capital gains tax and inflation
symbols
- \(\phi_t\)
- ratio to which the presence of inflation results in over-taxation of capital gains
- \(n\)
- number of periods
- \(r_r\)
- real return
- \(r\)
- nominal return
- \(\pi\)
- inflation
idea
\(\phi_t = \text{nominal gains} / \text{real gains}\)
\(\phi_t = \frac{\prod\limits_{}^{n} (1 + r)}{\prod\limits_{}^{n} (1 + r_r)} - 1\)
we will show is equal to:
\(\phi_t = \frac{(1 + \pi)^n(1 + r)^n - (1 + \pi)^n}{(1 + r)^n - (1 + \pi)^n}\)
derivation
nominal gains
\(r\)
nominal gains over n periods
\(\prod\limits_{}^{n} r = (1 + r)^n - 1\)
real gains
frequently approximated as \(r_r = r - \pi\)
we will be more precise with \(r = (1 + r_r)(1 + \pi) - 1\)
\((1 + r_r)(1 + \pi) = (1 + r)\)
\((1 + r_r) = \frac{1 + r}{1 + \pi}\)
\(r_r = \frac{1 + r - 1 - \pi}{1 + \pi}\)
\(r_r = \frac{r - \pi}{1 + \pi}\)
real gains after n periods
\(\prod\limits_{}^{n} r_r = (1 + r_r)^n - 1\)
\(\prod\limits_{}^{n} r_r = (1 + \frac{r - \pi}{1 + \pi})^n - 1\)
\(\prod\limits_{}^{n} r_r = (\frac{1 + \pi + r - \pi}{1 + \pi})^n - 1\)
\(\prod\limits_{}^{n} r_r = (\frac{1 + r}{1 + \pi})^n - 1\)
bring it together
\(\phi_t = \frac{\prod\limits_{}^{n} r}{\prod\limits_{}^{n} r_r}\)
\(\phi_t = \frac{(1 + r)^n - 1}{(\frac{1 + r}{1 + \pi})^n - 1}\)
\(\phi_t = \frac{(1 + r)^n - 1}{\frac{(1 + r)^n}{(1 + \pi)^n} - \frac{(1 + \pi)^n}{(1 + \pi)^n}}\)
\(\phi_t = \frac{(1 + r)^n - 1}{\frac{(1 + r)^n - (1 + \pi)^n}{(1 + \pi)^n}}\)
\(\phi_t = \frac{(1 + \pi)^n((1 + r)^n - 1)}{(1 + r)^n - (1 + \pi)^n}\)
\(\phi_t = \frac{(1 + \pi)^n(1 + r)^n - (1 + \pi)^n}{(1 + r)^n - (1 + \pi)^n}\)
π == 0 as sanity check
\(\phi_t = \frac{(1 + \pi)^n(1 + r)^n - (1 + \pi)^n}{(1 + r)^n - (1 + \pi)^n}\)
\(\phi_t = \frac{(1 + 0)^n(1 + r)^n - (1 + 0)^n}{(1 + r)^n - (1 + 0)^n}\)
\(\phi_t = \frac{(1 + r)^n - 1}{(1 + r)^n - 1}\)
\(\phi_t = 1\)
some sample tables
Φt for n=1
| nominal return | 0.00 | 0.01 | 0.02 | 0.04 | 0.08 | 0.16 | 0.32 |
|---|---|---|---|---|---|---|---|
| inflation | |||||||
| 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |
| 0.02 | 0.00 | -1.02 | 2.04 | 1.36 | 1.17 | 1.09 | |
| 0.04 | 0.00 | -0.35 | -1.04 | 2.08 | 1.39 | 1.19 | |
| 0.06 | 0.00 | -0.21 | -0.53 | -2.12 | 4.24 | 1.70 | 1.30 |
| 0.08 | 0.00 | -0.15 | -0.36 | -1.08 | 2.16 | 1.44 | |
| 0.10 | 0.00 | -0.12 | -0.28 | -0.73 | -4.40 | 2.93 | 1.60 |
Φt for n=2
| nominal return | 0.00 | 0.01 | 0.02 | 0.04 | 0.08 | 0.16 | 0.32 |
|---|---|---|---|---|---|---|---|
| inflation | |||||||
| 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |
| 0.02 | 0.00 | -1.03 | 2.06 | 1.37 | 1.18 | 1.10 | |
| 0.04 | 0.00 | -0.35 | -1.06 | 2.12 | 1.42 | 1.22 | |
| 0.06 | 0.00 | -0.22 | -0.55 | -2.18 | 4.37 | 1.75 | 1.35 |
| 0.08 | 0.00 | -0.16 | -0.37 | -1.12 | 2.25 | 1.50 | |
| 0.10 | 0.00 | -0.13 | -0.29 | -0.77 | -4.62 | 3.08 | 1.69 |
Φt for n=5
| nominal return | 0.00 | 0.01 | 0.02 | 0.04 | 0.08 | 0.16 | 0.32 |
|---|---|---|---|---|---|---|---|
| inflation | |||||||
| 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |
| 0.02 | 0.00 | -1.06 | 2.12 | 1.42 | 1.22 | 1.14 | |
| 0.04 | 0.00 | -0.37 | -1.12 | 2.26 | 1.51 | 1.31 | |
| 0.06 | 0.00 | -0.24 | -0.59 | -2.38 | 4.79 | 1.93 | 1.51 |
| 0.08 | 0.00 | -0.18 | -0.42 | -1.26 | 2.56 | 1.74 | |
| 0.10 | 0.00 | -0.15 | -0.33 | -0.89 | -5.35 | 3.62 | 2.02 |
Φt for n=10
| nominal return | 0.00 | 0.01 | 0.02 | 0.04 | 0.08 | 0.16 | 0.32 |
|---|---|---|---|---|---|---|---|
| inflation | |||||||
| 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | |
| 0.02 | 0.00 | -1.12 | 2.24 | 1.50 | 1.30 | 1.24 | |
| 0.04 | 0.00 | -0.41 | -1.24 | 2.53 | 1.72 | 1.53 | |
| 0.06 | 0.00 | -0.27 | -0.69 | -2.77 | 5.64 | 2.33 | 1.89 |
| 0.08 | 0.00 | -0.21 | -0.50 | -1.53 | 3.27 | 2.34 | |
| 0.10 | 0.00 | -0.18 | -0.41 | -1.12 | -6.91 | 4.87 | 2.90 |
stray thoughts and implications
- Paying tax less frequently is preferable if your returns are outpacing inflation.
- Paying out interest (like in a high-yield savings account) forces n = 1, the least favorable tax treatment.
- Holes in the table are where return is equal to inflation. Some tax on 0 gain or loss yields infinite tax ratio.
- Nominal gains that do not exceed inflation result in φt < 0. A tax burden is incurred by real losses.
- Investments that do not significantly outperform inflation are heavily penalized by capital gains taxes in high-inflation environments.