# Time Value of Money Question

For the following Question

First: Scenario 1 : I wonder how the customer pay as whole 12000 dollar after 12 years for a 8525 dollar product if the interest rate = 0%

I think that , there is about 3% interest

Second: What is the criteria upon which the company decide in first scenario to pay 1000$/year and for the second scenario to pay the full amount? ## 3 Answers This is a very poorly written question. The gist of the question is this: should a company take a lump-sum payment from a customer of \$8,525 or a \$12k payment that is spread uniformly over 12 years. The thing the question wants you to consider between 1a and 1b is that in 1a the company cannot earn interest (which the answer just assumes is compounding) and in 1b the company can earn 7% interest. For the given answer, part 1a is saying that, since the company cannot earn interest on investments, it is better to just take the \$12k. However, one ought to know something about how this company discounts to actually answer this question. That is, one should make this comparison by comparing \$8,525 to the discounted present value of the \$12k, taking into account that the \$12k is received in payments spread uniformly over 12 years. The real answer to this question as written is, ' IT DEPENDS'. Part 1b is saying that$8,525(1+.07)^{12} \equiv 19,199.93 > 12,000$and so the company should take the lump-sum payment and invest it. HOWEVER, this is again the wrong way to answer this question. Instead, one should compare$8,525(1+.07)^{12}$with the money the firm will have if it chooses the \$12k and subsequently invests each annual payment. This value is \$19,140.64 (link to calculate). The answer here is correct but for the wrong reason. Really, this is a very bad question and a very bad answer. One way to do this is with a table like this (assuming the$1000 has to be paid at the beginning of the year)

Year    Starting debt   Payment     After payment   Interest    End-year
1       8525.00         -1000.00    7525.00         526.75      8051.75
2       8051.75         -1000.00    7051.75         493.62      7545.37
3       7545.37         -1000.00    6545.37         458.18      7003.55
4       7003.55         -1000.00    6003.55         420.25      6423.80
5       6423.80         -1000.00    5423.80         379.67      5803.46
6       5803.46         -1000.00    4803.46         336.24      5139.71
7       5139.71         -1000.00    4139.71         289.78      4429.48
8       4429.48         -1000.00    3429.48         240.06      3669.55
9       3669.55         -1000.00    2669.55         186.87      2856.42
10      2856.42         -1000.00    1856.42         129.95      1986.37
11      1986.37         -1000.00     986.37          69.05      1055.41
12      1055.41         -1000.00      55.41           3.88        59.29


You can either read this as investing 8525, taking money out to pay the 1000 and then receiving interest of 7% on the remaining balance to leave 59.29 to your credit at the end (so the twelve payments is better), or borrowing 8525 to buy initially while repaying 1000 a year to the bank and facing 7% interest on the outstanding balance leaving a debt of 59.29 owing at the end (so buying initially is worse).

The surprising thing is the final amount is close to zero, and would be virtually zero if the interest rate had been about 6.93%

If the payments of 1000 were at the end of the year, the final figure would have been about 1311.48, meaning the twelve payments of 1000 would be an even better choice

If, instead of 7%, the interest rate were 0%, the table would end up at -3475, meaning the lump-sum upfront payment would be better

The question asks you to compare the value of an immediate payment and a series of payments in the future. In finance, every set of future payments has an immediate payment equivalent, which we get by dividing each future payment by a discount rate. When this question says "interest rate," it means "discount rate." The two terms are used interchangeably where there is little ambiguity about what is meant.

To answer your first question, it doesn't matter what the true risk-free or risky interest rates are now...that's not part of the question. It's a hypothetical world in which there is an interest rate of zero. In that case, future dollars are worth the same as present dollars, so we are comparing the value of \$8,525 with \$12,000. Clearly the latter is better.

For the second question, how do we find the present value (immediate lump sum equivalent) of these payments?

$$PV = \frac{1000}{0.07}\left(1-\frac{1}{1.07^{12}}\right) = 7,942.69$$

Using the formula for the present value of an annuity. We could also do it using the direct approach

$$PV = \sum_{i=1}^{12}\frac{1000}{1.07^{i}}$$

to get the same result. We see that \$8,525 > \$7,942.69 so it's better to take the immediate payment.

The answers seem like they were hastily and erroneously written--better to ignore them and rely on my solution than to try and figure them out. The question here is a typical and straightforward introductory undergraduate finance question, though, and well worth understanding.

• Why have you assumed a relationship between the interest rate specified in the problem and the discount rate this firm uses when evaluating future earnings? Discount rates and interest rates are not interchangeable concepts.
– 123
Apr 9, 2017 at 22:15
• Standard practice for introductory questions of this sort. If you asked this question in an advanced MBA or higher class, then the problem would make a distinction. But you wouldn't ask such a basic question to students at that level. I agree with your objection in principle, but it's like every other simplifying assumption made in order to make introductory classes accessible--something useful but technically a fudge. Apr 9, 2017 at 22:24
• No. Discount rates and interest rates are different concepts. Further, the answer provided to the OP clearly implies that the question intends for 'interest rate' to mean interest rate and not discount rate.
– 123
Apr 9, 2017 at 22:44
• In introductory classes, the simplifying assumption that there is a single interest rate that is also used for discounting is always used. I'm sorry you don't like it. If you ever teach an intro finance class, you can try it a different way. Apr 9, 2017 at 22:47
• BTW the "answer" is not evidence as it is very wrong in several ways. Probably writing the solution was outsourced to a TA. Also common practice, unfortunately. Apr 9, 2017 at 22:49