An annuity is a financial contract, typically issued by an insurance company, that involves making periodic payments to an investor or lender until the end of the agreement.
Calculating the present value of future payments from an annuity helps an investor understand the amount of money that an annuity represents in today’s dollars to make informed investment decisions.
What Is the Time Value of Money?
To better understand present value, let’s explain the time value of money.
Which would you prefer: Having $10,000 in your pocket now or receiving $10,000 in 5 years? Common sense would tell you it’s better to take the money now rather than wait 5 years to get the same amount. Instinctively, that’s an application of the time value of money.
That’s because money has the potential to increase in value over a period of time. Deposits in a bank savings account, for example, earn interest and increase in value. Money in your hand is worth more than money received in the future.
What Is the Present Value of an Annuity?
The present value of an annuity is the application of the time value of money.
The present value of an annuity is the cash amount that a future stream of payments would be worth today when discounted to the present at a given interest rate. (Check out this present value of annuity table.)
Present Value of Annuity: an Example
Suppose you have an annuity that will make annual payments of $30,000 at the beginning of each year for the next 20 years. That would be a total of $600,000 (20 years x $30,000).
Now suppose you calculate the present value of that future stream of payments by discounting them at a current interest rate of 3%. Using Investopedia’s calculator, the present value is $459,713.97.
This means that the $600,000 in future payments is worth $459,714 in today’s dollars.
Present Value of an Annuity: Formulas
Annuities have two types: ordinary annuities and annuities due
With ordinary annuities, payments are received at the end of each time period. Annuities due, on the other hand, make payments at the beginning of each time period.
We’ll explain how to find present value of annuity as well as how to calculate the present value of an annuity due.
Here’s how to calculate the present value of each type of annuity:
Present value of ordinary annuity = (PMT [(1 – (1 / (1 + r)n)) / r])
Present value of an annuity due = (PMT [(1 – (1 / (1 + r)n)) / r]) x (1+r)
- PMT = the amount of each payment
- r = the discount rate
- n = the number of periods that payments are made
To clarify the differences in these two formulas, consider this example. Suppose you have the opportunity to receive a lump-sum payment of $325,000, or 25 annual payments of $25,000 each. You decide to use a discount rate of 6% to calculate the present value of each type of annuity.
Present value of ordinary annuity:
($25,000 [(1 – (1 / (1 + .06)25)) / .06]) = $319,583.90
In the case of an ordinary annuity, you would take the lump-sum payment of $325,000 since it is higher than the present value.
Present value of an annuity due:
($25,000 [(1 – (1 / (1 + .06)25)) / .06]) x (1+.06) = $338,758.94
The present value for an annuity due is $338,758.94 is higher than the lump-sum amount. It would make sense to choose the annuity due contract.
Present value of annuity calculations can be used to compare the attractiveness of various investments or types of annuity contracts.
Using Present Value of Annuity in Business Decisions
Suppose Fred, the owner of ABC Corp, developed a new device that helps plumbers solder copper pipe faster, saving a lot of labor man-hours and reducing the cost of installation. Fred has received a patent but doesn’t have the desire or money it would take to manufacture and market the product himself.
A large plumbing supply company, Giant Tools, has made Fred an offer. Giant has offered him an up-front sum of $750,000 to purchase the patent and all rights, or the company will pay him $100,000 per year for the next 15 years, for a total of $1.5 million. Which offer should Fred accept?
Fred believes he is a savvy investor and can achieve a return of 11% per annum by investing in stocks and bonds. The present value of Giant’s offer for future payments at a discount rate of 11% is $719,086.96. This amount is less than the up-front offer, so Fred takes the lump-sum payment of $750,000.
If Fred accepts the $750,000 and is fortunate enough to realize an 11% return per annum, his investments would realize a future value of $3,588, 442 in 15 years.
What Is the Effect of Different Discount Rates?
The choice of which discount rate to use when calculating a present value is highly subjective. An investor could decide to use an average inflation rate, for example, because the investor wants to preserve his capital. Or an investor could feel that he could earn a higher return of 11% by investing in the stock market.
Let’s see how different discount rates will affect present values. To illustrate, take an ordinary annuity that offers annual payments of $30,000 for 20 years. From the formula, the present value of this annuity at a projected inflation rate of 2% will be $490,543. Alternatively, the present value at a discount rate of 11% would be $238,899.84.
As you can see, higher discount rates result in lower present values.
Let’s revisit Fred’s offer from Giant Tools. Suppose Fred doesn’t want to spend that much time studying the stock market and would probably just put the money in a savings account paying 3%. How does that change Fred’s decision? The present value of 15 annual payments of $100,000 discounted at 3% is $1,193,793.51. The decision is obvious: Fred should accept the payments over 15 years.
As you can see, the discount rate chosen is subjective and depends on the view of the individual.
Present Value of an Annuity Applied to Capital Investments
The concept of the time value of money and present value can also be applied when evaluating various capital projects.
Suppose XYZ Corp. has a cost of capital of 8% and is considering a $400,000 expansion of its plant. It will produce an additional $75,000 a year in cash flow over the next 7 years. The present value of this project discounted at 8% is $390,477.75.
The other option is to build a new facility for $500,000 to manufacture a new product. This new plant is projected to produce $100,000 a year for 7 years. The present value of the investment in a new plant discounted at 8% is $520,637.01.
On the basis of present value analysis, the $400,000 plant expansion isn’t attractive because its present value of $390,477.75 is less than the initial investment. On the other hand, the construction of the new plant to make a new product will have a present value of $520,637.01, exceeding its $500,000 cost. On this basis, this project should be accepted.
The calculation for the present value of annuities is a useful financial metric to analyze investments and make informed decisions. It is a versatile tool that has numerous applications in business.