## The CFA L3 Tax Equations - FVIF and tax drag

In our coverage of taxes for Level 3 we've talked about the key principles to be aware of, the different types of tax regimes, and the difference between marginal and average tax rates.

That is all vital, testable stuff.

But when it comes to specifics and calculations for L3 **this is where the rubber meets the road** and we need to apply our general understanding to specific calculations.

You should expect to be tested on:

- Specific return calculations under a given scenario (i.e. using one of the FVIF equations)
- How tax drag on either a percentage or dollar basis varies depending on the type of tax
- Calculating blended tax impacts: the effective capital gains tax, the annual return after realized taxes, or accrual equivalent tax rates or returns
- How government taxes impact investment risk

Before we dive in you should know that this is not the most fun section for Candidates. There is inevitably some memorization involved. But if you keep the basic principles in mind you should be able to minimize some of the pain and earn all the available points on the level 3 exam.

### What are the tax equations really telling us?

All of the equations we are about to cover deal with the impact of taxes on the rate at which we compound some initial asset value. In other words, this is where you need to figure out how the rate at which we compound our initial assets changes depending on when we take out taxes.

As you go through each equation you want to try to move from pure memorization standpoint to an understanding about what the order of operations in the equation is actually telling us.

Ultimately this comes down to the timing of taxes and the rate at which you compound wealth. If you get stuck always default back to the core principle that the more we can defer paying taxes the higher our terminal value will be.

### The Future Value Interest Factor (FVIF)

In terms of solving equations in the section we are generally solving for the *Future Value Interest Factor* (FVIF). __The FVIF is the number you multiply by your PV to get the future value of a given investment__. The FVIF number accounts for the rate of return, the length of the time period, and the impact of tax withdrawals along the way.

So say we start with $2 dollar and your FVIF for the given time period is 2.25. That means that at the end of the period we would have $4.50 ($2 x 2.25). You can also compare this to the total compounded return without taxes in order to calculate the ** tax drag**.

**Tax Drag**

Tax drag is the percent or dollar amount of your gain that is lost to taxes. While it's pretty straightforward in terms of how you calculate it, there are some nuances in terms of understanding it on a dollar versus percentage basis, and most importantly how the different tax mechanisms we're about to cover affect tax drag as returns or the length of time in question increase or decrease.

**Calculating tax drag**

To calculate tax drag you just compare the return without taxes to the total gain with taxes and that gives you your dollar tax drag value. If you want your percentage, you just take the tax drag dollar value and divide it by what the total gain would have been without taxes. Note that Candidates often make the mistake of dividing by the total amount without taxes, but you just want to take the gain.

This is a vital concept so let's run through a quick example. Say:

- N = 2
- Return = 5%
- The annual tax rate = 25%
- PV = $100

Without taxes, you'd end up with 110.25 (which is 100 x (1.05^2)), with taxes, you'd end up with 107.64. So, the difference of $2.61 is your tax drag on a dollar basis, and the tax drag percentage is 25.46% ($2.61/$10.25).

This type of problem is definitely testable and you'll see it in the CFA Curriculum end of chapter questions. That said, however, you should also memorize if tax drag increases, stays the same or decreases over time on either a dollar or percentage basis under the different types of taxes.

## Calculating FVIF for different tax systems

There are four main types of taxes we'll cover here:

- Accrual taxes
- Deferred capital gains taxes - with cost basis = market value
- Deferred capital gains taxes with the cost basis ≠ market value
- Wealth based taxes

In addition to the calculations of each FVIF expect the CFA level three exam to have a question or two about tax drag as a function of higher returns or longer time horizons.

### Accrual Taxes

Annual accrual taxes are taxes which are taken out before any compounding. In other words accrual taxes are taxes that are paid annually:

**$\text{FVIF}_i=[1+r(1-T)]^n$**

With annual accrual, taxes are taken out each year BEFORE they are compounded. There are several implications for this:

- The % of total taxes paid is > than the stated tax rate (due to compounding)
- As you increase the investment time horizon (N), the tax drag increases in both $ and % terms
- As you increase the investment returns (R), the tax drag increases in both $ and % terms

Of all of the tax systems we cover here accrual taxes is the most negative in terms of its overall impact.

### Deferred Capital Gains Tax (MV = Cost Basis)

Deferred capital gains allow the investment to compound without taxes being taken out until the very end. In the principles of taxes blog post we saw how this tax-free compounding grew the portfolio at a more rapid rate compared to accrual taxes. There are two equivalent equations for calculating the deferred capital gains tax rate.

**$\text{FVIF}_{\text{CG}}= \left ( 1+r \right )^n-\left [ \left ( 1+r \right )^n-1 \right ]t_{\text{cg}}$**

**or**

**$\text{FVIF}_{\text{CG}}= \left ( 1+r \right )^n\left ( 1-t_{\text{cg}} \right )+t_{\text{cg}}$**

In both equations what we are showing is that returns compound tax-free and we take cumulative taxes for that time period out at the end. We also are only taxing the actual capital gains and not our initial starting value. In the equations, that's why we're adding back in capital gains taxes on our intial value.

To unpack both equations a bit more:

In the first equation, the first term is the pretax accumulation and the bracketed term that follows (that one plus “r” to the “nth” minus one) is the capital gain. So our future accumulation is the total return minus the original cost basis. And then the third term is the tax on the investment gain, i.e. our tax rate.

In the second equation we represent this slightly differently. Here the first term is the future accumulation as if the entire amount was subject to tax and then we're kind of adding back in that second term (the tax) to account for the fact that we don't pay taxes on the original cost basis.

### Deferred Capital Gains Tax (MV ≠ Cost Basis)

Above we assumed that the market value is equal to our cost basis which is a special case. Now, let's look at what it is if the market value is not equal to our cost basis. You'll notice the equation is very similar except for the cost basis term at the end:

**$\text{FVIF}_{\text{CGB}}= \left [ \left ( 1+r \right )^n\left ( 1-t_{\text{cg}} \right ) \right ]+t_{\text{cg}}B$**

Where** $B$ = Cost Basis/Market Value**

**Implications**

The *lower* the cost basis relative to current market value the *more* taxes you will pay and the lower the FVIF will be (as you can see since the last term in the equation has a + sign). This should make intuitive sense—the lower you bought the stock relative to where it is now, the more gains you already have and the more deferred capital gains taxes you will have to pay eventually. You can just memorize this last equation and set B = 1 when MV = Cost basis.

#### Tax Drag with Deferred Capital Gains Tax

So, we've mentioned over and over again that tax drag and how different types of taxes affect it is very testable for L3. Now with accrual taxes, the length of time and higher investment returns kind of "compounded the negative" of taxes and led to both greater tax drag on a dollar and percentage basis.

With deferred capital gains taxes, because we're not paying taxes along the way, **the tax drag is a constant rate**, regardless of either time or return. So the tax rate equals tax drag percentage. And the implication as well is that the longer we defer the taxes, the more value there is.

**To summarize, with deferred capital gains taxes:**

- The loss to deferred taxes is always a constant rate regardless of time or return
- Tax rate = tax drag %
- Thus the value of deferring taxes increases as the time or return increase

### Wealth Based Taxes

Wealth based taxes are very different. First, they are paid on both the principal AND the return. This is a huge contrast to accrual and deferred capital gains where we were just talking about taxing the delta, or the growth of the asset.

But with wealth-based taxes you're levied a flat tax on everything. This system is much less common, but it does exist (for example in UK real estate).

The formula is:

$\text{FVIF}_{\text{WT}}= \left [ \left ( 1+r \right )\left ( 1-\text{TW} \right ) \right ]^n$

__Implications__

- As with accrual taxes the longer the time period the greater the tax drag %
- UNLIKE accrual taxes, the greater the returns the lower the tax drag %

### Summarizing the FVIF Calculations and Tax Drag Implications

So, this table is really crucial. I think it's quite testable.

Know that:

- Your annual accrual leads to greater tax drag in dollar and percentage terms, based on both returns and horizon
- With deferred capital gains, the more and more you kind of can defer your payment over time, the more and more your portfolio will benefit. So, in terms of dollar value, greater returns, longer time horizon are actually reducing the tax drag and the percentage is a constant equal to your tax rate
- With the wealth tax, the higher your returns, the lower the tax drag percentage. It's still of course going to be higher in overall dollar terms because it's just whatever percentage times your overall value, but the percentage will go down as returns go up

In our next post we'll cover the blended effects of being subject to different types of taxes and how that leads to our overall tax rate.