MT481 Unit 5 Treasury Inventory Problem Set Example

Parallel Shift, Then a Twist: An MT481 Unit 5 Bond Market Problem Set on One Dealer's Treasury Inventory

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Purdue University Global

MT481: Financial Markets

Unit 5 Bond Market Problem Set

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[Date]

The dealer, its positions and the yield moves are composite; the sources cited are real.

The Position Table

A composite primary dealer holds three Treasury positions, each bought at par: $300 million face of a two-year note with a 4.00 percent coupon, $150 million of a ten-year note at 4.25 percent and $60 million of a thirty-year bond at 4.50 percent. One table carries the positions through all five problems, so each new calculation can be read against the last. All coupons are paid semiannually, and prices are quoted per 100 of face value to three decimals.

Problem 1: A Parallel Rise of 50 Basis Points

If every yield rises half a point, each position is repriced at its new yield. The two-year note falls to 99.054, a loss of about $2.84 million. The ten-year falls to 96.056, about $5.92 million. The thirty-year falls to 92.273, about $4.64 million. The total loss is about $13.4 million. The thirty-year position is the smallest by face value but loses nearly as much as the ten-year, because long bonds are far more sensitive to yield changes.

Problem 2: Value of a Basis Point

Trading desks summarize rate exposure as the dollar value of a one-basis-point change in yield. For the two-year position it is about $57,116; for the ten-year, about $121,169; for the thirty-year, about $98,247. The firm's total exposure is about $276,500 per basis point. Reading the positions this way shows that the ten-year holding, not the largest by face value, carries the most rate risk.

Problem 3: Duration Against Full Repricing

Modified duration gives the approximate percent move in price per point of yield: about 1.90 for the two-year, 8.08 for the ten-year and 16.37 for the thirty-year. Applying duration to the 50-basis-point move predicts prices of about 99.048, 95.961 and 91.813. Full repricing gave 99.054, 96.056 and 92.273. For the two-year note the estimate is almost exact. For the thirty-year bond it overstates the fall by 0.46 points, about $276,000 on the position.

Duration is a straight-line guess at a curved relationship, and the long bond is where the curve bends most.

Fabozzi (2015) describes this gap as the effect of convexity: price gains from falling yields outrun losses from rising ones, so a straight-line estimate is too pessimistic in both directions, more so for long bonds and big moves.

Problem 4: A Twist

Yields rarely move in parallel. Suppose the two-year yield falls 25 basis points while the ten-year rises 10 and the thirty-year rises 40. The two-year position gains, the other two lose, and the net loss is about $3.53 million. A dealer who measured risk only by total dollars per basis point, assuming every yield moves together, would have missed that its exposure depends on the shape of the curve as well as its level.

Problem 5: A Hedge That Fails

Suppose the dealer wants to remove the thirty-year bond's rate risk by selling short two-year notes. Each $1 million of two-year notes has a basis-point value of about $190, so offsetting the long bond's $98,247 would take a short position of about $516 million in two-year notes. Against a parallel move, the hedge works: the two exposures cancel. Against the twist in Problem 4, it fails badly. The two-year yield falls, so the short position loses, while the thirty-year yield rises, so the bond loses too. Both sides of the hedge lose at once.

The lesson is that a hedge built on a single measure of rate risk protects only against the kind of move that measure assumes. Mishkin (2019) notes that the slope of the yield curve shifts with expectations about policy and growth, so dealers hedge long positions with instruments of similar maturity, such as long bond futures, rather than with short notes.

What the Dealer Should Report Daily

The problems suggest a better daily risk report. Instead of one total for dollars per basis point, the report should show exposure in maturity buckets, short, intermediate and long, so a manager can see how the book would behave if the curve steepened or flattened. It should also show the loss from a set of standard scenarios, including a parallel move, a steepening and a flattening, alongside the duration-based estimate, so that convexity errors on long positions are visible. Such a report takes little extra work once positions are repriced, and it would have warned the dealer in Problem 5 before the hedge was put on.

What the Set Shows

Taken together, the problems show that price sensitivity rises with maturity, that basis-point value is the practical measure desks use, that duration is accurate only for small moves and short bonds and that curve shape can matter as much as curve level. Litterman and Scheinkman (1991) found that most movements in Treasury yields can be described by three factors, level, slope and curvature, which is why a risk report built on level alone leaves a dealer exposed.

References

Fabozzi, F. J. (2015). Bond markets, analysis, and strategies (9th ed.). Pearson.

Litterman, R. B., & Scheinkman, J. (1991). Common factors affecting bond returns. The Journal of Fixed Income, 1(1), 54-61. https://doi.org/10.3905/jfi.1991.692347

Mishkin, F. S. (2019). The economics of money, banking, and financial markets (12th ed.). Pearson.