Lesson 1 of 9
A dollar later
Present value and future value are one formula read in two directions. Both are three keystrokes, and both are checkable by hand.
By Javier Sanz
Founder of ValueScreener. Built Ninety Nine, a securities services provider, and sold it to a tier-one global bank; ran operations and internationalization at Alpaca, the brokerage API.
Who writes thisEditorial processLinkedIn (opens in a new tab)X (opens in a new tab)
In one minute
- Future value multiplies by one plus the rate, once per period. Present value divides by the same thing.
- The two are the same equation. Which one you are doing depends only on which end of time you already know.
- The rate and the number of periods have to describe the same unit of time, or the answer is meaningless.
A dollar today and a dollar in ten years are not the same object. One of them can be lent, and by the tenth year it has become more than a dollar. Financial mathematics is the arithmetic of moving money along that line, and it is one formula.
One dollar moved ten years in each direction, at 4.96 percent
- Value today$1.00
- Grown for ten years$1.62
- One dollar arriving in ten years, valued today$0.62
Forward: multiply
Lend a dollar at a rate of r for one period and you get back 1 plus r. Leave it for a second period and the whole of that grows again, so you get 1 plus r, squared. After n periods you have 1 plus r, to the power n. That exponent is the whole of compounding.
At 4.96 percent a year for ten years, one dollar becomes 1.0496 to the tenth, which is 1.6227. A hundred dollars becomes 162.27.
Backward: divide
Now run it the other way. You know a cash flow arrives in year ten and you want what it is worth today. You are asking what amount, grown for ten years at the same rate, would produce it. So divide instead of multiplying.
Present value equals Future cash flow divided by (1 + r) to the power n.
Example100 / 1.0496 to the 10th = 61.63
One hundred dollars arriving in ten years is worth 61.63 today at that rate. Nothing about the company changed between the two paragraphs. Only the direction of the question.
The unit trap
The rate and the exponent must be the same unit of time. A 6 percent annual rate over 18 months is not 6 percent times 1.5. It is 1.06 to the power 1.5, because the half year compounds on the first year's growth.
This is where most hand errors live. Write the period down before you write the rate down, and the next lesson's factor table makes the habit automatic.
Why this is the whole of valuation
A business is a series of dated cash amounts. Every valuation method in this academy is this formula applied once per date and the results added up. The arguments that follow are about which cash flows and which rate, never about the arithmetic. The arithmetic was settled here.
Check yourself
At 5 percent a year, what does 1,000 dollars grow to in three years?
- 1,150.00
- 1,157.63
- 1,050.00
Show the answer
Answer: B.
1.05 to the power 3 is 1.157625, so 1,000 becomes 1,157.63. The extra 7.63 over a flat 150 is the growth on the growth.A cash flow of 100 arrives in ten years. You raise the discount rate. What happens to its present value?
- It rises
- It falls
- It does not move until the rate doubles
Show the answer
Answer: B.
A higher rate makes the divisor larger, so the value today is smaller. The effect is bigger the further out the cash flow sits.You have an annual rate and a cash flow that arrives in 18 months. What do you raise 1 plus the rate to?
- 1.5 multiplied by the rate
- The power 1.5
- The power 18
Show the answer
Answer: B.
The exponent counts periods in the same unit as the rate. Eighteen months is 1.5 annual periods, so it is an exponent of 1.5, not a multiplication.Sources
- Williams, The Theory of Investment Value, 1938: an asset is worth the present value of the cash it distributes.
- US Department of the Treasury, Daily Treasury Par Yield Curve Rates, session of 11 September 2026: the 4.96 percent ten-year par yield used throughout. ValueScreener stores the same figure for the same session.
What this leaves out
- The Treasury yield is used here as a rate that exists and has a date, not as the right rate for valuing a company. A business is not a Treasury.
- This lesson assumes a single rate for every year. Real curves are not flat, which is the subject of the bond course.
- Nothing here is advice about any security.