Private Wealth Management puzzles, solved step by step

Puzzles: 100
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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?Fixed income numeracyCorePrivate 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 relationship
$95 = \frac{7}{1+y} + \frac{7}{(1+y)^2} + \frac{107}{(1+y)^3} \;\Rightarrow\; y \approx 8.97\%$
95 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?
020A perpetual bond with a 9% coupon trades at 104 per 100 of face value. The issuer can call it at par in two years and is expected to. What yield is the client actually buying?Fixed income numeracyHardPrivate banking
Try it first
If the bond is called at par in two years, what is the yield?
Show the worked solution
About 6.8%, not 9%. The client pays 104 and, if the call is used, receives a coupon of 9 after one year and 9 plus 100 after two. The Rs 4 of premium above par is lost at the call. The yield that discounts 9 and 109 back to 104 is 6.79%. For a bond above par that is likely to be called, the yield to call is the honest figure.
Why does the call matter so much?
Think of paying extra for a flat on a lease you expect to run for ever, then learning the landlord can end it after two years and refund only the base price. The rent was fine; the premium you paid is gone. A bond bought above par loses the premium if it is called at par, so its yield must be measured to the call date, not as if the coupon ran for ever. Here 4 of premium is lost over just two years.
The client pays 104, receives 9 and then 109 at the call, losing the 4 of premium above par, so the yield to call is 6.79% against a coupon of 9% and a current yield of 8.65%.
How do you estimate it in your head?
Spread the premium loss over the years to the call. Coupons of 9 a year less 4 of premium over two years is about 7 a year, on an average price of about 102, which is roughly 6.9%, close to the exact 6.79%. The shorter the time to the call, the bigger the drag per year, because the same premium is lost over fewer coupons.
The relationship
$104 = \frac{9}{1+y} + \frac{109}{(1+y)^2} \;\Rightarrow\; y \approx 6.79\%$
104 the price paid
9 the annual coupon
109 the final coupon plus the 100 repaid at the call
y the yield to call
What it says in wordsThe yield to call is the rate that makes the coupons up to the call and the call price worth the price paid.
Name the second risk. A perpetual bond is callable, not must-call: if rates rise or the issuer weakens, it may not be called, and the client then holds a bond with no maturity at all. Desks quote the yield to worstThe lowest yield across all the dates on which the bond could be called or mature, the conservative figure for a callable bond. for exactly this reason, and a client needs both scenarios before buying.
Where candidates lose it
The trap is quoting the 9% coupon, or the 8.65% current yield, as the return. Both ignore that the client paid 104 for something that will most likely be repaid at 100 in two years.
The second loss is treating the call as certain. Say what happens if it is not called: a perpetual with no maturity, whose price can fall a long way.
What the interviewer asks next
What is the yield to call if the call is in one year instead of two?
The bond is not called and trades at 90. What is its current yield?
Why do issuers usually call a bond like this when rates fall?
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?Fixed income numeracyCorePrivate 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 relationship
$\text{spread left} = s - PD \times (1-R) = 2.5\% - 2\% \times 0.6 = 1.3\%$
s 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?
046A bullet portfolio holds 5-year zero-coupon bonds. A barbell holds equal amounts of 1-year and 9-year zeros, so both have a duration of 5 years at a 7% yield. If yields jump sharply, up or down, which portfolio comes out ahead, and why?Fixed income numeracyHardPrivate banking
Try it first
Yields move 4 points in parallel, either way. Which portfolio does better?
Show the worked solution
The barbell, in both directions, because it has more convexity. Equal duration makes the two portfolios move alike for small changes in yield. For large moves the price curve's bend matters, and a barbell's is sharper: convexity of about 40 against 26. On Rs 100, a 4-point fall in yields leaves the barbell 1.41 ahead; a 4-point rise leaves it 0.90 ahead.
Why does equal duration not mean equal behaviour?
Two cars can be doing 60 at this instant and still be in different places a minute from now if one is accelerating. Duration is the speed at the current yield; convexity is how that speed changes as yields move, and it decides who is ahead after a large move. A 9-year zero's price bends much more sharply than a 5-year zero's, far more than twice as much, while the 1-year zero barely bends at all, so the barbell's mix bends more than the bullet.
The bullet and barbell price curves both pass through 100 at 7% and nearly overlap, but the barbell's lead is positive for every move: +0.31 and +0.25 for 2-point moves, rising to +1.41 and +0.90 for 4-point moves.
Yield move Bullet value Barbell value Barbell lead
-4 points 120.99 122.39 +1.41
-2 points 109.89 110.21 +0.31
none 100.00 100.00 +0.00
+2 points 91.16 91.41 +0.25
+4 points 83.23 84.13 +0.90
Rs 100 in each portfolio of zero-coupon bonds, flat 7% yield curve, annual compounding, parallel moves.
If the barbell always wins, what is the catch?
Two catches. First, markets price convexity, so a barbell with the same duration usually yields a little less, and if yields sit still the bullet quietly earns more. Second, the edge assumes a parallel move. If short yields fall 1 point while 9-year yields rise 1 point, with the 5-year unchanged, the bullet is untouched and the barbell loses about Rs 3.54 per Rs 100. A barbell is a bet on large, parallel moves; a bullet is a bet on stability or on the curve's shape changing.
For a client portfolio, the practical reading is that duration alone does not describe interest rate risk. Both portfolios here have a modified duration of about 4.67, and a manager who reports only that number has hidden the shape of the bet.
Where candidates lose it
The common answer is that equal duration means equal behaviour, which is true only for small moves. Others say the barbell wins when rates fall and loses when they rise, treating the long bond as the only thing that matters.
Say convexity, give the direction, both ways, and then name the cost: a lower yield if nothing moves and a loss if the curve twists. The interviewer wants to hear that there is no free lunch.
What the interviewer asks next
Why do long-dated bonds have so much more convexity than short ones?
What happens to the barbell if the yield curve steepens?
How would you explain convexity to a client who holds only fixed deposits?
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?Fixed income numeracyCoreWealth 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 relationship
$P = 100 + (8 - 7) \times \frac{1 - 1.07^{-10}}{0.07} = 100 + 7.02 = 107.02$
8 - 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?Fixed income numeracyCoreIndian 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?
084A bank fixed deposit quotes 7% a year, compounded quarterly. What is the effective annual yield?Fixed income numeracyWarm upIndian wealth management
Try it first
Pick the effective yield.
Show the worked solution
About 7.19%. Quarterly compounding pays 7 / 4 = 1.75% each quarter, and each quarter's interest earns interest in the quarters that follow. Rs 100 grows to 101.75, 103.53, 105.34 and 107.19. The effective yield is 1.0175 to the fourth, less 1, which is 7.19%.
Why is the effective yield above the quoted 7%?
If a friend repays you in four instalments and you lend each instalment on at once, you end the year with more than if the whole amount came back at year end. The quoted rate is the per-period rate times the number of periods; the effective yield adds the interest earned on interest paid earlier in the year. The more often it compounds, the wider the gap.
Rs 100 grows by 1.75% a quarter to Rs 107.19 at the year end, each quarter's interest slightly larger than the last, so the effective yield is 7.19% against the quoted 7%.
The relationship
$\text{EAR} = \left(1 + \frac{0.07}{4}\right)^4 - 1 = 1.0175^4 - 1 = 7.19\%$
0.07 the quoted annual rate
4 compounding periods a year
EAR the effective annual rate, what Rs 100 actually becomes in a year
What it says in wordsDivide the quoted rate by the periods, compound it that many times, and subtract one.
When does the 7.19% not apply?
The effective yield assumes the interest stays in the deposit; on a payout deposit that sends interest to the client each quarter, he earns 7% unless he reinvests it himself. Tax is the second gap: interest on deposits is typically taxed as income each year at the client's slab rate, so the after-tax figure is lower still. Confirm the current treatment before comparing a deposit with a fund.
Use the effective rate whenever you compare products that compound at different frequencies, such as a quarterly deposit against an annual-coupon bond. Comparing quoted rates across frequencies is comparing different units.
Where candidates lose it
The common slip is to say 7% because that is what the bank quoted. The interviewer is checking whether you hear the words "compounded quarterly" and know they change the answer.
The rarer slip is reading 7% as a quarterly rate and quoting about 31%. Say the per-quarter rate, 1.75%, out loud before compounding.
What the interviewer asks next
What would the effective yield be with monthly compounding?
A bond pays 7.1% once a year. Which pays more, the bond or the deposit?
How does a 30% tax slab change the comparison?
097Interest rates rise by 1 point overnight. A client's bond fund has a modified duration of 6 and was yielding 7.5%. It loses about 6% at once. Roughly how long before the investor is back to even?Fixed income numeracyHardWealth management
Try it first
How long until the fund is as far ahead as it would have been with no rate rise?
Show the worked solution
Two answers: back to its starting value in about 9 months, but back level with the no-rise path only after about 6.7 years, close to the fund's duration. The fund drops to 94 and now earns 8.5%, so 94 grows past 100 in under a year. Against staying at 7.5%, it gains about 1 point a year, which takes roughly six to seven years to repay the 6 point loss.
Why does a rate rise hurt now and help later?
Imagine you have just locked into a rent of Rs 75,000 a month, and next day rents rise to Rs 85,000. Your lease is worth less to anyone buying it, but the next lease you sign will pay more. A rate rise marks down the bonds the fund holds today, and then lets it reinvest coupons and maturing money at the higher yield, so the loss comes at once and the payback comes year by year.
The relationship
$\Delta P \approx -D_{mod} \times \Delta y = -6 \times 1\% = -6\% \qquad 0.94 \times \left(\frac{1.085}{1.075}\right)^n = 1 \Rightarrow n \approx 6.7$
D_mod modified duration, the percentage price change for a 1 point move in yield
\Delta y the change in yield, here plus 1 point
n years until the fund catches the no-rise path
What it says in wordsDuration sizes the immediate loss; the extra yield then earns it back at about 1 point a year.
The fund drops from 100 to 94 at once and, growing at the new 8.5% yield, passes 100 again after about 9 months, but it only draws level with the path it would have followed without the rise after about 6.7 years, close to its duration of 6.45.
Which answer does the client need to hear?
Both, clearly labelled. "Back to what you put in" is under a year; "back to where you would have been" is about the duration. An investor whose holding period is longer than the fund's duration is left better off by a rate rise, because the higher yield more than repays the loss by the end of it. That is the logic of matching a bond fund's duration to the client's horizon.
State the approximations. The 6% ignores convexity, which makes the actual fall a little smaller. The payback assumes the fund keeps its duration steady and reinvests everything at 8.5%, and that rates do not move again. The rule that payback is about the duration uses the Macaulay duration, here 6 x 1.075, about 6.45 years; 6.7 is close to both.
Where candidates lose it
The common slips are "never" and "a year". The first forgets that the fund reinvests at the higher yield; the second counts the extra point of yield as if it repaid 6 points in one go.
The quieter loss is giving one number without saying what "even" means. Split it into back to cost and back to the no-rise path, then tie the second to duration.
What the interviewer asks next
The fund's duration is 2 instead of 6. How do the loss and the payback change?
Why does the fund fall slightly less than 6% in practice?
A client needs the money in 18 months. Which fund duration suits him?

95	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

104	the price paid
9	the annual coupon
109	the final coupon plus the 100 repaid at the call
y	the yield to call

s	the spread over the government bond, 2.5 points
PD	the yearly probability of default, 2%
R	the recovery rate, 40%

Yield move	Bullet value	Barbell value	Barbell lead
-4 points	120.99	122.39	+1.41
-2 points	109.89	110.21	+0.31
none	100.00	100.00	+0.00
+2 points	91.16	91.41	+0.25
+4 points	83.23	84.13	+0.90

Try it first

What does current yield leave out?

How do you get close to 8.97% without a spreadsheet?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the call matter so much?

How do you estimate it in your head?

Where candidates lose it

What the interviewer asks next

Try it first

Why is a spread not free money?

What does the 1.3 points actually pay for?

Where candidates lose it

What the interviewer asks next

Try it first

Why does equal duration not mean equal behaviour?

If the barbell always wins, what is the catch?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the price rise when yields fall?

How does duration give the same answer?

Where candidates lose it

What the interviewer asks next

Try it first

Why build a ladder instead of one deposit?

How do you compute the interest quickly?

Is the Rs 50,000 a permanent cost?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the effective yield above the quoted 7%?

When does the 7.19% not apply?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a rate rise hurt now and help later?

Which answer does the client need to hear?

Where candidates lose it

What the interviewer asks next