imagine your employer lets you deduct the wear on your car from taxable income. No money leaves your pocket for the wear itself, but your tax bill falls. Depreciation is an expense that costs no cash but reduces tax, so the only cash effect is the tax saved: 25% of Rs 10, Rs 2.5. That is the {term('depreciation tax shield', 'The tax saved because depreciation is deductible even though it uses no cash; equal to depreciation times the tax rate.')}, and it is the one number the question is testing.
A Rs 10 rise in depreciation at a 25% tax rate cuts net income by Rs 7.5, raises cash by Rs 2.5 through the tax saved, and lowers fixed assets by Rs 10, so total assets and retained earnings both fall by Rs 7.5 and the balance sheet balances. What order do you walk it in so nothing gets lost?
Income statement first, then cash flow, then balance sheet, one line each. Net income is the bridge: it closes the income statement, opens the cash flow statement, and lands in retained earnings on the balance sheet. Income statement: depreciation +10, pre-tax -10, tax -2.5, net income -7.5. Cash flow: -7.5 plus 10 added back, cash +2.5. Balance sheet: cash +2.5, fixed assets -10, so assets -7.5; retained earnings -7.5, so the two sides move together.
Add one sentence on why a hedge fund analyst cares. Two companies with identical operations can report different earnings because of depreciation choices, while their cash generation differs only by the tax effect. That is one reason investors look at cash flow alongside earnings before trusting a P/E.
Where candidates lose it
The common loss is saying cash is unchanged because depreciation is non-cash. That forgets the tax: depreciation is deductible, so the tax bill falls and cash rises by Rs 2.5.
The second loss is saying cash falls 7.5 by reading net income as cash. Walk the add-back out loud and check that assets and equity both fall by 7.5 before you stop.
What the interviewer asks next
- Now the depreciation rise comes from a Rs 10 write-down of an asset that is not tax deductible. What changes?
- What if the company is loss-making and pays no tax this year?
- Walk a Rs 10 rise in inventory, bought with cash, through the three statements.
Asked at Millennium Management, Investment Research, New York, 2024 (Wall Street Oasis):
Nothing as much, technical questions were super basic like $10 depreciation
089The equity risk premium is 4.5%, and a market's fair multiple is 1 divided by (real yield + premium - real growth). Real yields rise from 1.5% to 2.5% while expected real growth rises from 2.0% to 2.5%. What happens to the fair multiple?CitadelNew York · 2026
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Where does the fair multiple go?
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The fair multiple falls from 25x to about 22.2x, a compression of about 11%. The denominator is the real yield plus the premium minus growth: 1.5 + 4.5 - 2.0 = 4.0% before, and 2.5 + 4.5 - 2.5 = 4.5% after. Real yields rose by a full point and growth by only half a point, so the net rate rose by half a point, and the multiple, its inverse, fell.
Why is the multiple one over a spread?
Think of a shop that pays you rent forever, rising a little each year. What you would pay for it depends on the return you demand minus how fast the rent grows. For earnings paid out and growing forever, price over earnings is one divided by the required return minus growth, so the multiple depends only on the gap between the two. In real terms the required return is the real yield plus the equity risk premiumThe extra return investors demand for holding shares rather than government bonds.. The question treats all earnings as paid out, which is the assumption to name.
The net rate in the denominator rises from 4.0% to 4.5% because real yields climbed a full point while growth climbed half a point, so the fair multiple falls from 25x to 22.2x, about 11% lower. How do you work it out quickly?
Compute the denominator before and after. Before: 1.5 + 4.5 - 2.0 = 4.0%, a multiple of 25x. After: 2.5 + 4.5 - 2.5 = 4.5%, a multiple of 22.2x. Rates went up by 1.0 point and growth by 0.5, so the spread widened by 0.5 point. A 0.5-point rise on a 4.0% base is a 12.5% rise in the denominator, and the multiple falls by 1 minus 1/1.125, about 11.1%.
The relationshipr the real yield on government bonds ERP the equity risk premium, 4.5% g expected real growth of earnings What it says in wordsThe fair multiple is one over the net rate: what investors demand minus how fast the earnings grow.What does this teach beyond the arithmetic?
Higher yields do not hurt equities one for one if growth rises with them. What matters is whether real yields rise faster or slower than expected growth: faster compresses multiples, slower expands them. Had growth also risen a full point, to 3.0%, the net rate would be 4.0% again and the multiple 25x. The limitation to state is sensitivity: near a 4% net rate, a half-point move shifts the multiple by about 2.8 turns one way and 3.6 the other, so small errors in the premium or the growth guess swamp the answer.
Where candidates lose it
The quick wrong answer is that nothing happens because both rates went up. The question is built so that growth rises by only half as much as yields, and it is the spread, not the level, that sets the multiple.
The second loss is dropping the growth change and answering 20x. Write the denominator out in full, before and after; it takes ten seconds and removes both errors.
What the interviewer asks next
- Growth rises by a full point, to 3.0%. What is the multiple now?
- The equity risk premium also falls to 4.0%. What is the net effect?
- Why do shares whose value sits far in the future fall more than the market when real yields rise?
Asked at Citadel, Software, New York, 2026 (Wall Street Oasis):
real yields rising faster than growth expectations predicts equity multiple compression
