Bond Exercises
1. Calculate the yield-to-maturity, the Macaulay duration, and the modified duration for a 5% annual pay coupon bond with exactly five years to maturity that is currently selling for 105.
2. Your company is about to issue five year bonds that will be rated BBB by Standard and Poors. Your investment bankers tell you that you will have to pay 160 basis points over Treasuries. Look up the yield-to-maturity on five year Treasuries in the Wall Street Journal or online at wsjmarkets.com under Bonds and then Treasury Quotes and calculate the yield-to-maturity on the bonds you are about to issue.
3. Look up the price of the February 2037 4.75% US Treasury Bond (CUSIP: 912810PT9). You can find Treasury Price quotes at https://www.wsj.com/market-data/bonds/treasuries
a. Using the spreadsheet features in Excel, calculate the yield to maturity, the duration, and the modified duration of the bond.
b. Using Excel, calculate the accrued interest on the bond.
c. Using Excel, construct a graph of bond price versus yield to maturity for the bond.
d. Calculate the effective duration and convexity of the bond using the formulas in the book or the class notes.
e. Using only your effective duration estimate, what would be the price of the bond if the YTM increased by 100 basis points?
f. Using your effective duration and your effective convexity estimate, what would be the price of the bond if the YTM increased by 100 basis points.
g. Using Excel, calculate the exact price of the bond if the YTM increased by 100 basis points. How close is it to your answer in part g?
4. Your job is to construct a $100 million long-only diversified portfolio of U.S. Treasury Bonds with a modified duration of no more than 7.5 years but with the highest yield to maturity.
a. Construct a portfolio with at least 10 separate issues that meets this objective. What is the yield to maturity and modified duration of this portfolio?
b. What will the modified duration of your portfolio look like one year from now?
c. What trades will you have to make over the next year in order to maintain the target modified duration?
Hints:
· The spreadsheet Unit 4 Treasury Quotes September 20 2021 from WSJ for Bond Exercises.xlsx has Treasury Quotes that you can use for the exercise. You will have to calculate the modified duration of the bonds, and then calculate the modified duration and yield of your chosen portfolio. Use Solver to find the optimal portfolio.
· Use SUMPRODUCT function to calculate yield and modified duration of the portfolio.
· Count the total number of bonds in the portfolio with a weight greater than zero using COUNTIF function. Make the number of bonds one of the constraints in your optimization.
For the Institutional Asset Management Track
Tesla Motors, Inc. 2015
You work for a hedge fund that does convertible bond arbitrage and have been asked to analyze Tesla’s convertible bonds for potential arbitrage opportunities. Prepare a presentation in PowerPoint for your senior management committee with your recommendations, along with an Excel spreadsheet with your calculations. Bring your PowerPoint presentation to class and be prepared to present it to your classmates.
Tesla Motors, Inc. is a fast growing manufacturer of electric vehicles and energy storage devices with large expansion plans that will require massive capital investments. The stock price has appreciated rapidly in recent years and is now selling for more than 10 times the original 2010 IPO price of $17.
Fast growing capital-hungry issuers with high stock prices often turn to the convertible debt market as a means of raising capital. Convertible debt gives an issuer the ability to use investor enthusiasm for the issuer’s future prospects to raise capital at very low cost. Convertible debt is debt that can be converted into the common equity of the company. Such debt combines plain vanilla debt with a warrant to purchase the common shares of the company. The coupon rate on convertible debt is usually less than the issuer would otherwise have to pay as a result of the value of the warrant. A high perceived value of the future volatility of the stock increases the value of the warrant and thus decreases the required interest rate on the convertible. Convertibles give the investor the “best of both worlds” by giving the investor equity if the company performs well, and the safety of debt – or negotiating power in a debt restructuring– if the company does not perform well.
In general, holders of convertible debt will choose to hold the debt and not convert it into equity as long as the interest income on the debt is higher than the dividends paid, if any, on the equivalent amount of stock.
Your hedge fund looks for underpriced convertible bonds. When it finds one, the fund purchases the bond and usually hedges the risk of the embedded option by shorting the shares of the common stock. It uses a form of the Black-Scholes option pricing model in order to determine how many shares to short. As market conditions change, it is necessary to change the number of shares short. At times, however, the fund may choose not to hedge if it believes that the stock price actually will appreciate enough to compensate it for the risk.
In 2014, Tesla raised $2 billion by issuing two bonds. The first was $800 million of 0.25% bonds maturing on March 1, 2019, CUSIP 88160RAB7. These bonds are convertible into stock at a rate of 2.7788 shares per $1000 of face value, an effective conversion price of $359.87. At the time of the issue, Tesla common was selling for approximately $250 per share. In addition, Tesla raised $1.2 billion of 1.25% bonds maturing on March 1, 2021, also convertible at $359.87 (2.7788 shares per $1000), CUSIP 88160RAC5.
The bonds were not rated by any of the rating agencies at the time of issue. Given the continuing losses and high capital needs at Tesla Motors, along with the high degree of execution risk in the company’s ambitious plans, it is likely that the bonds would have been rated below investment grade (a/k/a “junk”). This was confirmed not long after the bonds were issued, when Standard and Poor’s issued an unsolicited rating of B-.[footnoteRef:1] Yields on bonds rated B- vary greatly based on the degree of financial distress of the company and option features of the bonds such as call and convertibility provisions. A typical 5- year B- bond was trading at about 600 basis points over comparable U.S. Treasury bonds. [1: Robinson, Michael, May 27, 2014, “Tesla Gets Unsolicited S&P Junk Rating on ‘Niche’ Position” http://www.bloomberg.com/news/articles/2014-05-27/tesla-gets-unsolicited-junk-grade-from-s-p-on-niche-position ]
The bonds contain virtually no restrictive covenants on the use of proceeds, issuance of additional debt, payment of dividends, or the maintenance of any particular financial ratios.
In addition to the usual conversion feature, the bonds contain a “make whole fundamental change” provision which would slightly increase the number of shares into which the bonds were convertible in the event of a merger or other fundamental change in the nature of the company. Such a fundamental change could force conversion of the shares and thus extinguish any remaining option premium, and the additional shares are intended to at least partially compensate for the lost option premium.
The bonds are not callable. Thus, Tesla would not be able to force conversion of the bonds prior to maturity.
Tesla has also hedged its exposure to the warrant feature in the bonds by spending $186 million to purchase a call from its underwriters at an exercise price of $359.87 and selling a warrant to the underwriters with an exercise price of $512.6562 for the 2019 notes and $560.6388. This had the effect of delaying the required issuance of additional shares until the stock price reached the higher exercise prices.
If all of the 2019 bonds were converted to equity, 2,223,040 shares would be issued. Likewise, if all of the 2021 bonds were converted, an additional 3,334,560 shares would be issued. However, these shares would be offset by the hedge, and thus dilution would not occur until the stock price reached the higher exercise prices of the warrants sold as part of the hedge. The company has 125.5 million shares outstanding. Employees hold options to purchase approximately 15 million shares, and the board has the authority to issue additional employee stock options at a rate of 4% of the shares outstanding each year.
Tesla common stock (TSLA) had an average daily trading volume of approximately 6 million shares per day. Approximately 25 million shares were reported as sold short, representing about 20% of the total outstanding and 26% of the public float. Despite the relatively large short interest, shares were readily available for borrowing at moderate cost. The annualized cost of shorting TSLA common was 0.43%. The bonds could be borrowed for shorting at an annualized cost of 0.25%.
Tesla shares were widely owned by institutional investors, who owned 57% of the shares. Tesla CEO Elon Musk, who accepted only $1 per year in base compensation, owned 27% of the shares. Mr. Musk, one of the founders of PayPal, also had a major interest in Solar City, a provider of solar energy systems.
On February 13, 2015, your trading screen displayed the following prices for the bonds and the common stock.
Finance.yahoo.com contains information about the Black-Scholes implied volatility of various exchange-traded options. Figure 2 contains a screen shot from February 13, 2015 with implied volatilities for various call options with an exercise price of $360 per share. However, such exchange-traded options do not extend out to the maturities of the Tesla bonds, so care must be taken in assessing the correct volatility.
Four year treasuries maturing in 2019 were yielding 1.25% while six year Treasuries maturing in 2021 are yielding 1.69%. B- bonds were typically yielding 475 basis points over Treasuries.
Be sure to cover the following points in your presentation:
a. Using the yield for a typical B- bond without a conversion feature, determine the value of the Tesla bonds as if they had no conversion feature. (Hint: Use the PRICE function in Excel.)
b. Calculate the implied option value by subtracting the value of the Tesla bonds without the conversion feature from the market price of the Tesla bonds.
c. What volatility do you think is appropriate for valuing the warrants embedded in the bonds according to the Black-Scholes model? (HINT: See Figure 2.)
d. Using the Black-Scholes option pricing formula for warrants, estimate the value of the warrant feature in each bond. (Recall that when a warrant is exercised, the company actually issues more shares, unlike a regular option. Thus, the Black-Scholes call value needs to be adjusted for the dilution caused by the issuance of the additional shares.) How does your warrant value compare with the implied value in part c?
e. Do you see an arbitrage opportunity between the convertible bonds and the stock? If so, what trades would you execute, and how much money could you make? How would you execute the trades? Who would monitor the position and adjust the hedge as needed?
Figure 1: Stock Price History of Tesla Motors
Figure 2: Option data from Yahoo! Finance.