# After tax nominal return.

Working through a question bank doing nominal after tax returns. What is the correct way of doing this? The question bank came up with the following three ways.

1- calculate real return add inflation, then apply tax.

2- calculate real after tax return, then add inflation.

3- in one case the question calculated next years living expense, adjusted for inflation ( meaning times 1 plus inflation) plus a fixed housing expense, all of which equaled real return. And then multiplied by inflation again. I thought this last part was counting inflation twice?

Any help would be appreciated.

either 1 or 3.

But you have made 1 error in 3. It should be “expenses this year plus inflation” and not expenses next year plus inflation.

First inflation term is to calculate nominal expense for next year , based on current year expense and inflation.

Second inflation term is to keep the purchasing power of the portfolio intact in the face of inflation.

Also another minor point is that your 3 does not mention taxes. If you have a taxable account where the portfolio is located you have to apply tax to 3 also

I think i now what is confusing me. On page 168 in the curriculum they do it the second way. However the book states that this way was used only for ease of presentation and should not be used. Could have sworn i did a few essays that did use the second formula…gnna stick to formula number 1 .

Janakisiri i meant what you are saying just worded differently. it was without taxes.

The answer depends on the source of the money: is it in a taxable account or a nontaxable account.

If the money comes from a taxable account, then the entire return (including the amount to cover inflation) will be taxed. In this case, you compute the after-tax return required to cover spending, add inflation, then gross it up for taxes (dividing by (1 - tax rate)).

If the money comes from a nontaxable account, then only the return required to cover spending will be taxed because that money is withdrawn from the account; the return to cover inflation isn’t taxed because that money remains in the account. In this case, the after-tax return to cover spending is grossed up for taxes (divided by (1 - tax rate)), then inflation is added to that.

You also have to be careful to determine when the expenses are paid. If today, use the values given; if at the end of the year, increase them by inflation. (And remember that different expenses may have different inflation rates.)

Why do we assume the inflation money stays in the account? If its a nominal expense one would need to pay real plus inflation. And i thought nontaxable means not taxed. Unless we are talking abt tax exmpt accounts which apply taxes at time 0…

The assumption is that we take from our account only the money we need to pay our expenses and taxes; the rest is left in the account to grow for future years.

Let’s go through a few examples.

We have \$1,000,000 in our account, annual expenses of \$50,000, and inflation of 3%. To simplify things, I’ll assume that we pay all of our expenses at the end of the year (we’ll charge everything on our credit card throughout the year).

Our nominal, after-tax, required rate of return (excluding inflation) is \$50,000 ÷ \$1,000,000 = 5%.

First, we’ll assume no taxes.

In this case, our total rate of return is 5% + 3% = 8%. Here’s what happens:

1. During year 1 our account earns \$80,000 (= \$1,000,000 × 8%), growing to \$1,080,000.

2. At the end of year 1 we take out \$50,000 to pay our bills, leaving \$1,030,000 (= \$1,000,000 × 1.03) in our account.

3. During year 2 our account earns \$82,400 (= \$1,030,000 × 8%), growing to \$1,112,400.

4. At the end of year 2 we take out \$51,500 (= \$50,000 × 1.03), leaving \$1,060,090 (= \$1,030,000 × 1.03) in our account.

And so on, every year.

Now, assume that our account is taxable, and our marginal tax rate is 30%. Because the account is taxable, we have to pay taxes on everything our account earns, whether we take it out (to pay expenses) or leave it in. Our total, pre-tax required rate of return is (5% + 3%) ÷ (1 - 0.30) = 11.4286% Here’s what happens:

1. During year 1 our account earns \$114,286 (= \$1,000,000 × 11.4286%), growing to \$1,114,286.

2. At the end of year 1 we pay \$34,286 (= \$114,286 × 30%) in taxes, leaving \$1,080,000 in our account.

3. At the end of year 1 we take out \$50,000 to pay our bills, leaving \$1,030,000 (= \$1,000,000 × 1.03) in our account.

4. During year 2 our account earns \$117,714 (= \$1,030,000 × 11.4286%), growing to \$1,147,714.

5. At the end of year 2 we pay \$35,314 (= \$117,714 × 30%) in taxes, leaving \$1,112,400 in our account.

6. At the end of year 2 we take out \$51,500 (= \$50,000 × 1.03), leaving \$1,060,090 (= \$1,030,000 × 1.03) in our account.

And so on, every year.

Now, assume that our account is nontaxable, and our marginal tax rate is 30%. Because the account is nontaxable, we have to pay taxes only on the amount we take out (to pay expenses); we pay no taxes on the gains we leave in the account. Our total, pre-tax required rate of return is (5% ÷ (1 - 0.30)) + 3% = 10.1429% Here’s what happens:

1. During year 1 our account earns \$101,429 (= \$1,000,000 × 10.1429%), growing to \$1,101,429.

2. At the end of year 1 we take out \$71,429 to pay our bills and taxes, leaving \$1,030,000 (= \$1,000,000 × 1.03) in our account.

3. At the end of year 1 we pay \$21,429 (= \$71,429 × 30%) in taxes, leaving \$50,000 to pay our bills.

4. During year 2 our account earns \$104,471 (= \$1,030,000 × 10.1429%), growing to \$1,134,471.

5. At the end of year 2 we take out \$73,571 to pay our bills and taxes, leaving \$1,060,900 (= \$1,030,000 × 1.03) in our account.

6. At the end of year 2 we pay \$22,071 (= \$73,571 × 30%) in taxes, leaving \$51,500 to pay our bills.

And so on, every year.

Note that at the end of each year we have the same amount in the account no matter whether the account is taxable or not, and that end-of-year amount grows by the inflation rate every year.

I hope that I made this clear.

Nice, it gets clearer when we seperate the after inflation principle from the after inflation return. Thanks for the long reply. Hope this did not complicate a simple issue