✏️ Present value of tax shield from permanent debt

✏️ Suppose your firm borrows $1M and pays only the interest, keeping the same amount of principal outstanding by rolling the debt over (permanent debt). The interest rate/yield is 5% and the marginal tax rate is 25%. What is the present value of this tax shield? Apply the tax shield at the end of the year.

Hint: we approach questions like this in three steps:

1. What is the annual interest payment?
2. What is the tax shield provided by this interest payment?
3. What is the present value of the annual tax shields?

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(A.) What is the annual interest payment?

$$\begin{aligned} \text{Interest Payment} & = \text{principal} \times r \\ & = \$1M \times 5\% \\ & = \$50,000 \\ \end{aligned}$$

(B.) What is the tax shield provided by this interest payment?

$$\begin{aligned} \text{annual tax shield} & = \text{interest payment}\times\text{marginal tax rate} \\ & = \$50,000 \times 25\% \\ & = \$12,500 \end{aligned}$$

If there are taxes of at least $12,500, the interest tax shield will decrease them by $12,500.

(C.) What is the present value of the annual tax shields?
We get the tax shield every year going forward, so we can use the perpetuity formula to calculate the PDV of the tax shield. The only question is what we should use as a discount factor. We will use the return on the firm’s debt because it is matched with the level of risk of cash flows from the firm. In other words, the tax shield is like “fixed income” for the firm, and, as we know, fixed income is a synonym for bonds and other debt instruments. ⇨ we use the interest rate as a discount factor.

$$\begin{aligned} \text{PV of tax shield} & = \frac{\text{annual year-end taxshield}}{\text{discount rate}} \\ & = \$12,500 / 5\% \\ & = \$250,000 \end{aligned}$$

But what if the yield on debt isn’t known?

It turns out that the yield on debt doesn’t matter for this problem.

Suppose that you aren’t given the yield on debt. Let’s just give the yield on debt a name. We’ll call it yield.

Suppose that you have a permanent debt of D and a marginal tax rate of MTR

Let’s go throught the same three steps.

1.) What is the annual interest payment?

Annual interest payment = D *yield.

2.) The tax shield, as above will be:

Annual Tax Shield (every year) = Annual interest payment * MTR = D *yield * MTR

3.) The PV of thes annual interest payment can be calculated using the perpetuity formula, like we did above.

For a discount rate, we will use the yield on your debt, for the same reason as we did above.

Perpetuity formula:
PV of cash flows you get every year, forever = CF/i

PV of permanent tax shield = value of annual tax shield / i = D *yield * MTR / yield

We can cancel yield out! Therefore, the yield of our debt will disappear from our formula!

PV of permanent tax shield = D * MTR

The PV of permanent debt is always just the amount of debt times your marginal tax rate.

You can use this to calculate the value of a permanent tax shield if you aren’t given the interest rate that the firm pays. It turns out that the interest rate that the firm pays doesn’t matter at all!