Private Wealth Management puzzles, solved step by step
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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?Private 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 relationship104 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?
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?Private 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?
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?Wealth 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 relationshipD_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?