Finance

Formula Summary

1. Total Dividend

Total Dividend = Dividend Per Share × Number of Shares

Where:

• Dividend Per Share = dividend paid for one share
• Number of Shares = number of shares owned

2. Dividend Yield

Dividend Yield = D / pB

Where:

• D = Dividend Per Share
• pB = Beginning Price Per Share

3. Capital Gain or Loss

Capital Gain/Loss = (pE - pB) / pB

Where:

• pB = Beginning Price
• pE = Ending Price

If pE > pB → Capital Gain

If pE < pB → Capital Loss

4. Simple Return

R = ((pE - pB) + D) / pB

Where:

• R = Simple Return
• pB = Beginning Price
• pE = Ending Price
• D = Dividend Per Share

Another way to write it:

Simple Return = Capital Gain/Loss + Dividend Yield

5. Convert Simple Return to Continuously Compounded Return

r = ln(1 + R)

Where:

• r = Continuously Compounded Return
• R = Simple Return
• ln = Natural Logarithm

Easy memory:

Simple → ln → Continuous

6. Convert Continuously Compounded Return to Simple Return

R = e^r - 1

Where:

• R = Simple Return
• r = Continuously Compounded Return
• e ≈ 2.71828

Easy memory:

Continuous → e → Simple

7. Multiperiod Simple Return

For several periods:

R(T) = (1 + R1) × (1 + R2) × … × (1 + RT) - 1

Where:

• R(T) = Total Simple Return over T periods
• R1 = Return in Period 1
• R2 = Return in Period 2
• RT = Return in the final period
• T = Number of periods

Easy memory:

Simple Returns = MULTIPLY across periods

8. Multiperiod Continuously Compounded Return

r(T) = r1 + r2 + … + rT

Where:

• r(T) = Total Continuously Compounded Return over T periods
• r1, r2, … rT = Continuously compounded returns for each period

Easy memory:

Continuously Compounded Returns = ADD across periods

9. Convert Multiperiod Simple Return to Multiperiod Continuously Compounded Return

r(T) = ln(1 + R(T))

10. Convert Multiperiod Continuously Compounded Return to Multiperiod Simple Return

R(T) = e^r(T) - 1

11. Terminal Capital Using Individual Simple Returns

CT = C0 × (1 + R1) × (1 + R2) × … × (1 + RT)

Where:

• C0 = Initial Capital / Starting Money
• CT = Terminal Capital / Ending Money

12. Terminal Capital Using Total Simple Return

Because:

(1 + R1) × (1 + R2) × … × (1 + RT) = 1 + R(T)

we can shorten the formula to:

CT = C0 × (1 + R(T))

So these two formulas mean the same thing:

CT = C0 × (1 + R1) × (1 + R2) × … × (1 + RT)

and

CT = C0 × (1 + R(T))

13. Terminal Capital Using Continuously Compounded Returns

Using all individual continuously compounded returns:

CT = C0 × e^(r1 + r2 + … + rT)

Because:

r(T) = r1 + r2 + … + rT

we can shorten it to:

CT = C0 × e^r(T)

14. Capital When the Same Return Happens Every Period

If the same return R happens every period:

CT = C0 × (1 + R)^T

Where:

• C0 = Starting Capital
• CT = Ending Capital
• R = Return per period
• T = Number of periods

Example with a -9.7% return for 4 years:

CT = $100 × (1 - 0.097)^4

15. Ending Value Using One Continuously Compounded Return

Ending Value = Beginning Value × e^r

Or:

CT = C0 × e^r

If there are T periods and r(T) is the total continuously compounded return:

CT = C0 × e^r(T)

Arithmetic Mean Return

16. Arithmetic Mean Return

AM = (R1 + R2 + … + RT) / T

Where:

• AM = Arithmetic Mean Return
• R1, R2, … RT = Individual Period Returns
• T = Number of Returns

Easy meaning:

Arithmetic Mean = Add all returns ÷ Number of returns

Arithmetic mean answers:

“What was the average of the individual returns?”

Geometric Mean Return

17. Geometric Mean Return

GM = [(1 + R1) × (1 + R2) × … × (1 + RT)]^(1/T) - 1

Where:

• GM = Geometric Mean Return
• R1, R2, … RT = Individual Period Returns
• T = Number of Periods
• 1/T = Take the T-th root

Easy meaning:

Geometric Mean = the constant return per period that would produce the same actual ending capital

Geometric mean answers:

“What constant return every period would give me the same final amount?”

18. Geometric Mean Using Starting and Ending Capital

The same idea can also be written as:

GM = (CT / C0)^(1/T) - 1

Where:

• C0 = Starting Capital
• CT = Ending Capital
• T = Number of Periods

For example, if:

C0 = $100

CT = $66.50

T = 4

Then:

GM = ($66.50 / $100)^(1/4) - 1

GM = 0.665^(1/4) - 1

GM ≈ -0.097 = -9.7%

If Dividends Are Taken Out Instead of Reinvested

19. Total Ending Wealth

If dividends are withdrawn and kept separately:

Total Ending Wealth = Ending Value of Investment + Dividends Taken Out

20. Total Simple Return With Withdrawn Dividends

Total Simple Return = (Total Ending Wealth - Beginning Investment) / Beginning Investment

This allows you to count the dividends as part of your total wealth even though they were not reinvested.

Percentage and Decimal Conversion

21. Percentage to Decimal

Decimal = Percentage / 100

Examples:

29.5% = 0.295

20% = 0.20

5% = 0.05

-9.7% = -0.097

22. Decimal to Percentage

Percentage = Decimal × 100

Examples:

0.295 × 100 = 29.5%

0.20 × 100 = 20%

-0.097 × 100 = -9.7%

Master Notes

• R = Simple Return
• r = Continuously Compounded Return
• R(T) = Total Simple Return over T periods
• r(T) = Total Continuously Compounded Return over T periods
• AM = Arithmetic Mean Return
• GM = Geometric Mean Return
• C0 = Initial Capital / Starting Money
• CT = Terminal Capital / Ending Money
• pB = Beginning Price
• pE = Ending Price
• D = Dividend Per Share
• T = Number of Periods
• ln = Natural Logarithm
• e ≈ 2.71828

Simplest Memory Rules

Simple Return:

R = ((pE - pB) + D) / pB

Simple → Continuous:

r = ln(1 + R)

Continuous → Simple:

R = e^r - 1

Multiple Simple Returns:

MULTIPLY

Multiple Continuously Compounded Returns:

ADD

Arithmetic Mean:

ADD returns, then DIVIDE by number of returns

Geometric Mean:

MULTIPLY growth factors, take the T-th root, then subtract 1

Arithmetic Mean = Average Return

Geometric Mean = Constant Compounded Growth Rate

Terminal Capital with Simple Return:

CT = C0 × (1 + R(T))

Terminal Capital with Continuously Compounded Return:

CT = C0 × e^r(T)