Present Value of Tax Shields on an Amortizing Loan
Summary
The document examines how to value interest tax shields when a loan balance declines through repayments. It contrasts two proposed shortcuts: applying a geometric series to a shrinking balance, and treating the shield as a growing perpetuity. An answer instead lays out a finite annuity-style calculation for a loan with fixed annual principal amortization, separates principal and interest cash flows, and values the tax benefit as a fraction of discounted interest payments. For the stated example, that method gives a tax-shield value of $11.8 million.
The example assumes annual payments, a fixed amortization amount, and a discount rate equal to the stated loan rate. The response explicitly cautions that repayment as a percentage of the remaining balance creates a perpetually shrinking loan and requires a different geometric-series derivation. The original question uses that percentage-repayment setup, so the fixed-amortization example does not directly resolve it. The note is useful for distinguishing repayment conventions and linking tax-shield value to the discounted interest schedule.
Key ideas
- An interest tax shield depends on the interest paid on the outstanding loan balance over time.
- For fixed annual principal amortization, discounted interest payments can be valued separately from principal repayments.
- The tax-shield present value is the tax rate multiplied by the discounted interest component.
- A percentage repayment of the remaining balance differs from a fixed annual amortization and needs a separate derivation.
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Full text
# Valuing interest tax shield with constant rate of loan redemption
# Valuing interest tax shield with constant rate of loan redemption
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 by Attack68 (score 1)
https://quant.stackexchange.com/a/50306
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 seriesShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.