A company seeks a $100,000 loan. Option A: Pay off the loan with one payment of $150,000 in five years. Option B. Pay $2000 per month for the next five years. Option C. Pay $10,000 at the end of each of the next 4 years plus $90,000 at the end of the fifth year.
(1) Rank the three options by their desirability if the interest rate is 3%.
(2) Rank the three options by their desirability if the interest rate is 9%.

You want the ranking of the desirability of the following payment schemes with respect to a $100,000 loan at interest rates of (1) 3% and (2) 9%.

A: $150,000 balloon payment in 5 years
B: $2000 per month for 5 years
C: $10,000 per year for 4 years, $90,000 balloon payment in 5 years

Present value

We can rank the payment schemes by looking at their present value at the different interest rates. The given sum of payments formula can be used with a slight modification.

For periodic payment p made at the end of the period assuming discount rate r, the present value is ...

PV = p/(1+r)·(1 -(1+r)^-n)/(1 -(1+r)^-1) = (p/r)((1 +r)^n -1)/(1+r)^n

For a balloon payment made at the end of n periods, the present value is ...

PV = P/(1+r)^n

We can sum these expressions to give the present value of a series of payments together with a balloon payment (as in option C):

[tex]PV=\dfrac{P+(p/r)((1+r)^n-1)}{(1+r)^n}=\dfrac{P+(p/r)(a -1)}{a}\qquad a=(1+r)^n[/tex]

This formula assumes that there is a periodic payment made at the end of each of the n periods prior to the balloon payment. That is, Option C is treated as 5 payments of $10,000 together with a 5th-year balloon payment of $80,000.

(1) 3% interest

The present value of Option A is ...

a = (1.03)^5 = 1.1592740743

PV = $150000/a = $129,391.32 . . . . PV of Option A

The present value of Option B is ...

a = (1 +.03/12)^60 ≈ 1.16161678156

PV = ($2000/0.0025)(a-1)/a ≈ $111,304.72 . . . PV of Option B

The present value of Option C is ...

'a' is the same as for Option A

PV = ($80000 +10000/.03(a -1))/a ≈ $114,805.77 . . . . PV of Option C

The present value represents the cost to repay the loan, so the lowest PV is the most desirable option. In order of decreasing desirability, the options are B, C, A.

(2) 9% Interest

The present value of Option A is ...

a = (1.09)^5 = 1.5386239549

PV = $150000/a = $97,489.71 . . . . PV of Option A

The present value of Option B is ...

a = (1 +.09/12)^60 ≈ 1.56568102694

PV = ($2000/0.0075)(a-1)/a ≈ $96,346.75 . . . . PV of Option B

The present value of Option C is ...

PV = ($80000 +10000/.09(a -1))/a ≈ $90,891.02 . . . . PV of Option C

In order of decreasing desirability, the options are C, B, A.

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Additional comment

We were curious about the interest rate(s) that change the order of option desirability. To that end, we created a graph of present value vs. interest rate for the three options. It shows that the desirability order of the options changes at interest rates between 5% and 17%, completely reversing at higher interest rates. (Y-axis is in thousands; X-axis is %. Functions g, h, p correspond to options A, B, C.)

The details of the algebra are tedious, so not shown here. Basically, we make use of the addition of fractions (a/b +c/d) = (ad+bc)/(bd), and the cancellation of like factors from numerator and denominator. The modification of the series sum formula is made to account for the discount period between the present and the first payment. The final form of the formula we used is an effort to minimize the number of times we have to compute 'a'. A calculator that can store that value is helpful.