When measuring returns over a single period the holding period return (HPR) is used. HPRs can be calculated using the following formula:
holding period return formula
If an investor started with a $1,000 portfolio, that is now worth $1,100, we could calculate the HPR as follows:
v1 = $1,100, v0 = $1,000
HPRs can also be adjusted to take into account dividends or interest received during the period. Assume in the example above the portfolio generated $25 in income:
v1 = $1,100, v0 = $1,000, income = $25
In this instance, the HPR increases from 10% to 12.5%.
In real life, it is often common for portfolio managers to make additional investments throughout the year. Let’s assume that a PM created a portfolio with an initial investment of $1,500,000. Throughout the course of the year the PM made three additional deposits of $60,000 each.
When calculating HPRs given the assumptions above, the beginning and ending values must be taken into account before and after the deposit was made. We can see how the $60,000 deposits affected the beginning and ending market values on the day the deposit was received in the table below:
Sub-Period
Cf
Beg MV
End MV
Jan 1, 2020 to Jan 31, 2020
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
Feb 1, 2020 to Feb 29, 2020
$ –
$ 1,615,000.00
$ 1,650,000.00
Mar 1, 2020 to Mar 31, 2020
$ –
$ 1,650,000.00
$ 1,625,000.00
Apr 1, 2020 to Apr 30, 2020
Apr 1, 2020 to Apr 7, 2020
$ –
$ 1,625,000.00
$ 1,630,000.00
Apr 8, 2020 to Apr 30, 2022
$ 60,000.00
$ 1,690,000.00
$ 1,685,000.00
May 1, 2020 to May 31, 2020
$ –
$ 1,685,000.00
$ 1,700,000.00
Jun 1, 2020 to Jun 30, 2020
$ –
$ 1,700,000.00
$ 1,710,000.00
Jul 1, 2020 to Jul 31, 2020
Jul 1, 2020 to Jul 5, 2020
$ –
$ 1,710,000.00
$ 1,712,000.00
Jul 6, 2020 to Jul 31, 2020
$ 60,000.00
$ 1,772,000.00
$ 1,760,000.00
Aug 1, 2020 to Aug 28, 2020
$ –
$ 1,760,000.00
$ 1,750,000.00
initial portfolio value and subsequent cash flows
Let’s examine the effect the deposits had in the month of January. On January 15th, the portfolio had an ending market value of $1,550,000, the deposit of $60,000 received on January 16th is credited to the ending market value on the 15th, for a total beginning market value of $1,610,000 on the 16th.
If the ending market value on January 31st was $1,615,000, in order to calculate the HPR for the month, we must calculate the HPR for each of these two sub-periods and geometrically link them together as follows:
Sub-Period
Cf
Beg MV
End MV
Monthly Ret
HPR
Jan 1, 2020 to Jan 31, 2020
3.65%
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
3.33%
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
0.31%
January HPR calculation
Based on the table above, the HPR return after geometrically linking the sub-periods of 1/1/20 – 1/15/20 and 1/6/20 to 1/31/20 is 3.65%. Geometrically linking returns can be done as follows:
geometric HPR
Let’s use the same methodology of geometrically linking returns to calculate the HPRs of this portfolio over the eight month period:
Sub-Period
Cf
Beg MV
End MV
Monthly Ret
HPR
Jan 1, 2020 to Jan 31, 2020
3.65%
Jan 1, 2020 to Jan 15, 2020
$ –
$ 1,500,000.00
$ 1,550,000.00
3.33%
Jan 16, 2020 to Jan 31, 2020
$ 60,000.00
$ 1,610,000.00
$ 1,615,000.00
0.31%
Feb 1, 2020 to Feb 29, 2020
$ –
$ 1,615,000.00
$ 1,650,000.00
2.17%
2.17%
Mar 1, 2020 to Mar 31, 2020
$ –
$ 1,650,000.00
$ 1,625,000.00
-1.52%
-1.52%
Apr 1, 2020 to Apr 30, 2020
0.01%
Apr 1, 2020 to Apr 7, 2020
$ –
$ 1,625,000.00
$ 1,630,000.00
0.31%
Apr 8, 2020 to Apr 30, 2022
$ 60,000.00
$ 1,690,000.00
$ 1,685,000.00
-0.30%
May 1, 2020 to May 31, 2020
$ –
$ 1,685,000.00
$ 1,700,000.00
0.89%
0.89%
Jun 1, 2020 to Jun 30, 2020
$ –
$ 1,700,000.00
$ 1,710,000.00
0.59%
0.59%
Jul 1, 2020 to Jul 31, 2020
-0.56%
Jul 1, 2020 to Jul 5, 2020
$ –
$ 1,710,000.00
$ 1,712,000.00
0.12%
Jul 6, 2020 to Jul 31, 2020
$ 60,000.00
$ 1,772,000.00
$ 1,760,000.00
-0.68%
Aug 1, 2020 to Aug 28, 2020
$ –
$ 1,760,000.00
$ 1,750,000.00
-0.57%
-0.57%
Jan 1, 2020 to Aug 28, 2020 HPR
4.66%
HPR table
Now, we can geometrically link each of the months together for a total HPR of 4.66% for the period of January 1st to August 28th.
The Excel model used to calculate multiple HPRs can be found here.
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:
Most people are familiar with the concept of net worth which is simply the sum of one’s assets less liabilities. Net worth is the amount reported on an individual’s traditional balance sheet.
Net wealth expands on the concept of net worth by taking into account human capital and the present value of future consumption needs. In other words, net wealth is the present value of all available marketable and non-marketable assets less the present value of all current and implied liabilities. Net wealth is the amount that is reported on the economic balance sheet, the formula is as follows:
net wealth formula
Let’s assume that an individual had the following assets on his traditional balance sheet:
Assets
Liquid Assets
Checking Account
$ 50,000.00
CDs
$ 250,000.00
Total Liquid Assets
$ 300,000.00
Investment Assets
Brokerage Account
$ 400,000.00
401(k)
$ 700,000.00
Cash value of life insurance
$ 32,000.00
Total Investment Assets
$ 1,132,000.00
Personal Property
House
$ 1,200,000.00
Cars
$ 50,000.00
House Contents
$ 200,000.00
Total Personal Property
$ 1,450,000.00
Total Assets
$ 2,882,000.00
traditional balance sheet assets
Based on this traditional balance sheet, this individual has $2,882,000 in traditional balance sheet assets. On the economic balance sheet, all of these entries would be consolidated into a single asset referred to as financial capital. From there, human capital and the present value of any pension assets would be added to financial capital in order to find the total dollar value of assets on the economic balance sheet.
Let’s assume that based on this individual’s profession, his human capital has a present value of $7,500,000 and the present value of future pension benefits is $500,000:
Assets
Financial Capital
$ 2,882,000.00
Human Capital
$ 7,500,000.00
PV Pension
$ 500,000.00
Total Assets
$ 10,882,000.00
economic balance sheet assets
Based on those assumptions, total assets on the economic balance sheet would amount to $10,882,000 compared to $2,882,000 in total traditional balance sheet assets.
Let’s also assume that this individual had the following liabilities on his traditional balance sheet:
Liabilities
Short-Term
Credit Cards
$ 15,000.00
Total Short Term
$ 15,000.00
Long Term
Mortgage
$ 400,000.00
HELOC
$ 125,000.00
Total Long Term
$ 525,000.00
Total Liabilities
$ 540,000.00
traditional balance sheet liabilities
Based on the traditional balance sheet, this individual’s net worth would be $2,882,000 – $525,000 = $2,342,000. Now, let’s calculate and compare the difference between net worth and net wealth.
On the economic balance sheet, the total dollar value of liabilities would be entered on the economic balance sheet as a single entry referred to as debt. In addition to debt, the economic balance sheet takes into account the present value of all future consumption needs.
Let’s assume that the present value of lifetime consumption needs amounts to $5,200,000, based on this individual’s lifestyle:
Liabilities
Debts
$ 540,000.00
PV Lifetime Consumption
$ 5,200,000.00
Total Liabilities
$ 5,740,000.00
economic balance sheet liabilities
Based on these assumptions, economic balance sheet assets amount to $5,740,000 compared to $540,000 in traditional balance sheet liabilities.
Given the numbers above, we can now calculate this individual’s net wealth, which amounts to $10,882,000 in economic balance sheet assets minus $5,740,000 in economic balance sheet liabilities, for a total net wealth of $5,142,000.
In short, give the assumptions above:
net worth = $2,342,000 net wealth = $5,142,000
Conceptually, two individuals could have the same exact net worth, but their economic net wealth could be vastly different after factoring in income potential and lifestyle needs. In essence, the difference in total net wealth between two individuals with identical net worth may result in different investment strategies, and tolerances and attitudes towards risk.
The Excel file used to calculate net worth and net wealth can be found here.
The concept of human capital can be thought of as the present value of an individual’s future earnings and wages. For most households, human capital represents the single largest asset on the economic balance sheet.
The formula used to calculate an individual’s human capital is as follows:
human capital formula
Where:
human capital variables
Depending on the profession, the wages used may be higher or lower and more or less sensitive to the business cycle. Additionally, the discount rates used in the model should be consistent with the risks of wage growth and consistency of the assumed profession.
Let’s assume an individual was 55 and planned on retiring when he or she reached the normal retirement age of 65. Further, this individual’s current salary is $100,000 and as a professor has consistently received a 3% cost-of-living adjustment (COLA) on an annual basis.
Since this individual has tenure, the discount rate assigned for occupational income volatility is 3%, and the risk free rate is currently 2%. What is the present value of human capital for this individual if he or she has an expected survival rate of 99% in the first year, declining at 1% thereafter on an annual basis?
First, let’s use Excel to model the future value of wage growth over the next ten years at a 3% annual COLA:
Year
FV Wages @ COLA
1
$103,000.00
2
$106,090.00
3
$109,272.70
4
$112,550.88
5
$115,927.41
6
$119,405.23
7
$122,987.39
8
$126,677.01
9
$130,477.32
10
$134,391.64
future value of wages
Next, we’ll need to discount the future value of wages in each year to the present period by the total discount rate, composed of the risk free rate and the discount rate assigned to occupational income volatility:
Risk-Free Rate
Income Volatility
Total Discount Rate
2.00%
3.00%
5.00%
rf rate + discount for occupational income volatility = total discount rate
Using the total discount rate of 5%, we can expand the table above as follows:
Year
FV Wages @ COLA
PV Wages
1
$103,000.00
$ 98,095.24
2
$106,090.00
$ 96,226.76
3
$109,272.70
$ 94,393.87
4
$112,550.88
$ 92,595.89
5
$115,927.41
$ 90,832.16
6
$119,405.23
$ 89,102.02
7
$122,987.39
$ 87,404.84
8
$126,677.01
$ 85,739.99
9
$130,477.32
$ 84,106.84
10
$134,391.64
$ 82,504.81
present value of future wages
Now, we will multiply the present value of wages in each year by the expected probability of survival in each given year:
Year
FV Wages @ COLA
PV Wages
p-survival
P-adjusted Wages
1
$103,000.00
$ 98,095.24
99%
$ 97,114.29
2
$106,090.00
$ 96,226.76
98%
$ 94,302.22
3
$109,272.70
$ 94,393.87
97%
$ 91,562.05
4
$112,550.88
$ 92,595.89
96%
$ 88,892.05
5
$115,927.41
$ 90,832.16
95%
$ 86,290.55
6
$119,405.23
$ 89,102.02
94%
$ 83,755.90
7
$122,987.39
$ 87,404.84
93%
$ 81,286.50
8
$126,677.01
$ 85,739.99
92%
$ 78,880.79
9
$130,477.32
$ 84,106.84
91%
$ 76,537.23
10
$134,391.64
$ 82,504.81
90%
$ 74,254.33
HC
$852,875.90
human capital table
Multiplying the present value of wages by the probability of survival in each year, yields the product which represents the probability weighted present value of wages. The summation of each of these values indicates this individual’s human capital is $852,875.90 under the given assumptions.
In other words, if this individual were to pass away today and had dependents who were counting on this income for survival, a total of $852,875.90 of life insurance would be required to replace his income if no life insurance policies were currently in force.
Keep this formula and model in mind the next time an insurance agent tries to randomly assign an arbitrary face amount to a policy when attempting to sell you life insurance.
A copy of the Excel model used to calculate the present value of human capital can be found here.
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:
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.
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:
tax-drag formula
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 = $1,000,000
We can now plug the ending values of the tax-free and taxable accounts into the tax-drag formula:
tax drag formula
The tax-drag in this particular scenario is 19.44%. In other words, a 1% tax on wealth every ten years eroded 19.44% of the appreciation relative to the tax-free account.
Constructing a tax-drag table that quantifies the effect wealth based taxation has over time will illustrate how destructive wealth taxes are even if wealth taxes are in the low single digits per annum.
First let’s construct the future value table for the tax-free account:
Rate
2%
3%
4%
5%
6%
7%
Year
($ 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.
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:
tax drag formula
Where:
We can define the ending value of a tax-free account as follows:
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 valueformulas:
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:
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.
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.
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.
