Private Wealth Management puzzles, solved step by step
- Puzzles
- 100
- Traced to a firm
- 3
- Topics
- 13
- Hard
- 30
008A 3-year bond pays a 7% annual coupon and trades at 95 per 100 of face value. What are its current yield and its yield to maturity?Private banking
Try it first
Which is higher, and roughly by how much?
Show the worked solution
Current yield is 7.37% and yield to maturity is about 8.97%. Current yield is the coupon over the price, 7 divided by 95. Yield to maturity is the single rate that discounts the coupons of 7, 7 and 107 back to the price of 95. It is higher because the bond also pulls up from 95 to 100 by maturity, a gain current yield leaves out.
What does current yield leave out?
Suppose you buy a gift voucher worth Rs 100 for Rs 95, and it also pays you Rs 7 a year until you redeem it. The Rs 7 a year is the income; the Rs 5 you gain at redemption is extra. Current yield counts only the income, so for a bond bought below par it understates the return, and for a bond bought above par it overstates it. Yield to maturity counts both, and discounts them properly.
The bond's price climbs from 95 to 100 over three years while it pays 7 a year, so its yield to maturity of 8.97% is the current yield of 7.37% plus about 1.61 points from the pull to par. How do you get close to 8.97% without a spreadsheet?
Use the approximation interviewers expect: yearly income plus the yearly share of the gain, over the average of price and par. Income of 7 plus 5 divided by 3 is 8.67, over an average price of 97.5, gives about 8.9%, within a tenth of the exact 8.97%. Then check the exact rate discounts the flows back to 95: at 8.97%, the three payments are worth 6.42, 5.89 and 82.68 today, which add back to 95 once the rounding is put back.
The relationship95 the price paid per 100 of face value 7 the annual coupon 107 the final coupon plus the 100 repaid at maturity y yield to maturity What it says in wordsYield to maturity is the one discount rate that makes the bond's future payments worth exactly its price today.Add the caveat a private banker would. Yield to maturity assumes the bond is held to the end, every coupon is paid, and each coupon is reinvested at the same 8.97%. For a client buying this bond in a portfolio, the credit of the issuer matters as much as the arithmetic.
Where candidates lose it
The common error is quoting the coupon, 7%, as the yield. The second is the current yield, 7.37%, offered as the full return. Both ignore the Rs 5 the client collects at maturity.
The opposite error is adding the whole Rs 5 to one year's income and getting a yield near 12%. The gain is spread over three years, so it adds about 1.6 points, not 5.
What the interviewer asks next
- The same bond trades at 105. Which is higher now, current yield or yield to maturity?
- If market yields fall to 7%, what does the bond trade at?
- What does yield to maturity assume about the coupons, and when does that assumption fail?
034A client is offered a corporate bond yielding 9.5% when government bonds of the same tenor yield 7%. The issuer has roughly a 2% chance of defaulting in any year, and bondholders would recover about 40% if it did. How much of the extra yield is really extra?Private banking
Try it first
Of the 2.5-point spread, roughly how much is left after expected defaults?
Show the worked solution
About 1.3 points of the 2.5-point spread is genuinely extra. Expected loss is the default chance times the share lost: 2% x (100% less 40% recovery) = 1.2 points a year. The spread of 9.5 less 7, or 2.5 points, minus that 1.2 leaves about 1.3 points. That remainder is the pay for bearing uncertainty about defaults and for lower liquidity, not free yield.
Why is a spread not free money?
A moneylender who charges 24% instead of the bank's 12% is not twice as profitable if one borrower in ten never pays. The extra yield on a risky bond is first a charge for the defaults you should expect, and only what is left over is a reward. Clients see 9.5% against 7% and hear 2.5 points of extra income; the adviser's job is to take the expected losses out first.
The corporate bond's 2.5-point spread over the 7% government bond shrinks to 1.3 points once 1.2 points of expected loss are taken out, leaving an expected return of about 8.3%. The spread only covers defaults up to a 4.2% annual default chance. The relationships the spread over the government bond, 2.5 points PD the yearly probability of default, 2% R the recovery rate, 40% What it says in wordsWhat the spread really pays is the spread minus the losses you should expect every year on average.What does the 1.3 points actually pay for?
Two things the average hides. Defaults cluster in bad years, exactly when the client also has equity losses, so the bond fails when he can least afford it. And a corporate bond is harder to sell than a government bond. If the true default chance is 4.2% rather than 2%, the whole spread is eaten and the client is holding extra risk for nothing. So the useful question is how confident anyone can be in the 2%.
The puzzle simplifies: a default also stops future coupons, recoveries take time to arrive, and the default chance is not constant over a bond's life. Treat 1.3 points as the order of magnitude, and read the issuer's actual rating history and covenants before any real comparison.
Where candidates lose it
The first trap is subtracting the full 2% default chance, forgetting that bondholders recover 40%. That gives 0.5 points left and makes the bond look worse than it is. The opposite error ignores defaults and calls the whole 2.5 points extra income.
Say the formula out loud, default chance times loss given default, then add the one sentence on why the remaining 1.3 points exists. That second part is what separates an adviser from a yield quoter.
What the interviewer asks next
- What default chance would make the client indifferent between the two bonds?
- If recovery falls to 20%, how much spread is left?
- Why do spreads widen sharply in a recession even if default rates have not yet risen?
060A 10-year bond paying an 8% annual coupon trades at par, Rs 100. Market yields for that bond fall to 7%. Roughly what is the new price?Wealth management
Try it first
Pick the closest new price.
Show the worked solution
About Rs 107, exactly Rs 107.02 per Rs 100. The bond pays Rs 8 a year when the market now accepts Rs 7, so it carries an extra Rs 1 a year for ten years. At 7%, ten years of Rs 1 is worth about Rs 7.02 today, which is the premium over par. The duration shortcut, a modified duration of 6.71 times a one-point fall, gives nearly the same, Rs 106.71.
Why does the price rise when yields fall?
Imagine you rent out a flat at Rs 8,000 a month on a ten-year lease, and new flats in the building now rent for Rs 7,000. Your lease has become more valuable, and a buyer would pay extra for it. A bond's coupon is fixed, so when the market yield falls below it, the bond is worth more than par by the value of the extra coupon it carries. Here the extra is Rs 1 a year for ten years, discounted at the new 7%: an annuity factor of 7.02, so about Rs 7.
The relationship8 - 7 the extra coupon, Rs a year, over what the market now demands annuity factor the value today of Rs 1 a year for 10 years at 7% What it says in wordsA bond's premium over par is its extra coupon valued as an annuity at the new yield.As the yield on a 10-year 8% bond falls from 8% to 7%, its price rises from 100 to 107.02; the straight duration line predicts 106.71, slightly less, because the true curve bends upward away from the line. How does duration give the same answer?
Modified durationThe approximate percentage change in a bond price for a one percentage point change in yield. for this bond at 8% is 6.71, so a one-point fall lifts the price by about 6.71%. Duration is the slope of the price-yield curve, so it is a good straight-line guide for small moves and a slight underestimate for large ones. The gap here is Rs 0.31, the curve's bend, called convexity. The same bend works in the holder's favour the other way: at 9% the price falls only to Rs 93.58, less than the duration line predicts.
For a client, turn it into rupees on his holding. Rs 50 lakh of this bond gains about Rs 3.5 lakh from a one-point fall in yields, and loses a little less than that from a one-point rise. That is interest rate risk said in a sentence he can act on.
Where candidates lose it
The fastest wrong answer is that the price falls because the yield fell. Candidates mix up the coupon, which is fixed, with the yield, which the market sets through the price.
The second miss is saying Rs 101, one point for one point, which ignores that the bond has ten years to run. The extra Rs 1 a year is paid ten times, so the price moves about seven times as much.
What the interviewer asks next
- What would the price be if the bond had only 2 years left?
- Yields rise to 9% instead. Is the price fall bigger or smaller than Rs 7, and why?
- Why does a zero-coupon 10-year bond move more than this one for the same fall in yield?
072A client puts Rs 1 crore into a ladder of fixed deposits: Rs 20 lakh each for 1, 2, 3, 4 and 5 years at 6.5%, 6.75%, 7%, 7.25% and 7.5%. What is his average yield, how much interest does he get a year, and how much money comes free each year?Indian wealth management
Try it first
What does the ladder cost him against putting everything in the 5-year deposit?
Show the worked solution
A 7.0% average yield, Rs 7 lakh of interest a year, and Rs 20 lakh free every year. Equal Rs 20 lakh rungs make the average yield the simple average of the five rates. Against locking the whole Rs 1 crore at 7.5%, the ladder gives up about Rs 50,000 a year at the start, in exchange for a fifth of the money maturing every year and a spread of reinvestment dates.
Why build a ladder instead of one deposit?
A family that stores rice in five sacks, opening one a year, never has to break into next year's supply for this year's needs. A ladder spreads maturities so that a slice of the money comes free every year, which gives liquidity without breaking a deposit early and spreads the risk of reinvesting all the money at one bad rate. The price is that the shorter rungs earn less than the longest one.
Five Rs 20 lakh rungs at 6.5% to 7.5% average 7.00% and pay Rs 7.00 lakh a year, with Rs 20 lakh maturing every year, against Rs 7.50 lakh a year if the whole Rs 1 crore were locked at 7.5% for five years. How do you compute the interest quickly?
Because every rung is the same size, the average yield is the plain average of the rates: 6.5, 6.75, 7, 7.25 and 7.5 add to 35, and 35 over 5 is 7.0%. Rs 1 crore at 7.0% is Rs 7 lakh a year, against Rs 7.5 lakh if all of it sat in the 5-year deposit, so the ladder costs about Rs 50,000 a year at the start. Rung by rung it is Rs 1.30, 1.35, 1.40, 1.45 and 1.50 lakh.
Is the Rs 50,000 a permanent cost?
Not if he keeps the ladder going and rates stay put. Each year the matured Rs 20 lakh goes into a new 5-year deposit at the top rate, so after five years every rung was bought at the 5-year rate while one still matures each year. A rolling ladder gives up yield mainly in its first years; once rebuilt, it earns close to the long rate and keeps its yearly liquidity. If rates fall, the rungs that come free get reinvested lower; if they rise, the ladder catches the rise sooner than one long deposit would. Tax on deposit interest is the same across rungs in this example; confirm the current treatment for the client's slab.
Where candidates lose it
The trap is weighting the rates wrongly or quoting the top rate, 7.5%, as the ladder's yield. With equal rungs the yield is the simple average, and it is lower.
The second miss is saying the ladder is costly without saying what it buys. The interviewer wants the Rs 50,000 and the Rs 20 lakh of yearly liquidity in the same sentence.
What the interviewer asks next
- If the client needs Rs 30 lakh in year 2, how would you reshape the ladder?
- Rates fall by one point across the curve. What happens to the ladder's income over the next five years?
- Why might a ladder of 1 to 5 year bonds behave differently from a ladder of deposits?