The Effects of Taxes or Annual Expenses On Returns

This article is about studyung the effects of tax or funds' expense ratio on portfolio performance. Taxes in some sense are just like annual expenses, if they are incurred periodically.

How Taxes are like expense ratio?

Taxes are like expense ratio to your investment portfolio in a time we need to sell and withdraw some money from it during which taxes are incrred. For example, if we sell 4% of our portfolio annunally and the tax we pay is 1% of it, the annual tax we pay with respect to total portfolio value is 4% X 1% = 0.04%, which is just like a expense ratio of 0.04%.

What about dividends?

If we need to pay taxes on dividends, they are just like expense ratio. For example, some stock pays annual dividend of $1 and its price is $100. Then, its yield is $1/$100 = 0.01. If the tax rate is 30%, we pay $1X30%=$0.3. If $0.3 is divided by the stock price of $100, we get 0.3%, which is the annunal cost rate.

How Taxes or expensese affect returns?

My first guess of how taxes or expenses affect portfolio returns is by the simple interest formul, which means if the expense ratio is 1%, the annual deduction of return is 1%. If the expense ratio is 2%, the deduction of return is 2% and so on and so forth.

Since the simple interst formlua does not make mathematic sense to me when I calcualate the actual effect of expense on reutrn, I have turned to make some artificial data to find the restuls. Suppose a fund's price grows 4% annually and the expense ratio is 2%, here is a table of date to use to see its effect on return.

Time	Price	Return (%)	Price Taxed	Return After Tax (%)	Ratio of Returns
0	100	0	100	0	0
1	104	4.00	101.92	1.92	0.480
2	108.16	8.16	103.87	3.87	0.474
3	112.49	12.49	105.87	5.87	0.470
4	116.99	16.99	107.90	7.9	0.465
5	121.67	21.67	109.97	9.97	0.460
6	126.53	26.53	112.09	12.09	0.456

Math Formula

According to the the above table, it seems like the effect of expennse ratio on return is close to the ratio of expense to return which is 2%/4%= 0.5 in the short run and the effect gets more sever in the long run.

During the course of makeing the above table, I have found the formula to return after expense.

1. At time 1, the value of the fund is 100*1.04 with no tax and 100*1.04*(1-0.02).
2. At time 2, the value of the fund is 100*1.04^2 with no tax and 100*1.04^2*(1-0.02)^2.
3. At time 3, the value of the fund is 100*1.04^3 with no tax and 100*1.04^3*(1-0.02)^3.

By derivation, the formula for future value of fund with expense is

FV = PV*((1+Return Rate)*(1-Expense Ratio))

where FV is the future value, PV is the present value and both return rate and expense ratio are annuallized value.

Here is a graph of different rates of returns and expense ratios. We can see the longer the duration of investment, the greater the lose of the portfolio value due to expenses.

Return After Expense Calculator

Initial Value: $.
Investiment Duration: year(s).
Annual Return Rate: %.
Annual Expense Ratio: %.

1. The investment result after 10 years with no expense is $1480.244 and the cumulative return is 48.024%.
2. The investment result after 10 years with an expense ratio of 1% is $1338.706 and the cumulative return is 33.871%.
3. The ratio of return with expense over the return with no expense is 0.705%.

Return With Tax on Quarterly Dividend Calculator

Suppose the stock price does not change.

Stock Value: $.
Investiment Duration: year(s).
Annualized Dividend Yield: %.
Tax Rate: %.

1. The quarterly dividends are $10
2. The investment result after 10 years with no taxes on dividends is $1488.864 and the cumulative return is 48.886%.
3. The investment result after 10 years with taxes on dividends of 1% is $1482.920 and the cumulative return is 48.292%.
4. The ratio of return with expense over the return with no expense is 0.988%.

Return With Tax on Monthlly Dividend Calculator

Suppose the stock price does not change.

Stock Value: $.
Investiment Duration: year(s).
Annualized Dividend Yield: %.
Tax Rate: %.

1. The monthly dividends are $3.3333333333333335
2. The investment result after 10 years with no taxes on dividends is $1490.833 and the cumulative return is 49.083%.
3. The investment result after 10 years with taxes on dividends of 1% is $1484.881 and the cumulative return is 48.488%.
4. The ratio of return with expense over the return with no expense is 0.988%.

In the real world

So far I have not touched random returns which happens all the time in the real world and I also want to know if a random array of returns would be affected the same way as with a fixed retun. The following app created a series of random return based on the assumption that stock returns follow arithmetic Brownian motion.

Mean Return Rate: %.
Return Rate Standard Deviation: %.
Annual Expense Ratio: %.
Investment Duration: year(s).

1. The investment result after 30 years with no expenses is $592.571 and the cumulative return is 492.571%.
2. The investment result after 30years with expenses of 1% is $438.325 and the cumulative return is 338.325%.
3. The ratio of return with expense over the return with no expense is 0.687%.

Run the Above App 1,000 Times

One last thing I want to do is to find out if the mean of all ratio of returs when return rate is random is equal to the one when the return rate is fixed.

1. The theoretical ratio of returns is .
2. The mean of simulated ratios of returns is .

Conclusion

The effects of taxes and expenses on investment could be sever, we can only try to avoid them to achieve max returns from investments.

The ratio of returns is the ratio of cumulative return with expenses over the one with no expense. The ratio of returns is a way I measure how much the expense ratio affects the total return. Taxes are just like expense ratio in this scenario.

The mean of the ratios of returns when portfolio values are fluctuatuing does is close to the one when they are not fluctuatuing, therefore the theoretical ratio of returns could be used in financial planning when taxes or expense ratio is considered,