The net present value (NPV) is the present value of a series of cash flows over a specified period of time. In the world of corporate finance, NPV is used to determine whether or not investment decisions in machinery or projects will add or subtract from shareholder wealth.
Assume that a manufacturer was looking to expand production in order to meet the need for increases in product demand. The new machinery would have an initial cash investment of $10,000; additionally, management makes the following projections for the incremental increase in annual cash flows once the machine is running:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
initial cash outlay and projected cash flows
Further, management expects that the required rate of return is 7%. Using these assumptions what is the NPV of the project? Utilizing the table above, we can discount each of the cash flows by the required return as follows:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
PV
$ (10,000.00)
$ 2,803.74
$ 2,838.68
$ 2,857.04
$ 2,860.86
$ 2,851.94
present values
In order to calculate NPV, we simply add all of the present values together then subject from the total the initial cash outlay:
0
1
2
3
4
5
Cashflow
$ (10,000.00)
$ 3,000.00
$ 3,250.00
$ 3,500.00
$ 3,750.00
$ 4,000.00
PV
$ (10,000.00)
$ 2,803.74
$ 2,838.68
$ 2,857.04
$ 2,860.86
$ 2,851.94
NPV
$ 4,212.26
NPV = $4,212.26
Generally speaking, projects that have a positive NPV add to shareholder wealth, while projects that have a negative NPV are detrimental to shareholder wealth.
Additionally, projects that have an NPV of $0 neither add or subtract to shareholder wealth and merely generate enough return to cover the costs of capital. The rate of return associated with an NPV of $0 is also referred to as the internal rate of return (IRR). In the world of fixed income investing, IRR is referred to as the yield-to-maturity (YTM).
Investment decisions may also be made by comparing IRR to the weighted average cost of capital (WACC). If the IRR is greater than the WACC, then management should move forward with the project. In instances where the decision made with IRR conflicts with NPV, then defer to NPV over IRR.
Using an HP12c, we can calculate the NPV of the project above using the following keystrokes:
The formula used to calculate the weighted mean is as follows:
weighted average formula
Let’s assume you have two portfolios, an Roth IRA and a Traditional IRA. The Roth IRA has an average expense ratio of 0.15% and a total portfolio value of $7,000, while the the Traditional IRA has an average expense ratio of 0.47% and a total portfolio value of $3,000.
Using these values we can calculate the weighted average expense ratio as follows:
weighted average calculation
Using an HP12C calculator, we can calculate the weighted average using the following keystrokes:
Compared to an ordinary annuity, the present value of an annuity due can be calculate by modifying the formula above with the addition of the quantity (1 + r) as follows:
present value of annuity due formula
Where: PMT = payment r = rate n = periods
Assume an individual won the lottery and the prize was to be a series of $1,000 payments received at the beginning of each year, over a ten year period. The winner has the option of choosing between the stream of payments or a lump sum discounted at a required rate of 7%, we can calculate what the present value of the stream of payments is as follows:
where; PMT = $1,000, r = 0.07, n = 10
Using an HP12C calculator, we can solve the equation above using the following keystrokes:
Compared to an ordinary annuity, payments for an annuity due are received at the beginning of each period. Due to this extra time that the payments have to compound, we can modify the ordinary annuity formula with the addition of the quantity (1 + r) to calculate the future value of an annuity due as follows:
future value of an annuity due formula
Where: PMT = payment r = rate n = periods
Assume that an individual was to invest $1,000 over a period of 10 years, in a security with a 7% rate of return, and the investment was made at the beginning of each year. We can calculate the future value of the investment as follows:
where; PMT = $1,000, r = 0.07, n = 10
Using an HP12C calculator, we can calculate the future value of an annuity due using the variables above as follows:
The present value of an ordinary annuity formula can be used to calculate the present value of a stream of income payments, the formula is as follows:
present value of an ordinary annuity formula
Where: PMT = payment r = rate n = periods
Assume you won the lottery and the prize is a $1,000 series of payments to be received over the next ten years, at the end of each year. As the winner you could choose either the $1,000 stream of payments or a lump sum discounted at a required rate of 7%. We can calculate the lump sum as follows:
where; PMT = $1,000, r = 0.07, n = 10
Give the result above, as long as the lump sum is exactly $7,023.58, and assuming you could realize a 7% return over ten years, you should be indifferent to receiving a lump sum or the stream of payments. If the lump sum offered is less than $7,023.58, you should chose the income; however, if the lump sum offered is greater than $7,023.58 you should choose the lump sum over the payment stream.
Using an HP12C calculator, you can calculate the present value of an ordinary annuity with the variables above using the following keystrokes:
The future value of an ordinary annuity formula can be used to calculate the future value of a stream of payments over time, the formula is as follows:
future value of ordinary annuity formula
Where: PMT = payment r = rate n = periods
Assume you were to invest $1,000 per year, in an investment that would grow at a 7% rate of return over ten years. What would the future value be at the end of the 10th year?
We can solver for the future value by plugging in the variables as follows:
where; PMT = $1,000, r = 0.07, n = 10
You can calculate the value above with an HP12C using the following keystrokes:
hp12c
[1000][PMT] [7][i] [10][n][FV]
Using Excel, we can model the growth of the investment at different PMTs and rates of growth over time.
Assuming the same variables above we can construct a table of values for each of the periods:
Period
PV
rate
FV
PMT
1
$ 1,000.00
2
$ 1,000.00
7.00%
$ 1,070.00
$ 2,070.00
3
$ 2,070.00
7.00%
$ 2,214.90
$ 3,214.90
4
$ 3,214.90
7.00%
$ 3,439.94
$ 4,439.94
5
$ 4,439.94
7.00%
$ 4,750.74
$ 5,750.74
6
$ 5,750.74
7.00%
$ 6,153.29
$ 7,153.29
7
$ 7,153.29
7.00%
$ 7,654.02
$ 8,654.02
8
$ 8,654.02
7.00%
$ 9,259.80
$ 10,259.80
9
$ 10,259.80
7.00%
$ 10,977.99
$ 11,977.99
10
$ 11,977.99
7.00%
$ 12,816.45
$ 13,816.45
FV of an annuity table; PMT = $1,000, r = 0.07
How would these values look if we reduced the return from 7% to 4%:
Period
PV
rate
FV
PMT
1
$ 1,000.00
2
$ 1,000.00
4.00%
$ 1,040.00
$ 2,040.00
3
$ 2,040.00
4.00%
$ 2,121.60
$ 3,121.60
4
$ 3,121.60
4.00%
$ 3,246.46
$ 4,246.46
5
$ 4,246.46
4.00%
$ 4,416.32
$ 5,416.32
6
$ 5,416.32
4.00%
$ 5,632.98
$ 6,632.98
7
$ 6,632.98
4.00%
$ 6,898.29
$ 7,898.29
8
$ 7,898.29
4.00%
$ 8,214.23
$ 9,214.23
9
$ 9,214.23
4.00%
$ 9,582.80
$ 10,582.80
10
$ 10,582.80
4.00%
$ 11,006.11
$ 12,006.11
FV of an annuity table; PMT = $1,000, rate = 0.04
We can illustrate the two tables graphically as well:
FV of an annuity chart
Generally speaking, over longer periods of time, the higher the rate of return, or the larger the annual contributions, the larger the difference between the two ending values will become.
In the world of financial planning, this formula can be applied to determine the approximate amount of money you will have at retirement on a pre-tax basis.
The variables will be defined by the amount of money you are contributing into your employer sponsored 401(k) plan (a type of tax-deferred account), Roth IRA or Traditional IRA, on an annual basis, any company matching contributions you may receive, and the number of years until you reach your retirement age.
A copy of the Excel model used to calculate the future value of an annuity can be found here:
The concept of future value and present value interest factors is an important one to understand when you begin to calculate future and present values which take into account more complex forms of taxation.
For instance, to determine the future value of an account that taxes interest or dividends on an annual basis at some specified rate would require you to calculate the future value interest factor using a formula that solves for such method of taxation.
Before the age of calculators and computers, solving future value and present value equations required the use of interest factor tables. Fortunately, solving for the factors is easier than in sounds.
The future value interest factor (FVIF) is simply the quantity that the present value is compounded by:
future value interest factor formula
Let’s construct a future value interest factor table using an assumed annual rate of 7%:
Rate
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
Year
1
1.0200
1.0300
1.0400
1.0500
1.0600
1.0700
2
1.0404
1.0609
1.0816
1.1025
1.1236
1.1449
3
1.0612
1.0927
1.1249
1.1576
1.1910
1.2250
4
1.0824
1.1255
1.1699
1.2155
1.2625
1.3108
5
1.1041
1.1593
1.2167
1.2763
1.3382
1.4026
6
1.1262
1.1941
1.2653
1.3401
1.4185
1.5007
7
1.1487
1.2299
1.3159
1.4071
1.5036
1.6058
8
1.1717
1.2668
1.3686
1.4775
1.5938
1.7182
9
1.1951
1.3048
1.4233
1.5513
1.6895
1.8385
10
1.2190
1.3439
1.4802
1.6289
1.7908
1.9672
future value interest factor table
Let’s assume you wanted to calculate the future value interest factor for an investment that will grow at 7% for ten years, plugging those values into the future value interest factor equation will yield the following:
where; r = 0.07, n = 10
Using the table above and finding the area where n = 10 and r = 7% intersect indicates that the FVIF is 1.9672. Any dollar amount multiplied by the FVIF of 1.9672 will yield the future value of an investment that grew at 7% for ten years.
Present value interest factors (PVIF) are simply the inverse of FVIFs:
present value interest factor formula
We can construct a PVIF table in the same manner:
Rate
2%
3%
4%
5%
6%
7%
Year
1
0.9804
0.9709
0.9615
0.9524
0.9434
0.9346
2
0.9612
0.9426
0.9246
0.9070
0.8900
0.8734
3
0.9423
0.9151
0.8890
0.8638
0.8396
0.8163
4
0.9238
0.8885
0.8548
0.8227
0.7921
0.7629
5
0.9057
0.8626
0.8219
0.7835
0.7473
0.7130
6
0.8880
0.8375
0.7903
0.7462
0.7050
0.6663
7
0.8706
0.8131
0.7599
0.7107
0.6651
0.6227
8
0.8535
0.7894
0.7307
0.6768
0.6274
0.5820
9
0.8368
0.7664
0.7026
0.6446
0.5919
0.5439
10
0.8203
0.7441
0.6756
0.6139
0.5584
0.5083
present value interest factor table
Notice, if you multiply any FVIF by its corresponding PVIF the product of the two numbers will equal one:
FVIF * PVIF
Using an HP12C calculator, we can solve for the FVIF in the example above using the following keystrokes:
hp12c
[1][.][0][7][ENTER] [10][y^x]
The Excel model for FVIF and PVIF table construction can be found here.
