TVM – modviz

How to Calculate Net Present Value

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.

We can solve for NPV using the following formula:

where:

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:

[10000][CHS][CF0]
[3000][CFj]
[3250][CFj]
[3500][CFj]
[3750][CFj]
[4000][CFj]
[7][i][f][NPV]

The Excel model used to calculate NPV can be found here.

How to Calculate the Present Value of a Perpetuity

A perpetuity is a series of cash flow payments occurring in equal amounts forever. The formula used to calculate the present value of a perpetuity is as follows:

where:

CF = the periodic cash flow of the perpetuity
i = the discount rate

Assume an investor wanted to purchase a preferred stock that paid an annual dividend of $3.50, using a discount rate of 7%, what is the value of the preferred stock?

We can calculate the present value using the formula above as follows:

In this particular scenario, the present value of the stream of dividend payments for this particular security would be $50.00.

How to Calculate the Present Value of an Annuity Due

Recall, that the present value of an ordinary annuity formula can be used to calculate the present value of a stream of payments received at the end of each year. The formula is as follows:

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:

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:

Using an HP12C calculator, we can solve the equation above using the following keystrokes:

[g][BEG]
[1000][PMT]
[7][i]
[10][n][PV]

How to Calculate the Future Value of an Annuity Due

Recall, that the future value of an ordinary annuity can be used to calculate the future value of a stream of payments that are received at the end of each year, using the following formula:

future value of an ordinary annuity formula

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:

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:

Using an HP12C calculator, we can calculate the future value of an annuity due using the variables above as follows:

[g][BEG]
[1000][PMT]
[7][i]
[10][n][FV]

How to Calculate the Present Value of an Ordinary Annuity

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:

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:

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:

[1000][PMT]
[7][i]
[10][n][PV]

How to Calculate the Tax-Drag of Wealth Based Taxation

In the United States, there has been talk of implementing a wealth tax in order to generate revenue off of wealthier individuals. While the politics of a wealth tax are beyond the scope of this post, we can still examine the effects of wealth based taxation on a quantitative basis.

The future value interest factor formula that accounts for an annual tax on wealth is as follows:

FVIF of wealth based taxation formula

Where:

Assume that an individuals portfolio which comprises their entire net worth will grow at a required rate of 7% for 10 years and the tax rate on wealth is 1% annually, plugging those values into the formula will yield the following:

wheres; r = 0.07, wealth tax = 1%, n = 10

An individual with an initial net worth of $1,000,000 would be worth $1,779,100 in ten years.

On the surface, a 1% tax on wealth doesn’t seem like much when this individual’s portfolio is now worth $1.7mm dollars; however, let’s quantify the tax-drag that wealth based taxation has on net worth.

Recall that the formula used to calculate tax-drag is as follows:

where:

In order to calculate the tax-drag we need to compute the value of the portfolio without taxes as follows:

where; r = 0.07, n = 10, PV = $1,000,000

We can now plug the ending values of the tax-free and taxable accounts into the tax-drag formula:

The tax-drag in this particular scenario is 19.44%. In other words, a 1% tax on wealth every ten years eroded 19.44% of the appreciation relative to the tax-free account.

Constructing a tax-drag table that quantifies the effect wealth based taxation has over time will illustrate how destructive wealth taxes are even if wealth taxes are in the low single digits per annum.

First let’s construct the future value table for the tax-free account:

	Rate
	2%	3%	4%	5%	6%	7%
Year	($ 000s)
1	$ 1,020.00	$ 1,030.00	$ 1,040.00	$ 1,050.00	$ 1,060.00	$ 1,070.00
2	$ 1,040.40	$ 1,060.90	$ 1,081.60	$ 1,102.50	$ 1,123.60	$ 1,144.90
3	$ 1,061.21	$ 1,092.73	$ 1,124.86	$ 1,157.63	$ 1,191.02	$ 1,225.04
4	$ 1,082.43	$ 1,125.51	$ 1,169.86	$ 1,215.51	$ 1,262.48	$ 1,310.80
5	$ 1,104.08	$ 1,159.27	$ 1,216.65	$ 1,276.28	$ 1,338.23	$ 1,402.55
6	$ 1,126.16	$ 1,194.05	$ 1,265.32	$ 1,340.10	$ 1,418.52	$ 1,500.73
7	$ 1,148.69	$ 1,229.87	$ 1,315.93	$ 1,407.10	$ 1,503.63	$ 1,605.78
8	$ 1,171.66	$ 1,266.77	$ 1,368.57	$ 1,477.46	$ 1,593.85	$ 1,718.19
9	$ 1,195.09	$ 1,304.77	$ 1,423.31	$ 1,551.33	$ 1,689.48	$ 1,838.46
10	$ 1,218.99	$ 1,343.92	$ 1,480.24	$ 1,628.89	$ 1,790.85	$ 1,967.15

future value of tax-free account table

We’ll do the same for the taxable account assuming an annual wealth tax of 1%:

	Rate
	2%	3%	4%	5%	6%	7%
Year	($ 000s)
1	$ 1,009.80	$ 1,019.70	$ 1,029.60	$ 1,039.50	$ 1,049.40	$ 1,059.30
2	$ 1,019.70	$ 1,039.79	$ 1,060.08	$ 1,080.56	$ 1,101.24	$ 1,122.12
3	$ 1,029.69	$ 1,060.27	$ 1,091.45	$ 1,123.24	$ 1,155.64	$ 1,188.66
4	$ 1,039.78	$ 1,081.16	$ 1,123.76	$ 1,167.61	$ 1,212.73	$ 1,259.15
5	$ 1,049.97	$ 1,102.46	$ 1,157.02	$ 1,213.73	$ 1,272.64	$ 1,333.81
6	$ 1,060.26	$ 1,124.18	$ 1,191.27	$ 1,261.67	$ 1,335.51	$ 1,412.91
7	$ 1,070.65	$ 1,146.32	$ 1,226.53	$ 1,311.51	$ 1,401.48	$ 1,496.69
8	$ 1,081.14	$ 1,168.91	$ 1,262.84	$ 1,363.31	$ 1,470.71	$ 1,585.45
9	$ 1,091.74	$ 1,191.93	$ 1,300.22	$ 1,417.17	$ 1,543.37	$ 1,679.46
10	$ 1,102.44	$ 1,215.41	$ 1,338.71	$ 1,473.14	$ 1,619.61	$ 1,779.06

future value of taxable account

Now we can calculate the tax-drag table using the two tables above:

	Rate
	2%	3%	4%	5%	6%	7%
Year
1	51.00%	34.33%	26.00%	21.00%	17.67%	15.29%
2	51.25%	34.67%	26.38%	21.40%	18.09%	15.72%
3	51.49%	35.00%	26.76%	21.81%	18.52%	16.17%
4	51.74%	35.34%	27.14%	22.22%	18.95%	16.62%
5	51.99%	35.67%	27.52%	22.64%	19.39%	17.08%
6	52.24%	36.01%	27.91%	23.06%	19.83%	17.54%
7	52.48%	36.35%	28.30%	23.48%	20.28%	18.01%
8	52.73%	36.69%	28.69%	23.91%	20.73%	18.48%
9	52.98%	37.02%	29.08%	24.33%	21.19%	18.96%
10	53.22%	37.36%	29.47%	24.77%	21.65%	19.45%

tax-drag table

Notice how the tax-drag is much higher if your net worth grows at a lower assumed rate of return. Higher rates can somewhat offset the tax-drag; however, a 2% rate of return over ten years has a tax-drag of over 53.22%, and that’s if the annual wealth tax is “only” 1%. Imagine having a tax-drag of 53.22% over ten years before adjusting for inflation. Can you say, “Capital flight”?

The Excel model used to calculate the tax-drag of annual wealth taxes can be found here.

How to Calculate the Tax-Drag on a Taxable Account

In order to calculate the tax-drag on a taxable account, you need to compare the net after-tax future value of a taxable account to the future value of a tax-free account. These values are then plugged into the following formula:

Where:

We can define the ending value of a tax-free account as follows:

ending value of a tax-free account formula

The formula above is essentially the future value formula with a different set of variables. Recall, that the future value of a taxable account adjusts the rate of return by a quantity that takes into account annual accrual based taxation:

ending value of a taxable account formula

Assume you made a $1,000 investment into a security that was projected to pay a 4% dividend for ten years, with an annual dividend tax of 30%. What is the tax-drag on the taxable account relative to the tax-free account?

In order to calculate the tax-drag we need to plug the variables into both future value formulas:

where; V-naught = $1,000, r = 0.04, n = 10

Plugging those same values to solve for the ending value of the taxable account yields the following:

where; V-naught = $1,000, r = 0.04, t = 30%, n = 10

Now that we have calculated the ending values for both the tax-free and taxable accounts, we can plug those values into the tax-drag formula:

Notice that the tax-drag of 33.7% is higher than the annual tax rate of 30%. The additional tax erosion above and beyond the 30% annual rate of taxation is due to paying 30% every year in the taxable account, relative to not paying any taxes in the tax-free account.

We can illustrate the longer term detrimental effects of tax-drag by constructing a tax-drag table. Using Excel, let’s compute a future value table for a tax-free account, assuming an annual tax rate of 30%:

	Rate
	2%	3%	4%	5%	6%	7%
Year
1	$ 1,020.00	$ 1,030.00	$ 1,040.00	$ 1,050.00	$ 1,060.00	$ 1,070.00
2	$ 1,040.40	$ 1,060.90	$ 1,081.60	$ 1,102.50	$ 1,123.60	$ 1,144.90
3	$ 1,061.21	$ 1,092.73	$ 1,124.86	$ 1,157.63	$ 1,191.02	$ 1,225.04
4	$ 1,082.43	$ 1,125.51	$ 1,169.86	$ 1,215.51	$ 1,262.48	$ 1,310.80
5	$ 1,104.08	$ 1,159.27	$ 1,216.65	$ 1,276.28	$ 1,338.23	$ 1,402.55
6	$ 1,126.16	$ 1,194.05	$ 1,265.32	$ 1,340.10	$ 1,418.52	$ 1,500.73
7	$ 1,148.69	$ 1,229.87	$ 1,315.93	$ 1,407.10	$ 1,503.63	$ 1,605.78
8	$ 1,171.66	$ 1,266.77	$ 1,368.57	$ 1,477.46	$ 1,593.85	$ 1,718.19
9	$ 1,195.09	$ 1,304.77	$ 1,423.31	$ 1,551.33	$ 1,689.48	$ 1,838.46
10	$ 1,218.99	$ 1,343.92	$ 1,480.24	$ 1,628.89	$ 1,790.85	$ 1,967.15

future value of a tax-free account table

Let’s do the same for a taxable account:

	Rate
	2%	3%	4%	5%	6%	7%
Year
1	$ 1,014.00	$ 1,021.00	$ 1,028.00	$ 1,035.00	$ 1,042.00	$ 1,049.00
2	$ 1,028.20	$ 1,042.44	$ 1,056.78	$ 1,071.23	$ 1,085.76	$ 1,100.40
3	$ 1,042.59	$ 1,064.33	$ 1,086.37	$ 1,108.72	$ 1,131.37	$ 1,154.32
4	$ 1,057.19	$ 1,086.68	$ 1,116.79	$ 1,147.52	$ 1,178.88	$ 1,210.88
5	$ 1,071.99	$ 1,109.50	$ 1,148.06	$ 1,187.69	$ 1,228.40	$ 1,270.22
6	$ 1,087.00	$ 1,132.80	$ 1,180.21	$ 1,229.26	$ 1,279.99	$ 1,332.46
7	$ 1,102.21	$ 1,156.59	$ 1,213.25	$ 1,272.28	$ 1,333.75	$ 1,397.75
8	$ 1,117.64	$ 1,180.88	$ 1,247.23	$ 1,316.81	$ 1,389.77	$ 1,466.24
9	$ 1,133.29	$ 1,205.68	$ 1,282.15	$ 1,362.90	$ 1,448.14	$ 1,538.08
10	$ 1,149.16	$ 1,231.00	$ 1,318.05	$ 1,410.60	$ 1,508.96	$ 1,613.45

future value of a taxable account table

With these two tables, we can now compute a tax-drag table using the values above:

	Rate
	2%	3%	4%	5%	6%	7%
Year
1	30.00%	30.00%	30.00%	30.00%	30.00%	30.00%
2	30.21%	30.31%	30.41%	30.51%	30.61%	30.71%
3	30.42%	30.62%	30.83%	31.03%	31.23%	31.43%
4	30.63%	30.93%	31.24%	31.55%	31.85%	32.15%
5	30.83%	31.25%	31.66%	32.07%	32.47%	32.87%
6	31.04%	31.56%	32.08%	32.59%	33.10%	33.61%
7	31.26%	31.88%	32.50%	33.12%	33.73%	34.34%
8	31.47%	32.20%	32.92%	33.65%	34.37%	35.08%
9	31.68%	32.51%	33.35%	34.18%	35.00%	35.82%
10	31.89%	32.83%	33.77%	34.71%	35.64%	36.57%

30% tax-drag table

As you can see, the importance of sheltering gains from taxation (in a legal manner of course) becomes extremely important the longer the time frame, the higher the tax rate, and the higher the expected level of return. Based on the table above, the tax-drag on an investment with a 7% rate of return over 10 years is 36.57%, which is much higher than 30%.

Understanding the impact of tax-drag illustrates the importance of having a tax-diversification strategy in addition to an asset diversification strategy. For instance, hold longer term capital appreciating securities which pay no dividends or interest in a taxable account, while investing in interest bearing and dividend paying securities inside of tax-sheltered accounts such as Traditional IRAs and Roth IRAs or employer sponsored retirement plans.

Generally speaking, if taxes are paid on an annual basis the tax-drag will be greater than the tax rate. Conversely, if taxes are deferred until the end of the period the tax-drag will be equal to the tax rate.

A copy of the Excel model used to construct the tax-drag table can be found here.

How to Calculate Accrual Based Taxes on Interest and Dividends

Calculating the future value of an account that taxes interest and dividends on an annual basis requires a basic understanding of future value interest factors (FVIFs).

The formula utilized to calculate the FVIF which takes into account this method of taxation is as follows:

FVIF accrual based dividends formula

The annual taxation on dividends is reflected in the formula above by adjusting the rate of return by the quantity (1 – tax on dividends). The formula for accrual based taxes on interest is identical:

FVIF accrual based interest taxes formula

The only difference between the two formulas is the use of subscript “d” versus subscript “i”.

Let’s assume that an investment paid an annualized rate of interest of 4% for ten years, and the annual tax on interest was 15%. Plugging those values into the formula above would yield the following:

where; r = 0.04, t-interest = 15%, n = 10

If your initial investment was $1,000, you could calculate the future value in the tenth year by multiplying $1,000 by the FVIF of 1.3970:

future value

Using Excel we can construct a FVIF table that takes into account different rates of taxation on interest or dividends, the FVIF table below is constructed with a discount rate of 4%:

	t-interest
year	10%	15%	20%	25%	30%
1	1.03600	1.03400	1.03200	1.03000	1.02800
2	1.07330	1.06916	1.06502	1.06090	1.05678
3	1.11193	1.10551	1.09910	1.09273	1.08637
4	1.15196	1.14309	1.13428	1.12551	1.11679
5	1.19344	1.18196	1.17057	1.15927	1.14806
6	1.23640	1.22215	1.20803	1.19405	1.18021
7	1.28091	1.26370	1.24669	1.22987	1.21325
8	1.32702	1.30667	1.28658	1.26677	1.24723
9	1.37479	1.35109	1.32775	1.30477	1.28215
10	1.42429	1.39703	1.37024	1.34392	1.31805

FVIF accrual based taxes on interest table, where; r = 0.04

Obviously, the higher the rate of taxation the more it will reduce the FVIF.

The Excel model for FVIFs based on annual accrual taxation can be found here.

How to Calculate the Future Value of a Tax-Deferred Account

Recall, that the formula utilized to calculate the future value of a lump sum is as follows:

future value formula

Where:
FV = Future Value
PV = Present Value
r = rate
n = periods

Calculating the future value of a tax-deferred account incorporates the tax paid on the money when it is withdrawn during the final period. We can account for the taxes paid by adjusting the present value after it has been compounded by the specified rate and number of periods:

future value of a tax-deferred account

The addition of the quantity (1 – t) adjusts the future value in the final period by the tax that is owed.

Assume you have a present value of $1,000, the will grow at a rate of 7% for ten years, with an assumed tax rate of 30%. Plugging those values into the formula will yield the following:

where; PV = $1,000, rate = 0.07, n = 10, t = 30%

Using Excel, we can model what happens during each of the ten periods:

Year	PV	rate	FV	Tax (30%)
1	$ 1,000.00	7%	$ 1,070.00
2	$ 1,070.00	7%	$ 1,144.90
3	$ 1,144.90	7%	$ 1,225.04
4	$ 1,225.04	7%	$ 1,310.80
5	$ 1,310.80	7%	$ 1,402.55
6	$ 1,402.55	7%	$ 1,500.73
7	$ 1,500.73	7%	$ 1,605.78
8	$ 1,605.78	7%	$ 1,718.19
9	$ 1,718.19	7%	$ 1,838.46
10	$ 1,838.46	7%	$ 1,967.15
Tax				$ 590.15
Net ATFV			$ 1,377.01

future value of a tax-deferred account table

Notice how the tax is paid during the final period. In the United States, this is how the future value of a Traditional IRA would be calculated. We can represent the table above visually with the following chart:

future value of a tax-deferred account chart

Using an HP12C calculator, we can calculate the future value of a tax-deferred account with the following keystrokes:

[1000][PV]
[7][i]
[10][n][FV]
[.][7][*]

The formula can be rearranged as follows to find the present value of a tax-deferred account:

The present value of a tax-deferred account formula is usually only seen on tests which require you to calculate the present value of a tax-deferred account based on an initial investment an investor made in the past, given some current value in the future.

A copy of the Excel model can be found here.

How to Calculate the Future Value of a Tax-Free Account

Recall, that the formula utilized to calculate the future value of a lump sum is as follows:

Where:
FV = Future Value
PV = Present Value
r = rate
n = periods

Calculating the future value of a tax-free account incorporates the tax paid on the money prior to investing it in the tax-free account. We can account for the taxes paid by adjusting the present value for taxes:

future value of a tax-free account

In essence, the present value is reduced by the tax owed today and becomes the net amount invested. This net amount is then grown tax-free through all periods and no tax liability is owed when the money is withdrawn.

Assume you have a present value of $1,000, that will grow at a 7% rate for 10 years, and the initial tax owed is 30%. We can calculate the future value of the tax-free account by plugging those variables into the formula as follows:

where; PV = $1,000, rate = 0.07, n = 10, t = 30%

Using Excel, we can model what occurs during each of the ten periods:

Year	PV	rate	FV
1	$ 700.00	7.00%	$ 749.00
2	$ 749.00	7.00%	$ 801.43
3	$ 801.43	7.00%	$ 857.53
4	$ 857.53	7.00%	$ 917.56
5	$ 917.56	7.00%	$ 981.79
6	$ 981.79	7.00%	$ 1,050.51
7	$ 1,050.51	7.00%	$ 1,124.05
8	$ 1,124.05	7.00%	$ 1,202.73
9	$ 1,202.73	7.00%	$ 1,286.92
10	$ 1,286.92	7.00%	$ 1,377.01

future value of a tax-free account table

Notice how the initial present value is reduced by the current tax rate. In the United States, this is how the future value of a Roth IRA would be calculated. We can illustrate the table above visually with the following chart:

future value of a tax-free account chart

Using an HP12C calculator, you can calculate the future value of a tax-free account using the following keystrokes:

[1000][ENTER]
[.][7][*][PV]
[7][i]
[10][n]
[FV]

The formula can be rearranged as follows to find the present value of a tax-free account:

The present value version of the tax-free account formula is usually only seen on tests which require you to calculate the initial investment an investor made in the past, given some current value in the future.

A copy of the Excel model can be found here.