Chapter 3: The Time Value of Money
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1. You want to buy an ordinary annuity that will pay you $4,000 a year for the next 20 years. You expect annual interest rates will be 8 percent over that time period. The maximum price you would be willing to pay for the annuity is closest to$32,000.$39,272.
$40,000.
$80,000.
2. With continuous compounding at 10 percent for 30 years, the future value of an initial investment of $2,000 is closest to$34,898.
$40,171.
$164,500.
$328,282.
3. In 3 years you are to receive $5,000. If the interest rate were to suddenly increase, the present value of that future amount to you wouldfall.
rise.
remain unchanged.
cannot be determined without more information.
4. Assume that the interest rate is greater than zero. Which of the following cash-inflow streams should you prefer? Year1 Year2 Year3 Year4 $400 $300 $200 $100
$100 $200 $300 $400
$250 $250 $250 $250
Any of the above, since they each sum to $1,000.
5. You are considering investing in a zero-coupon bond that sells for $250. At maturity in 16 years it will be redeemed for $1,000. What approximate annual rate of growth does this represent?8 percent.
9 percent.
12 percent.
25 percent.
6. To increase a given present value, the discount rate should be adjustedupward.
downward.
True.
Fred.
7. For $1,000 you can purchase a 5-year ordinary annuity that will pay you a yearly payment of $263.80 for 5 years. The compound annual interest rate implied by this arrangement is closest to8 percent.
9 percent.
10 percent.
11 percent.
8. You are considering borrowing $10,000 for 3 years at an annual interest rate of 6%. The loan agreement calls for 3 equal payments, to be paid at the end of each of the next 3 years. (Payments include both principal and interest.) The annual payment that will fully pay off (amortize) the loan is closest to$2,674.
$2,890.
$3,741.
$4,020.
9. When n = 1, this interest factor equals one for any positive rate of interest.PVIF
FVIF
PVIFA
FVIFA
None of the above (you can't fool me!)
10. (1 + i)nPVIF
FVIF
PVIFA
FVIFA
11.You can use to roughly estimate how many years a given sum of money must earn at a given compound annual interest rate in order to double that initial amount .Rule 415
the Rule of 72
the Rule of 78
Rule 144
12.In a typical loan amortization schedule, the dollar amount of interest paid each period . increases with each payment
decreases with each payment
remains constant with each payment
13.In a typical loan amortization schedule, the total dollar amount of money paid each period .increases with each payment
decreases with each payment
remains constant with each payment
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Most of us have had the experience of making a series of fixed payments over a period of time—such as rent or car payments—or receiving a series of payments for a period of time, such as interest from a bond or certificate of deposit (CD). These recurring or ongoing payments are technically referred to as "annuities" (not to be confused with the financial product called an annuity, though the two are related).
There are several ways to measure the cost of making such payments or what they're ultimately worth. Here's what you need to know about calculating the present value (PV) or future value (FV) of an annuity.
Key Takeaways
- Recurring payments, such as the rent on an apartment or interest on a bond, are sometimes referred to as "annuities."
- In ordinary annuities, payments are made at the end of each period. With annuities due, they're made at the beginning of the period.
- The future value of an annuity is the total value of payments at a specific point in time.
- The present value is how much money would be required now to produce those future payments.
Two Types of Annuities
Annuities, in this sense of the word, break down into two basic types: ordinary annuities and annuities due.
- Ordinary annuities: An ordinary annuity makes (or requires) payments at the end of each period. For example, bonds generally pay interest at the end of every six months.
- Annuities due: With an annuity due, by contrast, payments come at the beginning of each period. Rent, which landlords typically require at the beginning of each month, is a common example.
You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas.
Calculating the Future Value of an Ordinary Annuity
Future value (FV) is a measure of how much a series of regular payments will be worth at some point in the future, given a specified interest rate. So, for example, if you plan to invest a certain amount each month or year, it will tell you how much you'll have accumulated as of a future date. If you are making regular payments on a loan, the future value is useful in determining the total cost of the loan.
Consider, for example, a series of five $1,000 payments made at regular intervals.
Image by Julie Bang © Investopedia 2019Because of the time value of money—the concept that any given sum is worth more now than it will be in the future because it can be invested in the meantime—the first $1,000 payment is worth more than the second, and so on. So, let's assume that you invest $1,000 every year for the next five years, at 5% interest. Below is how much you would have at the end of the five-year period.
Image by Julie Bang © Investopedia 2019Rather than calculating each payment individually and then adding them all up, however, you can use the following formula, which will tell you how much money you'd have in the end:
FV Ordinary Annuity = C × [ ( 1 + i ) n − 1 i ] where: C = cash flow per period i = interest rate n = number of payments \begin{aligned} &\text{FV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [\frac { (1 + i) ^ n - 1 }{ i } \right] \\ &\textbf{where:} \\ &\text{C} = \text{cash flow per period} \\ &i = \text{interest rate} \\ &n = \text{number of payments} \\ \end{aligned} FVOrdinary Annuity=C×[i(1+i)n−1]where:C=cash flow per periodi=interest raten=number of payments
Using the example above, here's how it would work:
FV Ordinary Annuity = $ 1 , 0 0 0 × [ ( 1 + 0 . 0 5 ) 5 − 1 0 . 0 5 ] = $ 1 , 0 0 0 × 5 . 5 3 = $ 5 , 5 2 5 . 6 3 \begin{aligned} \text{FV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [\frac { (1 + 0.05) ^ 5 -1 }{ 0.05 } \right ] \\ &= \$1,000 \times 5.53 \\ &= \$5,525.63 \\ \end{aligned} FVOrdinary Annuity=$1,000×[0.05(1+0.05)5−1]=$1,000×5.53=$5,525.63
Note that the one-cent difference in these results, $5,525.64 vs. $5,525.63, is due to rounding in the first calculation.
Calculating the Present Value of an Ordinary Annuity
In contrast to the future value calculation, a present value (PV) calculation tells you how much money would be required now to produce a series of payments in the future, again assuming a set interest rate.
Using the same example of five $1,000 payments made over a period of five years, here is how a present value calculation would look. It shows that $4,329.58, invested at 5% interest, would be sufficient to produce those five $1,000 payments.
This is the applicable formula:
PV Ordinary Annuity = C × [ 1 − ( 1 + i ) − n i ] \begin{aligned} &\text{PV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [ \frac { 1 - (1 + i) ^ { -n }}{ i } \right ] \\ \end{aligned} PVOrdinary Annuity=C×[i1−(1+i)−n]
If we plug the same numbers as above into the equation, here is the result:
PV Ordinary Annuity = $ 1 , 0 0 0 × [ 1 − ( 1 + 0 . 0 5 ) − 5 0 . 0 5 ] = $ 1 , 0 0 0 × 4 . 3 3 = $ 4 , 3 2 9 . 4 8 \begin{aligned} \text{PV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [ \frac {1 - (1 + 0.05) ^ { -5 } }{ 0.05 } \right ] \\ &=\$1,000 \times 4.33 \\ &=\$4,329.48 \\ \end{aligned} PVOrdinary Annuity=$1,000×[0.051−(1+0.05)−5]=$1,000×4.33=$4,329.48
Calculating the Future Value of an Annuity Due
An annuity due, you may recall, differs from an ordinary annuity in that the annuity due's payments are made at the beginning, rather than the end, of each period.
Image by Julie Bang © Investopedia 2019To account for payments occurring at the beginning of each period, it requires a slight modification to the formula used to calculate the future value of an ordinary annuity and results in higher values, as shown below.
Image by Julie Bang © Investopedia 2019The reason the values are higher is that payments made at the beginning of the period have more time to earn interest. For example, if the $1,000 was invested on January 1 rather than January 31 it would have an additional month to grow.
The formula for the future value of an annuity due is as follows:
FV Annuity Due = C × [ ( 1 + i ) n − 1 i ] × ( 1 + i ) \begin{aligned} \text{FV}_{\text{Annuity Due}} &= \text{C} \times \left [ \frac{ (1 + i) ^ n - 1}{ i } \right ] \times (1 + i) \\ \end{aligned} FVAnnuity Due=C×[i(1+i)n−1]×(1+i)
Here, we use the same numbers, as in our previous examples:
FV Annuity Due = $ 1 , 0 0 0 × [ ( 1 + 0 . 0 5 ) 5 − 1 0 . 0 5 ] × ( 1 + 0 . 0 5 ) = $ 1 , 0 0 0 × 5 . 5 3 × 1 . 0 5 = $ 5 , 8 0 1 . 9 1 \begin{aligned} \text{FV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \frac{ (1 + 0.05)^5 - 1}{ 0.05 } \right ] \times (1 + 0.05) \\ &= \$1,000 \times 5.53 \times 1.05 \\ &= \$5,801.91 \\ \end{aligned} FVAnnuity Due=$1,000×[0.05(1+0.05)5−1]×(1+0.05)=$1,000×5.53×1.05=$5,801.91
Again, please note that the one-cent difference in these results, $5,801.92 vs. $5,801.91, is due to rounding in the first calculation.
Calculating the Present Value of an Annuity Due
Similarly, the formula for calculating the present value of an annuity due takes into account the fact that payments are made at the beginning rather than the end of each period.
For example, you could use this formula to calculate the present value of your future rent payments as specified in your lease. Let's say you pay $1,000 a month in rent. Below, we can see what the next five months would cost you, in terms of present value, assuming you kept your money in an account earning 5% interest.
Image by Julie Bang © Investopedia 2019This is the formula for calculating the present value of an annuity due:
PV Annuity Due = C × [ 1 − ( 1 + i ) − n i ] × ( 1 + i ) \begin{aligned} \text{PV}_{\text{Annuity Due}} = \text{C} \times \left [ \frac{1 - (1 + i) ^ { -n } }{ i } \right ] \times (1 + i) \\ \end{aligned} PVAnnuity Due=C×[i1−(1+i)−n]×(1+i)
So, in this example:
PV Annuity Due = $ 1 , 0 0 0 × [ ( 1 − ( 1 + 0 . 0 5 ) − 5 0 . 0 5 ] × ( 1 + 0 . 0 5 ) = $ 1 , 0 0 0 × 4 . 3 3 × 1 . 0 5 = $ 4 , 5 4 5 . 9 5 \begin{aligned} \text{PV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \tfrac{ (1 - (1 + 0.05) ^{ -5 } }{ 0.05 } \right] \times (1 + 0.05) \\ &= \$1,000 \times 4.33 \times1.05 \\ &= \$4,545.95 \\ \end{aligned} PVAnnuity Due=$1,000×[0.05(1−(1+0.05)−5]×(1+0.05)=$1,000×4.33×1.05=$4,545.95
Present Value of an Annuity
The Bottom Line
The formulas described above make it possible—and relatively easy, if you don't mind the math—to determine the present or future value of either an ordinary annuity or an annuity due. Financial calculators (you can find them online) also have the ability to calculate these for you with the correct inputs.