MT480 Unit 3 Bond Valuation Problem Set Example

$921.89, $1,000 and $1,086.46: An MT480 Unit 3 Bond Valuation Problem Set on Price Against Required Return

[Student Name]

Purdue University Global

MT480: Corporate Finance

Unit 3 Bond Valuation Problems

[Instructor Name]

[Date]

Brandywine Hollow Mushrooms, its notes and every figure are composite; the sources cited are real.

The Layout

Each problem states the bond's terms, draws a timeline of semiannual coupons and principal, applies the price formula with its semiannual adjustments and closes with a sentence on what the answer means for Brandywine Hollow Mushrooms, the composite grower whose financing the course follows. Semiannual payment matters throughout: a 5.40 percent annual coupon on a $1,000 note pays $27 every six months, and yields are divided by two and periods doubled before discounting.

Problem 1: The Grower's Notes at Three Yields

The grower's 5.40 percent notes have eleven years, 22 semiannual periods, left to maturity. Each period pays $27, and $1,000 is repaid at the end. Its value today adds the discounted worth of all 22 coupons to the discounted worth of the final $1,000.

At a required yield of 5.40 percent, the coupon equals the yield, and the notes sell at par, $1,000. At 6.40 percent, discounting at 3.2 percent a period, the price falls to about $921.89. At 4.40 percent, discounting at 2.2 percent, it rises to about $1,086.46.

A one-point rise in yield costs $78.11; a one-point fall gains $86.46. The asymmetry is convexity: a drop in yield lifts the price more than an equal rise in yield lowers it, which favors holders of long bonds when rates are uncertain. Fabozzi (2015) describes convexity as the curvature of the price and yield relationship, and these three prices show it in a way a formula alone does not.

Problem 2: A Shorter Note Moves Less

The grower also has a three-year note with a 4.90 percent coupon, priced at par. A one-point rise in yield, to 5.90 percent, lowers its price by about $27.13; a one-point drop lifts it about $28.05. The same one-point change that moved the eleven-year notes by roughly $80 moves the three-year note by under $30.

The same one-point move in rates shifts the eleven-year notes about three times as much as the three-year note.

Longer maturities put more of a bond's value in distant payments, which are discounted most heavily, so they respond more to changes in yield. A treasurer worried about rising rates would prefer short notes; one expecting rates to fall would prefer long ones.

Problem 3: From a Quoted Price to a Yield

A dealer quotes the grower's eleven-year notes at 96.35, meaning 96.35 percent of face value, $963.50. Solving the price formula for the yield that makes the present value of the payments equal $963.50 gives a yield to maturity of about 5.855 percent a year. Dividing the $54 of annual coupons by the $963.50 price gives the current yield, about 5.605 percent. The yield to maturity is higher than the current yield because the buyer also gains $36.50 when the note is repaid at par.

Problem 4: A Zero-Coupon Bond

An eight-year zero-coupon bond makes a single payment, at the end. Using semiannual compounding, a rise in its yield from 6.1 percent to 7.1 percent lowers its price by about 7.45 percent. A zero is the most sensitive bond of a given maturity, because all of its value sits in one distant payment.

Problem 5: A Callable Issue

The grower is considering a new ten-year bond with a 6.25 percent coupon, callable after four years at $1,030. If it sells for $1,062.50, its yield to maturity is about 5.43 percent, but its yield to call, assuming the grower redeems it at the first opportunity, is about 5.18 percent. Brealey et al. (2020) note that a call option held by the issuer lowers the bond's value to investors, so they should expect the lower figure, because a borrower calls bonds when rates have fallen, exactly when the bond would otherwise rise in value. The call protects the grower and costs the investor; the price reflects that trade.

What the Results Mean for the Grower's Refinancing

The grower's notes now trade below par because market yields have risen above their 5.40 percent coupon. That has two consequences. Investors buying the notes today earn more than the coupon, through the gain at maturity. And if the grower issues new debt, it must pay the current market rate, not the old coupon, which is why Unit 7 values the notes at market price rather than face value. A treasurer reading these problems would also note that a callable structure buys the company flexibility to refinance if rates fall, at the price of a higher coupon or a lower issue price today.

What the Set Shows

Taken together, the problems show four things a treasurer needs: price and yield move in opposite directions, longer and zero-coupon bonds move more, a market price always implies a yield and a call feature limits an investor's gain. Ross et al. (2022) present bond valuation as the simplest application of discounting, and these problems use it to answer the questions the grower will face when it refinances its notes in Unit 7.

References

Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of corporate finance (13th ed.). McGraw-Hill Education.

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

Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2022). Fundamentals of corporate finance (13th ed.). McGraw Hill.