# Deriving Cost of Debt formula

1. Jun 9, 2015

### Square1

Hi,

There is a cost of debt formula that goes c = ib(1-T), where c is the cost of issuing debt that firm incurs ( a percentage like interest) i is the interest earned by the investor, and T is the tax rate a firm pays. The key is that the interest rate alone is not really the cost of taking on debt. Although the investor will obtain the interest rate i from the company, the effective rate being paid by the company is less than i. This is because at the end of financial period, a firm can always deduct interest payments on debt from taxable income.

I understand the logic that the effective cost of debt to the firm will be less than some nominal interest rate, but I don't understand the the mechanics of this formula. Can someone help? One thing that is confusing me is that the interest rate would get applied to one quantity, whereas the tax rate would be applied to another quantity. How can we establish a relationship between the two based on that while not referencing these quantities in the formula?

Thanks

2. Jun 9, 2015

### PAllen

As long as the company has > ib income, the tax it would pay on some matching income would be ibT. Given the deduction, it doesn't pay this tax. Thus the cost of paying the interest has been reduced by this amount. Thus the given formula.

And if the company has inadequate income to pay the interest ... it is in big trouble. There is the possibility that it has enough income, but not enough taxable income. Then the benefit would be less. Such a special case is not captured in that formula.

3. Jun 9, 2015

### Square1

I'm not following this :(

ib is the interest rate earned by the person providing the funds to the company. It's just an interest rate, not a tax rate.

I'll reword it maybe. c = ib(1-T)
where:
c, is the cost of borrowing (like interest on a mortgage a homeowner pays)
ib, is the interest rate the investor is receiving
T, is the tax rate

Normally you would think that c should equal ib, but it is not the case for a business since they will always deduct the payments of interest from their taxable income, thus effectively guaranteeing a cost of debt, c, that is less than the nominal rate ib.

I'm expanding the formula and looking at it like this: c = ib - ibT. I dunno....I think it's better to look at this way but I'm still lost....

EDIT: I'm still looking into this and the concept surrounding this is called "tax shield", or "debt tax shield". Quickly googling some articles on this and I think it may be the right direction to take.

Last edited: Jun 9, 2015
4. Jun 9, 2015

### PAllen

Just think about what I wrote. It is a complete explanation.

5. Jun 9, 2015

### Square1

Hmm. Assuming you swapped symbols ib for income instead of interest rate, I think your first paragraph is saying the part which I understand. The second paragraph is basically describing bankruptcy, which I understand.

The point is that I am looking for the math for deriving the formula. I think I might have got it now, but I really had to approach it from a "totals" sense (meaning I multiplied the rates by the principal of a loan). It's too hard for me thinking about it only with percentages :( Any tips?

Total Cost of Debt = rb*P - rb*P*T...(1)
Where:
P, is the principal of the loan

I factor out P so that I am just left with
Total Cost of Debt/P = rb - rbT...(2)
c = rb(1-T)...(3)
Where:
c, is the after tax cost of debt rate

6. Jun 9, 2015

### PAllen

No, I didn't mix up symbols. Your way of looking at it is ok, but not sure why you have trouble with the simple logic as given.

Suppose company X borrows money for which they pay investor 5%. As long as taxable income exceeds their obligations to investors, they will incur a tax benefit of T * (5% of principle). This means the actual cost to them of paying 5% interest, looking at their overall balance sheet, is only 5% * (1-T) [of principle]. Percentages avoids worrying about principle in every statement and describes the effective cost of money for the company: not 5% that investors want, but 5% *(1-T).

7. Jun 10, 2015

### Square1

But you see you are also naturally first thinking about it in terms totals (or at least when talking about it here).

"incur a tax benefit of T * (5% of principle)"
"looking at their overall balance sheet"
"5% * (1-T) [of principle]"

It's just not intuitive enough just start dealing with plain percentages. After all, they are meant to be applied to something. First look at totals. Once you have "5% * (1-T) [of principle]" and choose you don't want to deal with the whole dollar value, you can easily take it out of both sides of the equation, and you are left with a percentage. You may be very experienced with math so perhaps it comes more automatic to you and that's why I struggle with it.

Last edited: Jun 10, 2015
8. Jun 13, 2015

### Fredrik

Staff Emeritus
Here's a more detailed explanation.

Income: X
Loan: L
Tax rate: T
Interest rate: ib

If they don't take the loan, they have to pay a tax of XT. The money they will have left after paying that tax is X-XT = X(1-T).

If they take the loan, they have to pay an interest of Lib. This effectively lowers their income to X-Lib, so now they now only have to pay a tax of (X-Lib)T. The money they will have left (not including the money they borrowed) after paying the tax and the interest rate is X-Lib-(X-Lib)T = (X-Lib)(1-T).

The cost of taking the loan is the difference between these two results, i.e. X(1-T)-(X-Lib)(1-T) = (X-(X-Lib))(1-T) = Lib(1-T). What percentage of the total loan is this? The answer is Lib(1-T)/L = ib(1-T).

9. Jun 13, 2015

### Square1

Thank you and Allen for your replies. I've cleared it up already :)

10. Jun 13, 2015

### Staff: Mentor

In my judgement, there is nothing wrong with thinking about in terms of the total amounts, as you did. But, after you finished that, you saw how the equation derivation came about in terms of the interest rates. Do I think that it is intuitively obvious? Probably not, although people with some experience with this kind of thing would realize it immediately.

Chet

11. Jun 13, 2015

### Square1

Yes my thoughts exactly. Within the time of OP and now I've noticed myself playing out other scenarios like this in my mind , and the mental leap is becoming automatic.

12. Jun 13, 2015

### Fredrik

Staff Emeritus
Right, once you're used to it, you can probably just think "the cost would be ib without the tax deduction, and the tax deduction is ibT, so it's ib-ibT=ib(1-T)".