Lesson 4.1 35 min
Compounding and discounting
Future value, present value, annuities and effective annual rates, then a lease against a cash price.
A dollar today and a dollar next year are different amounts of money, because the dollar today can be put to work. Every investment decision in this module rests on one idea: to compare cash on different dates, move it all to the same date.
Future value
If $10,000 earns 6% a year and the interest stays invested, the balance after one year is $10,000 × 1.06. After two years it is that figure × 1.06 again.
Future value = Present value × (1 + r)^n
Here r is the rate for one period and n is the number of periods. The rate and the period must match: a monthly rate goes with a count of months.
| Opening $ | Interest $ | Closing $ | |
|---|---|---|---|
| Year 1 | 10,000 | 600.00 | 10,600.00 |
| Year 2 | 10,600.00 | 636.00 | 11,236.00 |
| Year 3 | 11,236.00 | 674.16 | 11,910.16 |
| Year 4 | 11,910.16 | 714.61 | 12,624.77 |
| Year 5 | 12,624.77 | 757.49 | 13,382.26 |
After 5 years the balance is $13,382.26. Simple interest, which pays 6% on the original $10,000 only, would give $13,000. The $382.26 difference is interest earned on interest, which is what compounding means. It grows faster the longer the money stays in.
Present value
Present value runs the same formula backwards. To have $20,000 in 5 years at 6%, the amount to invest today is $20,000 ÷ 1.06^5 = $14,945.16.
Present value = Future value ÷ (1 + r)^n
Moving cash back to today is called discounting, and r is then called the discount rate. A higher rate gives a smaller present value, because a higher rate means more can be earned while waiting.
Where the rate comes from
For an investment decision the discount rate is the return the money could earn elsewhere at the same level of risk. It is an opportunity cost. A business that borrows at 9% and is unsure of a project should use a rate above 9%. Lesson 4.2 returns to this. In these lessons the rates are stated as assumptions.
Annuities
An annuity is a series of equal payments at equal intervals. A lease with five payments of $4,000 is an annuity. Discounting each payment one by one works, and so does a formula.
Present value of an annuity = Payment × (1 − (1 + r)^−n) ÷ r
The term in brackets, divided by r, is the annuity factor. At 8% for five years it is 3.9927. The formula assumes the first payment comes at the end of the first period. If payments come at the start of each period, the value is higher by one period of interest, so multiply by (1 + r).
Effective annual rate
A lender who quotes 12% a year, compounded monthly, charges 1% a month. Over a year that compounds to (1.01)^12 − 1, which is 12.68%. The quoted rate is the nominal rate. The effective annual rate is what the money actually earns or costs over a year, and it is the figure for comparing offers that compound at different intervals.
A quick check
The rule of 72 estimates the years to double money: divide 72 by the rate in percent. At 8% that gives 9 years. The exact answer from the formula is 9.01 years. The rule is an estimate, not a substitute for the formula.
Worked case
Lease or pay cash
Northside is offered a grinder and a refrigerated display case, which can be bought for $16,500 in cash or leased for $4,000 a year for 5 years, paid at the end of each year. Northside's alternative use of cash is worth 8% a year, so it discounts at 8%. The lease payments add up to $20,000, which is more than the cash price, but they are paid over five years.
| Payment $ | Factor at 8% | Present value $ | |
|---|---|---|---|
| Payment 1, end of year 1 | 4,000 | 0.9259 | 3,703.70 |
| Payment 2, end of year 2 | 4,000 | 0.8573 | 3,429.36 |
| Payment 3, end of year 3 | 4,000 | 0.7938 | 3,175.33 |
| Payment 4, end of year 4 | 4,000 | 0.7350 | 2,940.12 |
| Payment 5, end of year 5 | 4,000 | 0.6806 | 2,722.33 |
| Present value of the lease | 15,970.84 |
The lease costs $15,970.84 in today's money against $16,500 for cash, so it is cheaper by $529.16 at an 8% discount rate. At a higher rate the lease looks better still, because late payments are worth less. At a lower rate the cash price wins. The decision depends on the discount rate, which is why the rate has to be stated.
If the payments were due at the start of each year, the present value would be $15,970.84 × 1.08 = $17,248.51, which is $748.51 above the cash price. The lease would then be dearer, so payment timing is part of the price.
Template
Time value workbook
Three sheets: future and present value of a single sum with a year-by-year growth table, an annuity sheet that values a lease against a cash price, and an effective annual rate sheet for any compounding frequency. The case figures are loaded.
The full working file, 4-1-time-value.xlsx, is in the Accounting and Finance Toolkit, Complete Track 1. The lesson above teaches the method and shows the worked numbers, so you can build it yourself in any spreadsheet.
Exercises
Work these, then open the answers
Find the present value of $15,000 due in 3 years at 8% a year.
Answer
$15,000 ÷ 1.08^3 = $11,907.48.
A savings plan puts $3,000 at the end of each year into an account earning 5%. What is the balance after 4 years?
Answer
Future value of an annuity = 3,000 × (1.05^4 − 1) ÷ 0.05 = $12,930.38.
A card is quoted at 12% a year compounded quarterly. What is the effective annual rate?
Answer
(1 + 0.12 ÷ 4)^4 − 1 = 12.55%. Monthly compounding of the same quoted rate gives 12.68%.
Check
4 questions
Sources
Sources
- Principles of Finance, OpenStax, Rice University