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Time Value of Money Calculations

Time Value of Money Calculations

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Calculate the present and future value for money that gains interest using time value of money tables.

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Tutorial

what's covered
In this lesson, you will calculate the effect of time and interest on financial decisions pertaining to planning, investing, and borrowing. Specifically, this lesson will cover:
  1. TVM: Plan with Confidence
  2. Solving TVM Problems
    1. Future Value of a Lump Sum
    2. Present Value of a Lump Sum
    3. Future Value of an Annuity
    4. Payments

1. TVM: Plan with Confidence

During your lifetime financial journey, you will be faced with many unique financial opportunities to invest, borrow, and plan for the future. Knowing how much you can afford to borrow before you apply for a loan or talk with a lender is a powerful tool. Additionally, having the ability to determine your loan payment based on the amount borrowed and interest rate charged might just protect you from being overcharged by thousands of dollars. Finally, calculating how much you should be saving for long-term goals will give you confidence in your plans or the opportunity to be agile and adjust your plans while you still have time to make changes. In this topic, you’ll learn how to perform your own time value of money (TVM) calculations, which is an essential tool for building a strong financial future.


2. Solving TVM Problems

There are multiple ways to solve TVM problems. Most often people prefer to use a spreadsheet program like Microsoft® Excel or Google® Sheets, a financial calculator, or an app designed to solve these types of problems. These tools can help improve your productivity and they can make it easier to adjust plans when needed. However, there is value in learning how to do TVM calculations yourself either using formulas or compound interest tables. We will provide some basic steps to follow when using a calculator to solve TVM problems. However, you should consult the user manual specific to your calculator for more detailed instructions.

Technology: Skill Reflect
Are there ways you currently use programs like Excel? If so, what are they? If not, how can you begin to use technology tools like Excel in your daily life to help increase your comfort and confidence with spreadsheets?

hint
When you see the term "annuity due," remember that the payments or deposits come at the beginning of the period. The annuity due formula accounts for this by inflating the traditional result for the calculation’s rate of return. All of the other formulas use end-of-period assumptions.

2a. Future Value of a Lump Sum
Here’s an example showing how to calculate a future value.

EXAMPLE

Let’s say that you receive $1,000 at your college graduation. If you invest the gift and earn 8% annually, how much will you have in 20 years? Let’s first solve using the following formula.
F V equals P V left parenthesis 1 plus I right parenthesis to the power of N
F V equals $ 1 comma 000 cross times left parenthesis 1.08 right parenthesis to the power of 20
F V equals $ 1 comma 000 cross times 4.66096
F V equals $ 4 comma 660.96
Next, let’s solve the problem using a TVM table. The table below shows the Future Value of $1 table that you’ll need to use. Here’s how to use the table.

step by step
  1. Find the number of periods (20) in the first column.
  2. Move to the right until you find the future value factor that corresponds to the 8% column: 4.66096.
  3. Multiply the factor by your present value ($1,000): $1,000 x 4.6609, or $4,660.96.

You should notice two things. First, the solution is the same whether calculated with the formula or the table. Second, the future value factor in the table (4.66096) is basically the same as left parenthesis 1.08 right parenthesis to the power of 20 in the formula.

Table: Future Value of $1

Periods 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 12.00%
1 1.04000 1.05000 1.06000 1.07000 1.08000 1.09000 1.01000 1.11000 1.12000
2 1.08160 1.10250 1.12360 1.14490 1.16640 1.18810 1.21000 1.23210 1.25440
3 1.12486 1.15763 1.19102 1.22504 1.25971 1.29503 1.33100 1.36763 1.40493
4 1.16986 1.21551 1.26248 1.31080 1.36049 1.41158 1.46410 1.51807 1.57352
5 1.21665 1.27628 1.33823 1.40255 1.46933 1.53862 1.61051 1.68506 1.76234
6 1.26532 1.34010 1.41852 1.50073 1.58687 1.67710 1.77156 1.87041 1.97382
7 1.31593 1.40710 1.50363 1.60578 1.71382 1.82804 1.94872 2.07616 2.21068
8 1.36857 1.47746 1.59385 1.71819 1.85093 1.99256 2.14359 2.30454 2.47596
9 1.42331 1.55133 1.68948 1.83846 1.99900 2.17189 2.35795 2.55804 2.77308
10 1.48024 1.62889 1.79085 1.96715 2.15892 2.36736 2.59374 2.83942 3.10585
11 1.53945 1.71034 1.89830 2.10485 2.33164 2.58043 2.85312 3.15176 3.47855
12 1.60103 1.79586 2.01220 2.25219 2.51817 2.81266 3.13843 3.49845 3.89598
13 1.66507 1.88565 2.13293 2.40985 2.71962 3.06580 3.45227 3.88328 4.36349
14 1.73168 1.97993 2.26090 2.57853 2.93719 3.34173 3.79750 4.31044 4.88711
15 1.80094 2.07893 2.39656 2.75903 3.17217 3.64248 4.17725 4.78459 5.47357
16 1.87298 2.18287 2.54035 2.95216 3.42594 3.97031 4.59497 5.31089 6.13039
17 1.94790 2.29202 2.69277 3.15882 3.70002 4.32763 5.05447 5.89509 6.86604
18 2.02582 2.40662 2.85434 3.37993 3.99602 4.71712 5.55992 6.54355 7.68997
19 2.10685 2.52695 3.02560 3.61653 4.31570 5.14166 6.11591 7.26334 8.61276
20 2.19112 2.65330 3.20714 3.86968 4.66096 5.60441 6.72750 8.06231 9.64629

2b. Present Value of a Lump Sum
Imagine you are offered the choice of taking $1,000 today or $1,200 in 5 years. Before you can make a decision, you also need to know what rate of return you can earn on your savings (sometimes called the discount rate). You determine that you can earn 5%. What should you do? You should use the present value of a lump sum formula to compare the $1,000 that you can receive today to the present value of $1,200 as follows:

P V equals fraction numerator F V over denominator left parenthesis 1 plus I right parenthesis to the power of N end fraction
P V equals fraction numerator $ 1 comma 200 over denominator 1.05 to the power of 5 end fraction
P V equals fraction numerator $ 1 comma 200 over denominator 1.27628 end fraction
P V equals $ 940.23
As you can see, the present value of receiving $1,200 in 5 years is only $940.22. So, it’s better to take the $1,000 right now.

The table below shows the Present Value of $1 table that you can also use to solve the problem.

hint
Note that the Present Value of $1 table is set up exactly the same as the Future Value of $1 table.

Table: Present Value of $1

Periods 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 12.00%
1 0.96154 0.95238 0.94340 0.93458 0.92593 0.91743 0.90909 0.90090 0.89286
2 0.92456 0.90703 0.89000 0.87344 0.85734 0.84168 0.82645 0.81162 0.79719
3 0.88900 0.86384 0.83962 0.81630 0.79383 0.77218 0.75131 0.73119 0.71178
4 0.85480 0.82270 0.79209 0.76290 0.73503 0.70843 0.68301 0.65873 0.63552
5 0.82193 0.78353 0.74726 0.71299 0.68058 0.64993 0.62092 0.59345 0.56743
6 0.79031 0.74622 0.70496 0.66634 0.63017 0.59627 0.56447 0.53464 0.50663
7 0.75992 0.71068 0.66506 0.62275 0.58349 0.54703 0.51316 0.48166 0.45235
8 0.73069 0.67684 0.62741 0.58201 0.54027 0.50187 0.46651 0.43393 0.40388
9 0.70259 0.64461 0.59190 0.54393 0.50025 0.46043 0.42410 0.39092 0.36061
10 0.67556 0.61391 0.55839 0.50835 0.46319 0.42241 0.38554 0.35218 0.32197
11 0.64958 0.58468 0.52679 0.47509 0.42888 0.38753 0.35049 0.31728 0.28748
12 0.62460 0.55684 0.49697 0.44401 0.39711 0.35553 0.31863 0.28584 0.25668
13 0.60057 0.53032 0.46884 0.41496 0.36770 0.32618 0.28966 0.25751 0.22917
14 0.57748 0.50507 0.44230 0.38782 0.34046 0.29925 0.26333 0.23199 0.20462
15 0.55526 0.48102 0.41727 0.36245 0.31524 0.27454 0.23939 0.20900 0.18270
16 0.53391 0.45811 0.39365 0.33873 0.29189 0.25187 0.21763 0.18829 0.16312
17 0.51337 0.43630 0.37136 0.31657 0.27027 0.23107 0.19784 0.16963 0.14564
18 0.49363 0.41552 0.35034 0.29586 0.25025 0.21199 0.17986 0.15282 0.13004
19 0.47464 0.39573 0.33051 0.27651 0.23171 0.19449 0.16351 0.13768 0.11611
20 0.45639 0.37689 0.31180 0.25842 0.21455 0.17843 0.14864 0.12403 0.10367

step by step
  1. Find the number of periods in column 1, which is 5.
  2. Go to the right until you find the present value factor in the 5% column: 0.78353.
  3. Multiply $1,200 by 0.78353 and you will get $940.24 (which is very close to the formula calculation).

term to know

Discount Rate
Rate of return you can earn on your savings.
2c. Future Value of an Annuity
By now, you might be asking how you can determine how much money you’ll have in the future if you save money every year instead of starting with a lump sum. When you start saving money on a regular basis, this is called an annuity. In TVM lingo, the amount saved or paid each period is referred to as a payment (PMT). Payments are different than present or future values because, as the name implies, a payment happens more than once.

EXAMPLE

How much, for example, will you accumulate in 20 years if you start saving $1,000 every year (each payment is the same amount) and earn 9% on your savings? The following formula can be used to answer this question:

F V A equals fraction numerator P M T over denominator I end fraction left parenthesis 1 plus I right parenthesis to the power of N minus 1
F V A equals fraction numerator $ 1 comma 000 over denominator 0.09 end fraction cross times left square bracket left parenthesis 1 plus 0.09 right parenthesis to the power of 20 minus 1 right square bracket
F V A equals $ 11 comma 111.11 cross times 4.60441
F V A equals $ 51 comma 160.11
You can also solve this problem with a TVM table. The table below shows the Future Value of an Annuity of $1 table that can be used to solve the problem as follows:

step by step
  1. Just like the other table examples, start by finding the number of payments (20) in the first column.
  2. Go to the right until you find the future value factor corresponding to the 9% column: 51.16012.
  3. Multiply $1,000 by 51.16012. This gives you a future value of $51,160.12, which is nearly the same as what you estimated using the formula.

Table: Future Value of Annuity for $1 at the End of Each Period

Per 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00% 11.00% 12.00%
1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
2 2.04000 2.05000 2.06000 2.07000 2.08000 2.09000 2.10000 2.11000 2.12000
3 3.12160 3.15250 3.18360 3.21490 3.24640 3.27810 3.31000 3.34210 3.37440
4 4.24646 4.31013 4.37462 4.43994 4.50611 4.57313 4.64100 4.70973 4.77933
5 5.41632 5.52563 5.63709 5.75074 5.86660 5.98471 6.10510 6.22780 6.35285
6 6.63298 6.80191 6.97532 7.15329 7.33592 7.52334 7.71561 7.91286 8.11519
7 7.89829 8.14201 8.39384 8.65402 8.92280 9.20044 9.48717 9.78327 10.08901
8 9.21423 9.54911 9.89747 10.25980 10.63663 11.02847 11.43589 11.85943 12.29969
9 10.58280 11.02656 11.49132 11.97799 12.48756 13.02104 13.57948 14.16397 14.77566
10 12.00611 12.57789 13.18079 13.81645 14.48656 15.19293 15.93743 16.72201 17.54874
11 13.48635 14.20679 14.97164 15.78360 16.64549 17.56029 18.53117 19.56143 20.65458
12 15.02581 15.91713 16.86994 17.88845 18.97713 20.14072 21.38428 22.71319 24.13313
13 16.62684 17.71298 18.88214 20.14064 21.49530 22.95339 24.52271 26.21164 28.02911
14 18.29191 19.59863 21.01507 22.55049 24.21492 26.01919 27.97498 30.09492 32.39260
15 20.02359 21.57856 23.27597 25.12902 27.15211 29.36092 31.77248 34.40536 37.27972
16 21.82453 23.65749 25.67253 27.88805 30.32428 33.00340 35.94973 39.18995 42.75328
17 23.69751 25.84037 28.21288 30.84021 33.75023 36.97351 40.54470 44.50084 48.88367
18 25.64541 28.13238 30.90565 33.99903 37.45024 41.30134 45.59917 50.39593 55.74972
19 27.67123 30.53900 33.75999 37.37896 41.44626 46.01846 51.15909 56.93949 63.43968
20 29.77808 33.06595 36.78559 40.99549 45.76196 51.16012 57.27500 64.20283 72.05244

think about it
The present value of an annuity is another TVM calculation that’s useful when planning your finances. What do you think are similarities and differences between the calculations for the present value of an annuity and the present value of a lump sum?

Note: calculations for present value of an annuity are beyond the scope of this tutorial.

2d. Payments
There are times you’ll need to calculate what is called an amortized payment – a payment of the same amount for a set number of months or years – such as for a car loan or mortgage. To do so, you should use the following formula:

formula

Monthly Payment = P V open square brackets fraction numerator I cross times left parenthesis 1 plus I right parenthesis to the power of N over denominator left parenthesis 1 plus I right parenthesis to the power of N minus 1 end fraction close square brackets
In the formula, PV is the amount borrowed, I = interest rate, and N = number of payments.

EXAMPLE

For example, say that Max wants to borrow $250,000 to purchase a home. He can get a 30-year loan at a 6% interest rate. To help Max determine if he can afford this loan, you first need to convert the years to months and the yearly rate of interest to monthly interest because Max will be making monthly payments.

  • Number of periods (N) = 30 years × 12 months = 360
  • Monthly interest rate (I) = 6%/12 = 0.5% = 0.005
You can now use the following formula to determine how much Max’s monthly payment will be (that is, you can calculate his principal and interest payment).

Monthly Payment = P V open square brackets fraction numerator I cross times left parenthesis 1 plus I right parenthesis to the power of N over denominator left parenthesis 1 plus I right parenthesis to the power of N minus 1 end fraction close square brackets
Monthly Payment = $ 250 comma 000 open square brackets fraction numerator 0.005 cross times left parenthesis 1 plus 0.005 right parenthesis to the power of 360 over denominator left parenthesis 1 plus 0.005 right parenthesis to the power of 360 minus 1 end fraction close square brackets
Monthly Payment = $ 250 comma 000 open square brackets fraction numerator 0.005 cross times 6.0226 over denominator 6.0226 minus 1 end fraction close square brackets
Monthly Payment = $ 1 comma 498.88
If you round the answer, Max will need to pay approximately $1,500 per month for the next 30 years to pay off the home mortgage loan.

You can also solve this problem using an amortization schedule. The factors shown in the table below show the monthly dollar payment needed to pay off a $1,000 loan.

Table: Amortization Schedule for Monthly Payments for Every $1,000 Borrowed

Years rightwards arrow 5 10 15 20 25 30
3.00% $17.97 $9.66 $6.91 $5.55 $4.74 $4.22
3.50% $16.67 $8.33 $5.56 $4.17 $3.33 $2.78
4.00% $18.42 $10.12 $7.40 $6.06 $5.28 $4.77
4.50% $18.64 $10.36 $7.65 $6.33 $5.56 $5.07
5.00% $18.87 $10.61 $7.91 $6.60 $5.85 $5.37
5.50% $19.10 $10.85 $ 8.17 $6.88 $6.14 $5.68
6.00% $19.33 $11.10 $8.44 $7.16 $6.44 $6.00
6.50% $19.57 $11.35 $8.71 $7.46 $6.75 $6.32
7.00% $19.80 $11.61 $8.99 $7.75 $7.07 $6.65
7.50% $20.04 $11.87 $9.27 $8.06 $7.39 $6.99
8.00% $20.28 $12.13 $9.56 $8.36 $7.72 $7.34
8.50% $20.52 $12.40 $9.85 $8.68 $8.05 $7.69
9.00% $20.76 $12.67 $10.14 $9.00 $8.39 $8.05
9.50% $21.00 $12.94 $10.44 $9.32 $8.74 $8.41
10.00% $21.25 $13.22 $10.75 $9.65 $9.09 $8.78

To estimate the monthly payment needed by Max:

step by step
  1. Find the interest rate of the loan in the first column.
  2. Go to the right until you find the amortization factor matching 30 years: $6.00 (Max will pay $6.00 for every $1,000 borrowed).
  3. Divide the loan amount by $1,000: ($250,000/$1,000) = $250 (you need to do this to keep the factors consistent with the loan amount).
  4. Multiply 250 by $6.00 to get $1,500, which is a very close approximation of the actual amount of the loan payment.
term to know

Amortized Payment
A payment of the same amount for a set number of months or years, such as for a car loan or mortgage.
summary
When you can mathematically solve time value of money (TVM) problems, it allows you to plan your finances with confidence when you’re saving, investing, and borrowing. Several types of TVM calculations were covered in this tutorial:
  • Future Value of a Lump Sum
  • Present Value of a Lump Sum
  • Future Value of an Annuity
  • Payment

Using technology, like spreadsheets, for TVM calculations can help you increase productivity and gain the ability to be agile when needed.

Source: This content has been adapted from Chapter 2.3 of Introduction to Personal Finance: Beginning Your Financial Journey. Copyright © 2019 John Wiley & Sons, Inc. All rights reserved. Used by arrangement with John Wiley & Sons, Inc.

Wiley and the Wiley logo are trademarks or registered trademarks of John Wiley & Sons, Inc. and/or its affiliates in the United States and other countries.

Terms to Know
Amortized Payment

A payment of the same amount for a set number of months or years, such as for a car loan or mortgage.

Discount Rate

Rate of return you can earn on your savings.