Portfolio Management puzzles, solved step by step

Macaulay duration about 2.83 years and modified duration about 2.67. At par the yield equals the 6% coupon, so the cash flows discount to 5.66, 5.34 and 89.00, which add to 100. Weight each year by its share of the price: 1 x 0.0566 + 2 x 0.0534 + 3 x 0.8900 gives 2.83. Divide by 1.06 for modified duration: a one point rise in yield cuts the price by roughly 2.67%.

What is duration, if not the time to maturity?

Picture a seesaw with three children sitting at the one, two and three metre marks. If the child at three metres is much heavier, the pivot that balances the seesaw sits close to three, not at the middle. Macaulay duration is that pivot: the average time you wait for your money, with each payment weighted by its present value. Here the three weights are the discounted coupons of 5.66 and 5.34 and the discounted final payment of 89.00. The last payment is so heavy that the balance point, 2.83 years, sits only two months short of maturity.

Why divide by 1.06 to get modified duration?

Macaulay duration is a time. Modified duration is a price sensitivity: the percentage change in price for a one point change in yield. With annual compounding the two differ by a factor of one plus the yield, so 2.833 divided by 1.06 is 2.673. Say what it means: if yields rise from 6% to 7%, the bond loses about 2.67% of its price, a little less in reality because the price-yield curve bends. That bend is convexity, and it is the natural next question.

About 7.7%. You collect the 7 coupon, and a year later the bond is a two-year bond. If the curve has not moved, it is priced at the two-year yield of 6.6%, which makes it worth 100.73. Coupon 7.00 plus a price gain of 0.73 is 7.73%. A shortcut gives the same: 7% plus the two-year duration of about 1.82 times the 0.4 point fall in yield.

Why does an unchanged curve still move the bond's price?

Think of walking down a gentle slope while standing still relative to the hillside: the ground stays put, but you end up lower because you moved along it. A bond ages along the curve; if the curve slopes up and stays still, the bond's yield falls as its maturity shortens, and its price rises. Today it is a three-year bond at 7.0%. In a year it will be a two-year bond, and two-year bonds yield 6.6%. That 0.4 point fall in yield is the roll downThe price gain a bond earns as it ages into a shorter maturity with a lower yield on an upward sloping curve..

How do you check it without a calculator?

Use duration. A two-year bond with a 7% coupon has a modified duration of about 1.82, so a 0.4 point fall in yield lifts its price by about 1.82 x 0.4, which is 0.73. Add the 7 coupon and the one-year return is about 7.73%, within a hair of the exact 7.73%. Say the limitation: the answer depends entirely on the curve staying put. If two-year yields rise to 7.0% by next year, the roll down disappears and the return is the coupon alone.

About 8.2%. The bond pays 7 a year and also rises from 95 to 100 by maturity, which is about 1 point a year over five years. That is 8 a year of return on money that averages about 97.5 invested, and 8 over 97.5 is 8.21%. The exact yield with annual coupons is 8.26%, so the shortcut is within a few hundredths of a point.

Where does the return on a discount bond come from?

Buy a Rs 100 gift voucher for Rs 95 that also pays you Rs 7 each year until it can be cashed at full value in five years. You collect the Rs 7 every year, and you also pocket the Rs 5 discount at the end. A discount bond's yield is its coupon plus the discount it recovers as the price is pulled to par, spread across the years to maturity. Spread evenly, that is 1 point a year, so the bond earns roughly 8 a year.

Why divide by the average price rather than 95?

The amount you have invested is not fixed at 95: in this rough picture the bond's value drifts up towards 100 over the five years, so the capital at work averages about 97.5. Dividing by the average price corrects most of the error from spreading the discount in a straight line. Dividing 8 by 95 instead gives 8.42%, too high. As a check, a bond priced at 8.21% comes out at 95.21, within a fraction of 95.

Know when the shortcut drifts. It is close for short bonds near par and gets worse for long maturities or deep discounts, where the true discounting curve bends away from a straight line. For a quick answer in the room, 8.2% with the one-line derivation is what the interviewer is after; then say that the exact figure is a touch higher.

Duration is 5.55, and Rs 14.5 crore must move from the 2-year bond to the 12-year bond. Portfolio duration is the value-weighted average: (40 x 2 + 35 x 5 + 25 x 12) over 100, which is 555 over 100, or 5.55. Each Rs 1 crore switched gains 10 years of duration on a hundredth of the portfolio, adding 0.1. Closing a gap of 1.45 needs Rs 14.5 crore.

Why is portfolio duration a simple weighted average?

Think of the average age of people in a room: each person counts in proportion to how many of them there are. Duration measures how much a bond's price moves for a one-point change in yield, and for a small parallel move the portfolio's rupee loss is just the sum of each bond's rupee loss, so its duration is the value-weighted average of the bonds' durations. The Rs 25 crore in the 12-year bond is only a quarter of the money but supplies more than half the duration: 3.00 of the 5.55.

What changes for the portfolio when duration goes from 5.55 to 7?

Solve for the switch with one line: the shift x changes duration by x times (12 minus 2) over 100, and that must equal 1.45. After the switch, a one-point parallel rise in yields costs about 7% of the portfolio, Rs 7 crore, instead of about 5.55%, Rs 5.55 crore. The portfolio gains more if yields fall and loses more if they rise. Its cash-flow profile also becomes a barbell, heavier at the long end, which gives it more convexity than a single bond with the same duration but more exposure to the long end of the curve if the curve steepens.

Say the limits: a weighted average of durations describes small, parallel shifts in yields. If short and long yields move by different amounts, a portfolio at duration 7 built from a barbell behaves differently from one built from 7-year bonds, and the switch changes the portfolio's yield and credit mix as well. For large moves, convexity adds a second-order correction.

About 101.81. Discount each cash flow at 7%: the year 1 coupon of 8 is worth 7.48 today and the year 2 payment of 108 is worth 94.33, which add to 101.81. The bond sits above par because it pays 8 when the market only asks for 7, and a buyer pays up for that extra coupon until the return on the price paid falls back to 7%.

What does pricing a bond actually mean?

Think of a friend who promises you Rs 8 next year and Rs 108 the year after. What would you hand over today? If you can earn 7% elsewhere, each promised rupee is worth less the further away it sits. A bond's price is every promised cash flow divided by one plus the yield, once for each year you wait, and then added up. Here that is 8 divided by 1.07, which is 7.48, plus 108 divided by 1.07 twice, which is 94.33. The total is 101.81.

Why must a bond with a high coupon trade above par?

Suppose it traded at 100. A buyer would earn 8% on a bond when the market pays 7% for the same risk, so everyone would want it and the price would rise. The price climbs until the return on the price paid equals the market yield, and that happens at a premium over par. The premium is easy to see directly: compared with a 7% bond at 100, this one pays an extra rupee each year for two years, worth 0.93 plus 0.87, which is 1.81. That is a check on the long method, and it is quicker to say in the room.

The same logic runs the other way. A coupon below the market yield means a discount to par, and a coupon equal to the yield means exactly 100. If you are asked how you would price a bond in today's market, say that the yield comes from comparable bonds of the same credit and maturity; the arithmetic above is the easy part. A premium bondA bond whose price is above its face value, because its coupon is higher than the yield the market currently demands. also pulls back towards 100 as it nears maturity, so a buyer at 101.81 loses the premium slowly while collecting the fat coupon.

About 36 basis points. Over a year the bond earns 180 basis points more than the government bond. Every basis point the spread widens knocks about 5 basis points off its price, because its spread duration is 5. The extra yield is used up when widening times 5 equals 180, which is at 36 basis points. Beyond that the corporate bond does worse than the government bond.

What is the cushion, and what eats it?

Think of a shop that earns a steady margin on every sale but whose stock loses value when fashions change. The margin comes in slowly; a markdown hits all at once. A credit spread pays carry slowly over the year, while widening hits the price immediately, in proportion to spread duration. The carry is 180 basis points. A spread duration of 5 means a 1 basis point widening costs about 5 basis points of price.

What would you add to sound like a credit investor?

Two refinements, both worth a sentence. First, part of the spread pays for defaults, not risk. If expected default losses were 50 basis points a year, an illustration, only 130 is true cushion and the break-even falls to 26 basis points. The spread is not all profit, so the honest break-even uses the spread after expected losses. Second, the price loss is felt at the end of the year, when the bond is shorter; at a spread duration of about 4.2 then, the break-even is closer to 43. The 36 is the conservative, quick answer.

The ratio also compares bonds quickly. A short bond with a small spread can have a wider break-even than a long bond with a big one, because the long bond's duration magnifies every move. Credit portfolio managers often rank bonds by spread per unit of spread duration for exactly this reason.

Effective duration is about 1.7 and effective convexity about -200. Duration is the price difference across the two shocks over twice the price times the shock: 1.7 over 1.0, which is 1.7. Convexity is the sum of the shocked prices less twice the base, over the price times the shock squared: minus 0.5 over 0.0025, which is -200. The call caps the price, so it gains less than it loses.

Why effective duration rather than the usual formula?

Think of renting out a flat on a lease the tenant can cancel whenever cheaper flats appear. When rents fall, the tenant leaves and you do not keep the high rent; when rents rise, you are stuck. A callable bond's cash flows change with yields, because the issuer calls it when rates fall, so you measure its duration from how its price actually moves, not from a fixed schedule of coupons. That is effective duration: shock the yield both ways, reprice, and read the slope.

What does negative convexity cost the holder?

It means the bond loses more when yields rise than it gains when they fall. The holder has sold the issuer an option to refinance, and the price of that option is the upside given up when rates fall. A straight bond with a duration of about 4 would gain about 2.0 points for a 50 basis point fall; this one gains 0.6. The holder is paid for this through a higher yield than an equivalent straight bond, and the question for a portfolio manager is whether that extra yield covers the option given away.

Two more things are worth saying. The duration of 1.7 is not fixed: as yields fall towards the level where a call becomes likely, duration shrinks further, and as they rise, it lengthens towards the bond's straight duration. That shifting is why callable bonds, and mortgage securities with the same feature, need effective measures rather than the textbook formulas. The curve in the figure is a stylised fit around the three prices given; a real one would come from an option pricing model.

DV01 is Rs 5 lakh, so a 25 basis point rise costs about Rs 1.25 crore. Modified duration of 5 means the value moves about 5% for each percentage point of yield, or 0.05% per basis point. 0.05% of Rs 100 crore is Rs 5 lakh, and 25 basis points is 25 times that.

Why turn duration into rupees at all?

A shopkeeper who says the rent went up 4% has told you something; one who says it went up Rs 2,000 a month has told you what to do about it. Duration is the percentage version. DV01, the rupee change for a one basis point move, is the version a desk can add across positions, compare with a limit and put in a risk report. Two portfolios with the same duration but different sizes carry very different rupee risk, and only DV01 shows it.

Where does the straight-line answer stop being right?

The modified durationThe percentage change in a bond price for a one percentage point change in its yield, taken at the current yield. estimate is a tangent: it treats the price and yield relationship as a straight line. For 25 basis points the straight line is close enough, but for a 200 basis point shock the curve bends away from it and the true loss is smaller than DV01 times 200. That bend is convexity. Mention it in one sentence; the interviewer will often ask for it next.

Say also that DV01 assumes every yield in the portfolio moves by the same amount. If short rates rise and long rates stay still, a single DV01 number misses it, which is why desks also keep DV01 by maturity bucket.

About +7.3% if yields fall 100 basis points and -6.7% if they rise 100. Duration alone gives 7% either way. Convexity adds half of 60 times 0.01 squared, which is 0.3%, in both directions, because the yield change is squared. So the gain is bigger than the loss for the same size of move.

Why does convexity help in both directions?

A ball rolling in a bowl rises the same way whichever side you push it. The convexity term depends on the yield change squared, so it is positive whether yields rise or fall: it adds to the price gain when yields drop and cushions the loss when they climb. Duration is the straight tangent to the price curve; convexity measures how far the real curve bends above it.

How big does convexity get, and what does it cost?

At 100 basis points the bend is only 0.3%, but it grows with the square: at 400 basis points it is 4.8%, a large share of the move. That asymmetry is valuable in volatile markets, so bonds with more convexity usually trade at a slightly lower yield: you pay for it through carry. Callable bonds and many mortgage securities have negative convexity, and for them the sign flips: losses outrun gains.

Say the limit too. This is a second-order estimate. For very large moves the true price differs again, and an interviewer may ask you to reprice the bond from its cash flows instead.

The barbell wins both ways on an immediate parallel move, by about 0.12 points if yields fall and 0.10 if they rise, but it yields about 0.15 points less a year. With duration matched, convexity is the only difference: 71 for the barbell against 49 for the bullet. Over a year the barbell needs a move of roughly 113 to 123 basis points to earn back its lower yield, and it loses if the curve steepens.

Why does spreading money to the two ends add convexity?

Balancing a plank on a see-saw with one heavy child in the middle is steady; balancing it with a child at each end makes it swing more for the same push. Price sensitivity grows faster than linearly with maturity, so a mix of short and long bonds with the same average duration as a middle bond has more curvature than the middle bond: here about 71 against 49. More convexity means gaining a little more when yields fall and losing a little less when they rise.

What does the barbell give up, and when does it lose?

First, yield. Weighting the 2-year at about 50% to match duration gives a blended yield near 6.85%, about 0.15 points below the bullet's 7.0%. That is the price of convexity: if yields sit still, the bullet simply earns more, and over a year the barbell's edge only pays for itself on a parallel move of more than about 1.2 points. Second, curve shape. If the curve steepens, with short yields falling and long yields rising, the barbell's long leg loses and the bullet wins; if the 7-year sector rallies against the two ends, the bullet wins too.

State the assumptions: zero-coupon bonds, a blended yield taken as a weighted average, which is an approximation, and an instantaneous shift. The conclusion survives all three: the barbell buys protection against large parallel moves with yield, and takes on curve-shape risk.