Interest Rate Tax Deductions: Example

PanterSkill Corporation is deciding between issuing two different bonds. They can either issue a
zero-coupon, 30-year bond, with a YTM of 7.75% (Bond "A"), or, a 30-year, 20%-coupon bond,
with a YTM of 8.125% (Bond "B"). Both bonds have a \$1,000 face value.

(1) What are the prices that an investor would be willing to pay for these bonds?

Zero-coupon bond "A" = \$106.53
Level-coupon bond "B" = \$2,225.26 + \$95.99 = \$2,321.25

(2)  If the firm wants to raise approximately \$100,000 and wants to either issue 940 bonds "A", or 43 bonds "B", what are the total annual explicit interest expenses for the firm?

No explicit interest is paid on the zero-coupon bond (no coupons)
For the level-coupon-bond there are 43 x \$200 = \$8,600

(3)  If PanterSkill issues the 940 zero-coupon bonds ("A"), what is the total implicit interest being paid over the 30 years?

(\$1,000 - \$106.53) x 940 = \$839,861.80

(4)  If PanterSkill, instead, issues the 43 bonds "B", what is the total interest being paid over the 30 years?

30 x \$8,600 = \$258,000

(5) If the corporate tax rate is 35%, and the tax law allows PanterSkill to take the interest rate deduction on the implicit interest only at the end of the 30 years (the firm can take regular annual deductions on explicit interest payments), what is the present value of the tax deductions for both bonds if the firm's EBIT is \$1 million per year, and the appropriate discount rate is 9%? (Hint: the EBIT of \$1 million per year is really irrelevant, what matters is how much the firm can save each year on its tax bill based on interest rate deductions).

Zero-coupon bond: (\$839,861.80 x 0.35) / (1.09)30 => PV = \$22,155.47
Level-coupon: Annuity for 30-years, with t = 30, r = 0.09, and C = 0.35 x 8,600 = 3,010 => PV = \$30,923.70

(6) All else equal, which of the two bonds ("A" or "B"), based on your previous answer, is more attractive to PanterSkill and why?

In terms of tax benefits, the level coupon bond is more attractive. In this example the firm has two different YTM's (one for each bond) while they are raising money for the same project. The bottom line is that the financing costs are not taken into account (just like for any capital budgeting problem), only the tax-shields matter !!! The financing costs, will be taken into account when calculating the correct discount rate, later in the course.

Duration and Interest Rate Risk: Example

Consider the following two bonds with the same yield-to-maturity (YTM) of 6%: Bond A is a
15-year, 25% coupon bond, and bond B is a 5-year, 5% coupon bond.

(1) Calculate the prices for both bonds.

Bond A: \$2,428.06 + \$417.27 = \$2,845.33 (premium bond)
Bond B: \$210.62 + \$747.26 = \$957.88 (discount bond)

(2) What are the prices for the bonds next year, if everything remains the same?

Both will have one year less before maturity:

Bond A: \$2,323.75 + \$442.30 = \$2,766.05
Bond B: \$173.26 + \$792.09 = \$965.35

(3) What happens to the prices of these bonds if the YTM increases to 7% in the next year, everything else being the same? (Hint: calculate the price for next year with YTM = 7%)

Bond A: \$2,186.37 + \$387.82 = \$2,574.19
Bond B: \$169.36 + \$762.90 = \$932.26

(4) What happens to the prices of these bonds if the YTM decreases to 5% in the next year, everything else being the same? (Hint: calculate the price for next year with YTM = 5%)

Bond A: \$2,474.66 + \$505.07 = \$2,979.73
Bond B: \$177.30 + \$822.70 = \$1000 (par bond)

(5) Which of the two bonds, based on your previous answers, is the most sensitive to a change in the interest rate (YTM), or, in other words, which of the two has the highest interest rate risk?

Bond A:  if YTM increases by 1% => bond price decreases by (\$2,766.05 - \$2,574.19) = \$191.86 or by \$191.86 / \$2,766.05 = 6.94%
If YTM decreases by 1% => bond price increases by (\$2979.73 - \$2,766.05) = \$213.68
Or by \$213.68 / \$2,766.05 = 7.73%

Bond B: if YTM increases by 1% => bond price decreases by (\$965.35 - \$932.26) = \$33.09 or by \$33.09 / \$965.35 = 3.43%
If YTM decreases by 1% => bond price increases by (\$1,000 - \$965.35) = \$34.65 or by \$34.65 / \$965.35 = 3.59%

For bond A  the price uncertainty is much greater, and hence it has greater interest rate risk, interest rate sensitivity.

(6) Calculate the duration measure for both bonds as of now.

On an exam, you will NOT have to calculate the duration for bonds with more than 4 years to maturity!!

Bond A:
D = 1 * [\$250/1.06] + 2 * [250/(1.06)2] + 3 * [250/(1.06)3] +  . . .  + 15 * [1,250/(1.06)15] / 2,845.33 = 8.11

Bond B:
D = 1 * [\$50/1.06] + 2 * [50/(1.06)2] + 3 * [50/(1.06)3] + 4 * [50/(1.06)4] + 5 * [1.050/(1.06)5]  = 3.75

(7) Without calculations, what will happen to the duration for these bonds next year and why?

They will both go down, since a shorter-time to maturity implies less interest rate risk.

IPO's, Underpricing & Winner's Curse: Example

Firms A and B both have announced an IPO; each firm's stock will be sold at \$10 per share. One the firms is undervalued by \$1, while the other is overvalued by \$0.50, but you are unable to determine which of them is undervalued and which of them is overvalued. Informed investors like big banks and pension funds are able to make this distinction. You plan to buy 100 shares of each. If an issue is rationed, you will be able to purchase only half of your order.

(1) If you are able to buy 100 shares of each firm's stock, what is your profit?

Without any rationing, your profit would be (\$1*100) - (\$0.50*100) = \$50

(2) For which of the two IPO's would you expect to be rationed given that other investors have better information, and hence know how to distinguish between the two issues?

You'd expect the underpriced issue to be rationed, since informed investors will subscribe to this one, but they won't subscribe to the overpriced one.

(3) What profit do you expect in reality?

With rationing (and being an uninformed investor) we expect our profits to equal:
(\$1*50) -  (\$0.50*100) = \$0

(4) What is the average underpricing?

One IPO is overpriced by 5% (\$0.50 on a \$10 price) and the other is underpriced by 10% (\$1 on a \$10 stock), hence, on average we have
[10% - 5%] / 2 = 2.5% underpricing.

(5) What is your expected profit if the undervalued stock is undervalued by \$0.50, everything else being the same?

The underpriced stock will still be rationed: (\$0.50*50) - (\$0.50*100) = -\$25

(6) What is in this case the average underpricing?

[5%-5%] / 2 = 0%

(7) Why do we have a winner's curse in this situation?

Without any underpricing, uninformed investors would incur a loss if they participate in IPO's. They will either get all the shares they ordered, but the shares will be overvalued, or, they will receive less or no shares in case the shares are undervalued.

(8) What is the expected profit if the undervalued stock is undervalued by \$1.50, everything else being the same?

(\$1.50*50) - (\$0.50*100) = \$25

(9) What is the average underpricing now?

[15% - 5%] / 2 = 5%  => more underpricing on average results in more gains for uninformed (and informed) investors.

(10)  Does it make sense to underprice by this much if the goal of the firm is a) to get you interested
in the IPO, and b) to minimize its losses.

No, by underpricing by 2.5% or slightly more the uninformed investors are already willing to participate because on average they will not be punished for not being informed. Underpricing by more would create larger losses for the firm.

Rights Offering Example

The Pitt-Panther Corporation has announced a rights offering to obtain \$10 million of equity financing for a new project. The stock currently sells for \$80 per share; there are 2 million shares outstanding.

(1) If Pitt-Panther sets the subscription price at \$20 per share, how many shares must be sold?

\$10 million / \$20 = 500,000

(2) How many rights are required in order to buy one share?

2 million / 500,000 = 4 rights are required to buy one new share

(3) What is the ex-rights price?

4 shares at \$80 = \$320
1 share at \$20 = \$  20
total is 5 shares, worth \$340, or \$68 per share.

(4) What is the value of a right?

The price including the right is \$80, the price excluding the right (since you just exercised that right) is \$68, therefore, the value of the right is \$12  (Think about the bargain that you are offered; you can buy a share in the firm (excluding the right) for \$20, while we know that the real value of a share without a right is \$68. That is \$48 profit. You need, however 4 shares to realize this profit, or \$12 each.)

(5) Suppose you own 100 shares, but intend to sell the rights, instead of exercising the rights and buy new shares. Is this a good decision?

Does not matter. If you do not sell, you have a value of 100 shares at \$80 is \$8,000. If you sell your rights, you will have 100 shares worth \$68, plus you can sell 100 rights at \$12 each, your total is still \$8,000.

(6) What if you decide to exercise your rights, does this affect your total value?

No. You will have 125 shares now (100 plus the additional 25 you buy in the rights offering) valued at \$68 each, or \$8,500 in total. This is the same as having 100 shares at \$80 each, and buying 25 more shares at \$20 each.

(7) If the rights are trading at \$10 per right, how will this affect you if you own 100 shares and what would you recommend to someone who does not own shares?

If rights are trading at \$10 each, you prefer to exercise the rights, because than we know the rights have a \$12 value. You are not willing to sell something for \$10 that is worth \$12. If you sell, you make a loss of 100 ? \$2 = \$200.

If you could buy right s for \$10 each, you want to buy as many as possible.  If you buy 4 rights for \$40 and an additional share in the rights offering, you now have share worth \$68, but you paid just \$60.

In other words, at a price of \$10, nobody wants to sell rights, and everyone would want to buy as many as possible. This is not a sustainable situation: the rights have to trade at \$12 each given the current stock price of \$80.

In this case you want to sell your rights. You make a dollar profit on each right you can sell. However, nobody wants to buy rights for \$13. If you buy for rights for \$52 plus an additional share for \$20, you paid \$72 for a share that is worth only \$68. Again, the right can only trade at \$12 given the current stock price of \$80.

(9) What would be the minimum current share price (including the right) for you to be interested in buying rights at \$13 each?

Pin = Price including rights
Pex = Price excluding rights

First note that the higher the current stock price (which is now equal to \$80), the more attractive it becomes to be offered to buy shares for \$20. In other words, the higher the stock price, the more you would be willing to pay for a right.

If we know that a right is trading at \$13, it means that the difference in the price with and without rights of the stock is \$13, or, Pin - Pex = \$13, or that Pex = Pin - \$13.

We also know that if we buy four (4) shares, at Pin, we can buy one more share for \$20. Therefore, we can have 5 shares for the total price of
(4 * Pin) + \$20.

The average price, excluding the rights, is thus: Pex = [(4 * Pin )+ \$20] / 5  => substituting for Pex:

Pin - \$13 = [(4 * Pin) + \$20] / 5  => Pin = \$85

Which makes sense, because if you can buy shares for \$20, it becomes more attractive is the real stock price is higher. In other words, the value of the right, which allows you to benefit from the firm's bargain price, has become more valuable.