The Future Value and Present Value of an Annuity
Understanding annuities is crucial for understanding loans, and investments that require or yield periodic payments. For instance, how much of a mortgage can I afford if I can only pay $1,000 monthly? How much money will I have in my IRA account if I deposit $2,000 at the beginning of each year for 30 years, and earns an annual interest rate of 5%, but is compounded daily?
An annuity is a series of equal payments in equal time periods. Usually, the time period is 1 year, which is why it is called an annuity, but the time period can be shorter, or even longer. These equal payments are called the periodic rent. The amount of the annuity is the sum of all payments.
An annuity due is an annuity where the payments are made at the beginning of each time period; for an ordinary annuity, payments are made at the end of the time period. Most annuities are ordinary annuities.
Analogous to the future value and present value of a dollar, which is the future value and present value of a lump-sum payment, the future value of an annuity is the value of equally spaced payments at some point in the future. The present value of an annuity is the present value of equally spaced payments in the future.
The Future Value of an Annuity
The future value of an annuity is simply the sum of the future value of each payment. The equation for the future value of an annuity due is the sum of the geometric sequence:
FVAD = A(1 + r)1 + A(1 + r)2 + ...+ A(1 + r)n.
The equation for the future value of an ordinary annuity is the sum of the geometric sequence:
FVOA = A(1 + r)0 + A(1 + r)1 + ...+ A(1 + r)n-1.
Without going through an extensive derivation, just note that the future value of an annuity is the sum of the geometric sequences shown above, and these sums can be simplified to the following formulas, where A = the annuity payment or periodic rent, r = the interest rate per time period, and n = the number of time periods.
The future value of an ordinary annuity (FVOA) is:
|FVOA||=||A||×||(1 + r)n - 1|
And the future value of an annuity due (FVAD) is:
|FVAD||=||A||×||(1 + r)n - 1|
|+||A(1 + r)n||-||A|
Note that the difference between FVAD and FVOA is:
FVAD = 0 + A(1 + r)1 + A(1 + r)2 + ...+ A(1 + r)n-1+ A(1 + r)n.
FVOA = A(1 + r)0 + A(1 + r)1 + A(1 + r)2 +... + A(1 + r)n-1 + 0.
|FVAD||-||FVOA||=||A(1 + r)n||–||A(1 + r)0|
|=||A(1 + r)n||–||A|
|(Math note: x0 = 1.)|
In other words, the difference is merely the interest earned in the last compounding period. Because payments of an ordinary annuity are made at the end of the period, the last payment earns no interest, while the last payment of an annuity due earns interest during the last compounding period.
Example — Calculating the Amount of an Ordinary Annuity
If at the end of each month, a saver deposited $100 into a savings account that paid 6% compounded monthly, how much would he have at the end of 10 years?
A = $100
r = 6% per year compounded monthly, which = .5% interest per month = .005
n = the number of compounding time periods = 120 in 10 years.
Substituting these values into the equation for the future value of an ordinary annuity:
100 * ((1+.005)120 -1)/.005 = $16,387.93
Example — Calculating the Amount of an Annuity Due
If the saver deposited the money at the beginning of the month instead of the end, then there will be an additional amount of money = A(1 + r)n - A = 100(1.005)120 -100 = $81.94, which is the difference in this example between an annuity due and an ordinary annuity.
Example — Calculating the Annuity Payment, or the Periodic Rent
A 20 year old wants to retire as a millionaire by the time she turns 70. (With life spans increasing, and the social security fund being depleted by baby boomers, the retirement age will have invariably risen by the time she reaches 65 years of age, probably to something even higher than 70, actually.) How much will she have to save at the end of each month if she can earn 5% compounded annually, tax-free, to have $1,000,000 by the time she is 70?
Solution: Note that the equation for the future value of an annuity consists of 3 independent variables, and 1 dependent variable. In other words, if we know the value of 3 of the variables, then we can determine the remaining variable.
Since r = 5% = .05, and n = 50, the interest factor (1 + r)n - 1)/r = (1.0550 - 1)/.05 = 209.35, rounded to 2 decimal places. To find A, we divide both sides of the equation for the future value of an annuity by this interest factor, which yields 1,000,000/209.35 = $4,776.69. So she would have to save $4,776.69 dollars per year, or $398.06 per month, to have $1,000,000 in 50 years—assuming, of course, that she could save it tax-free!
Of course, using the formula for the present value of a dollar, we find that in 50 years, assuming 3% inflation, $1,000,000 will be worth about 1,000,000/1.0350 = $228,107.08! Ouch!
Since the current limit for IRA contributions is $2,000 per year for a young person, how much will this earn after 50 years, assuming that the $2,000 is deposited at the end of the year? FVOA = 2,000 * (1.0550 - 1)/.05 = $418,695.99.
What's that in today's dollars, assuming 3% inflation? 418,695.99/1.0350 = $95,507.52! Clearly, the IRA contribution limits must be raised substantially. Of course, you can save all the money at the beginning of each year instead of at the end, and this annuity due will yield an extra (using the Annuity Difference Formula above) 2,000 * 1.0550 - 2,000 = $20,934.80 which, in today's dollars, again assuming a 3% inflation rate, = $20,934.80/1.0350 = $4,775.38 more money in today's dollars over the ordinary annuity, but clearly, you'll still be eating dog food when you retire with this amount of cash, unless you are planning to die early! With the limitations on IRAs, stocks are the only viable choice for investments that could possibly yield anything decent to retire on!
The Present Value of an Annuity
The present value of an annuity (PVA) is the sum of the present value of each annuity payment. Since the present value of a lump sum payment is simply the future value of that payment divided by the interest factor (1 + r)n, the present value of an annuity is the sum of the present value of each of those payments:
(1 + i)k
|PVA = Present Value of Annuity Amount|
A = annuity payment
i = interest rate per time period
n = number of time periods
The sum of this geometric progression can be simplified to:
(1 + r)n
Example — Calculating the Present Value of an Annuity
You win a $1,000,000 lottery, which is paid in annual installments of $50,000 over 20 years. How much did you really win, assuming that you could earn 5% interest, compounded annually?
Solution: Since you are not receiving the full $1,000,000 payment right away, but in the form of an annuity, its actual worth is much less.
Present Value of Annuity = 50,000 * (1 - (1 + .05)-20)/.05 = $623,110.52
Example — How Much of a Loan Can you afford?
You want to get a mortgage, but can only afford to pay $1,000 per month. How much can you borrow, if the interest rate is 5% annually for a 30 year mortgage?
Solution: The monthly payments constitute an annuity, whose present value is the amount of the loan.
Loan Amount = 1,000 * (1 - (1 + .004166667)-360)/.004166667 = $186,281.62
r = the monthly interest rate = .05/12 = .004166667.
n = the number of months in 30 years = 12 × 30 = 360.
Math Reminder: y-x = 1/yx.
Example — Calculating Monthly Mortgage payments
You want to borrow $200,000 to buy a house. What are the monthly mortgage payments if the interest rate is 6% for 30 years?
Solution: In the above example, we asked how much one would have to save per month or per year to have $1,000,000 in 50 years. In other words, what periodic payments would we have to make to have a future value of $1,000,000? Here, we take out a loan, and thus, we already have the money, whose present value, or discounted value, is equal to the amount of the loan. The monthly payment would be the annuity payment, A. Thus, we use the equation for the present value, because the present value is already known, and what we need to know is how much are the payments going to be if the length of the loan is 30 years, and the interest rate is 6% annually.
Because we know 3 of the 4 variables, but not A, the monthly payment, we solve for A by dividing both sides of the present value of annuity equation by the factor (1 - (1 + r)-n)/r, but note that to divide by a fraction is the same as multiplying the numerator by the inverse of the fraction, and so, we can simplify further:
1 - (1 + r)-n
1 - (1 + r)-n
|Formula for the monthly payment of a loan.|
A = monthly payment, or annuity payment.
PV = present value, or the amount of the loan.
r = interest rate per time period.
n = number of time periods.
1 - (1 + .005)-360
The interest rate for each month = .06/12 =.005
The number of months in 30 years = 12 × 30 = 360.
Calculating the Interest rate
We end our discussion on annuities by noting that r cannot be solved algebraically in the formula for the present value of annuities, so, even if we know the annuity payment, the number of time periods, and the present value, we can only estimate r. It is possible to estimate r either by plugging in values with guesses, by looking it up in special tables that plot r against the annuity payment A, or by using a graphing calculator, and graphing the value of the annuity payment as a function of interest for a given present value. In the latter case, the interest rate is where the line representing the rate of interest intersects the line for the annuity payment.
Net Present Value and Internal Rate of Return
The present value of an annuity can be easily calculated because it consists of periodic payments of equal amounts. However, many times the payments are not equal in amount, and time intervals between payments may differ, in which case the present value of an annuity must be calculated by summing the present value of each payment. These unequal payments are sometimes referred to as a mixed stream:
Present Value of a Mixed Stream = Sum of the Present Value of Each Payment
Additionally, many business investments consist of both cash inflows and cash outflows. When a business wants to make an investment, one of the main factors in determining whether the investment should be made is to consider its return on investment. In most cases, not only will cash flows be uneven, but some of the cash flows will be received and some will be paid out. Additionally, some of the cash flows will be uncertain, and the taxation of some of the transactions could also have an effect on the present value of the inflows and outflows of the investment, especially over an extended period.
To decide whether to make a business investment, the business calculates what is called the net present value (NPV) of the investment, which is the net present value of all cash inflows minus the sum of the present value of the cash outflows, including the cost of the investment, using a discount rate (DR) that is judged to be a required rate of return. If the NPV is positive, then the investment is considered worthwhile. The NPV can also be calculated for a number of investments to see which investment yields the greatest return.
Net Present Value = Sum of Present Value of Cash Inflows – Sum of Present Value of Cash Outflows
In the capital budgeting of long-term investments in business, the required rate of return is called the hurdle rate or the discount rate, and should be equal to or greater than the incremental cost of capital (aka marginal cost of capital), which is the weighted average of costs to issue debt or equity to finance the investment.
Closely related to the net present value is the internal rate of return (IRR), calculated by setting the net present value to 0, then calculating the discount rate that would return that result. If the IRR ≥ required rate of return, then the project is worth investing in.
- If IRR ≥ DR, then invest.
- If IRR < DR, then forget about it.
|+ ... +||CFn|
|CF = Cash Flows|
CF0 = Initial Investment
n = number of cash flows
The IRR is difficult to calculate, but most spreadsheets have a formula that will return the discount rate.
Calculating Present and Future Values Using PV, NPV, and FV Functions in Microsoft Excel
Microsoft Office Excel and the free OpenOffice Calc have several formulas for calculating the present and future value of an investment as a lump-sum payment or as an annuity, and for calculating net present value.
Present Value = PV(rate,number of periods,payment,future value,type)
Net Present Value = NPV(rate, value1,value2,...)
Future Value = FV(rate,number of periods,payment,present value,type)
Examples—Using Microsoft Office Excel or OpenOffice Calc for Calculating Present Value and Future Value of Investments
The following formulas were computed using Microsoft Office Excel 2007, although previous versions of Excel also have these formulas. These same formulas will also work in the free OpenOffice Calc, but the values are separated by semicolons instead of commas. To summarize the general format:
- =PV(interest rate,number of periods,payment,FV,0 or 1)
- =FV(interest rate,number of periods,payment,PV,0 or 1)
- OpenOffice Calc:
- =PV(interest rate;number of periods;payment;FV;0 or 1)
- =FV(interest rate;number of periods;payment;PV;0 or 1)
- 0=payment at end of period; 1=payment at beginning of period; if omitted, then 0 is assumed.
- if a variable has no value, then simply insert an extra comma or semicolon to indicate no value for that variable.
You are 30 years old and want to have $1,000,000 when you retire at 65. But how much is that worth today, assuming a constant inflation rate of 3%?
Present Value of $1,000,000 at age 65 = PV(0.03,35,,-1000000) = $355,383.40
- Note that -1,000,000 was entered so that the PV is positive.
You are 25 years old and want to save $4,000 per year in your IRA. How much will you have when you retire at 65, paying at the end of each year, earning a constant interest rate of 5% compounded annually? At the beginning of each year? If you already had $10,000 in your IRA at 25?
Future Value, paying at the end of each year = FV(0.05,40,-4000,,) = $483,199.10.
- Note that the $4,000 payment is entered as a negative value, since you are paying that amount, not receiving it.
Future Value, paying at the beginning of each year = FV(0.05,40,-4000,,1) = $507,359.05
- Note that there is a comma placeholder for the present value since it is assumed that you had nothing in the account for the start.
Hence, if you pay at the beginning of each year instead of at the end, you will have $24,159.95 more for your retirement.
Future Value, paying at the beginning, but with $10,000 already saved = FV(0.05,40,-4000,-10000,1) = $577,758.94.
- Note that the $10,000 is also entered as a negative number, since you paid it in. In this example, you can see that both the payment and the present value are entered as negative values.