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five financial equations cheatsheet

Sometimes I read something useful that is worth noting down for myself as a reminder to get back on track.

1. Compound Growth #

$$FV = PV(1+r)^n$$

Purpose: Calculates what money invested today will become after compounding over time. Both time and the rate of return affect how quickly the money grows.

Layman terms: Your money earns money, and then the new money also starts earning money. It is like a snowball rolling downhill: the longer it rolls, the larger it becomes.

Example:

You invest $1,000 and earn 8% per year for 10 years:

$$FV = 1{,}000(1+0.08)^{10}$$

$$FV \approx \$2{,}159$$

Your original $1,000 grows to about $2,159 without adding any more money.

2. Rule of 72 #

$$\text{Doubling time in years} \approx \frac{72}{r_{\%}}$$

Purpose: Quickly estimates how long an investment, debt balance, price level, or other growing quantity will take to double at a given annual percentage rate.

Layman terms: Divide 72 by the yearly growth rate to estimate how many years it will take for something to become twice as large.

Example:

At an 8% annual return:

$$\frac{72}{8}=9$$

Your money will take approximately nine years to double.

A $10,000 investment earning roughly 8% per year would therefore grow to about $20,000 in nine years.

The Rule of 72 is a quick estimate, not an exact calculation.

3. Present Value #

$$PV = \frac{FV}{(1+r)^n}$$

Purpose: Converts a future amount of money into its equivalent value today. It is the compound-growth equation run backwards.

Layman terms: Money received in the future is worth less than money received today because money available today can be invested and allowed to grow.

Example:

Someone offers to pay you $10,000 in five years. Assuming you could earn 8% per year elsewhere:

$$PV = \frac{10{,}000}{(1+0.08)^5}$$

$$PV \approx \$6{,}806$$

Receiving $10,000 in five years is financially equivalent to receiving approximately $6,806 today, assuming an 8% annual return.

4. Geometric Mean Return #

$$r_g = \left[ \prod_{i=1}^{n}(1+r_i) \right]^{1/n}-1$$

Purpose: Calculates the average compounded return across multiple periods. Unlike a simple arithmetic average, it correctly accounts for compounding, volatility, and investment losses.

Layman terms: This tells you the steady yearly return that would have produced the same final result as the actual sequence of gains and losses.

Example:

Suppose an investment:

• Gains 50% in year one
• Loses 50% in year two

The simple arithmetic average is:

$$\frac{50\%+(-50\%)}{2}=0\%$$

That result is misleading. Starting with $100:

$$\$100 \times 1.50 = \$150$$

Then:

$$\$150 \times 0.50 = \$75$$

You finished with $75, meaning you lost 25% overall.

The geometric mean return is:

$$r_g = \left[(1.50)(0.50)\right]^{1/2}-1$$

$$r_g \approx -13.4\%$$

The investment performed as though it had lost approximately 13.4% per year for two years.

Spreadsheet form:

=GEOMEAN(1 + returns) - 1

Multiply all period growth factors, take the $n$-th root, and then subtract $1$.

5. Real Return #

$$r_{\text{real}} \approx r_{\text{nominal}}-\pi$$

Purpose: Estimates how much your purchasing power increased after accounting for inflation.

Layman terms: Your account balance may be increasing, but rising prices reduce what that money can actually buy. Real return measures whether you are genuinely becoming wealthier after inflation.

Example:

Your investment earns 10%, while inflation is 3%:

$$r_{\text{real}} \approx 10\%-3\%=7\%$$

Your account grew by 10%, but your purchasing power increased by only approximately 7%.

For example, if you began with $10,000:

• The investment grows to $11,000.
• Goods that previously cost $10,000 now cost approximately $10,300.
• Your purchasing power increased by roughly $700, or 7%.

This subtraction formula is an approximation. The exact real-return formula is:

$$r_{\text{real}} = \frac{1+r_{\text{nominal}}}{1+\pi}-1$$

Using a 10% nominal return and 3% inflation:

$$r_{\text{real}} = \frac{1.10}{1.03}-1 \approx 6.8\%$$

Symbols #

From @lumenxbt