Valuing interest tax shield with constant rate of loan redemption

Question

A $D=\$30mm$ loan at $r_D = 6.5\%$ and a tax rate of $\tau_c=40\%$ yields an annual tax shield of $$TS=D*r_D*\tau_c=\$0.78mm$$

If $\rho=5\%$ of the loan remainder in the current year is to be payed back in addition to the interest payments and the tax shield can be discounted at the loan interest rate, what is the present value of the tax shield?

My approach was to multiply the annual tax shield with the geometric series $\sum_{x=1}^{\infty} (1-\rho)^{x} = \frac{1}{1-(1-\rho)}=\frac{1}{\rho}$, where $x$ denotes the number of years since the loan inception. This method yields $$PV(TS)=\frac{D*r_D*\tau_c}{\rho}=\$15.6mm.$$

This result however is more than twice as large as a different solution working with the rate of repayment $\rho$ as a negative growth rate based on the Gordon Growth Model. $$PV(TS)=\frac{D*r_D*\tau_c}{r_D-\rho}=\frac{0.78Mio\$}{0.065-(-0.05)}=\$6.78mm$$

Based on my gut feeling, the second option seems more correct, however I am unable to find a mistake in the first method.
Thanks for any help!

Answer 1

Given an annually payable loan of:

$$ \begin{split} D &= \text{notional} \\ r_f &= \text{risk free rate for discount }\\ r_D &= \text{interest rate payable on outstanding balance}\\ A &= \rho D = \text{fixed yearly amortization amount}\\ N &= 1/\rho = \text{number of annual payments} \\ \end{split} $$

The PV of the loan is:

$$PV = \sum_{i=1}^{\frac{1}{\rho}} v_i \left ( A + r_DD(1-(i-1)A) \right )$$

where $v_i$ is the discount factor given by:

$$ v = \frac{1}{1+r_f}, \quad v_i = v^i$$

So:

$$ PV = (A + r_D D) \left ( \frac{v-v^{N+1}}{1-v} \right ) - r_DA \left( \frac{v^2-v^{N+1}}{(1-v)^2}-\frac{(N-1)v^{N+1}}{1-v} \right) $$

You can observe that this PV is separable into capital repayment and interest payments. Since the tax shield is applicable to a specific percentage of the interest payment calculation you can extract this directly:

$$ PV \text{ of tax shield} = \tau_c r_D \left ( D \left ( \frac{v-v^{N+1}}{1-v} \right ) - A \left ( \frac{(N-1)v^{N+1}}{1-v} \right ) \right )$$

For example with your numbers: $D=\$30mm, r_f=6.5\%, r_D=6.5\%, \rho=5\%, N=20yrs$ I calculate that your PV tax shield is worth $\$11.8mm$.

Note: if the yearly amount is not a fixed amortisation but a percentage of the outstanding balance (i.e. perpetually shrinking loan) then you have to rework the formulae for geometric series