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:
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.
