Part 1 - Measures of shares of return
29 September 2025 16:44
Arithmetic return
Suppose share j is priced at pj, t-1 per share at time t-1, and at pj,t per share at time t; and a dividend of dj,t per share is paid during period t. Then the
arithmetic return over a single period t is defined as:
- i.e. the return on share j at time t = closing price + dividend - opening share price then divided by opening price
- d is dividend in the period
- Arithmetic return will always be greater than or equal to -1 (cannot be less than -1)
○ In a worst case scenario if you were to buy a share is that you don’t get a dividend, your share is valued at 0 at the end of the period, meaning
you would have (0 + 0 - x)/x --> your capital gain collapses to 0 so arithmetic returns can't be below -1
The capital gain is the difference between what it is worth at the end less what it was worth at the beginning of the period
EXAMPLE
Holding period return
Suppose an investment in shares j is valued at vj,(t-N) at time t-N, and valued at vj,t at time t; with dividends paid during the intervening period reinvested in
the share
- Then the holding period return over N periods is:
- Similar to arithmetic returns, it cannot have a value less than -1 as the worst case scenario would mean the final value falls to 0 = (0 - x)/x = -1
EXAMPLE
- In this case, it is important to keep track of how many shares are held as you need to calculate what proportion of shares can be bought with the
dividend value
- For the first year, holding return should be the same as arithmetic return
- When working out holding period return vj,(t-N) is the price of the share at the start of the period
Given dividends are re-invested immediately on receipt in the same share,:
Whenever dealing with rate of return, always use the decimal representation
- Always write rates as a decimal then find the percentage
Annualised holding period return
Finance Page 1
, Suppose an investment in shares of j is valued vj,(t-N) at time t-N, and at vj,t at time t; with dividends paid during the intervening years reinvested in the share
- Then the annualised holding period return over each of these N years is:
- R prime from the period (t-N) to t = the nth root of (1 + Holding period return on a particular share) less 1
- Again, the annualised holding period return can't be less than -1
○ Worst case scenario is where the holding period return is -1 meaning (1+H) = 0 and R' = -1
- 1 year holding return is equal to arithmetic return which is also equal to the annualised holding period return for year 1
EXAMPLE
Logarithmic return
If share j is priced at pj,t-1 per share at time t-1, and at pj,t per share at time t; and a dividend of dj,t per share is paid during period t
- Then logarithmic return over a single period is:
EXAMPLE
Collated returns from the continued example:
- In penultimate column, cumulative returns are the sum of all logarithmic returns up to and including that period
Difference between arithmetic and logarithmic returns
Finance Page 2
29 September 2025 16:44
Arithmetic return
Suppose share j is priced at pj, t-1 per share at time t-1, and at pj,t per share at time t; and a dividend of dj,t per share is paid during period t. Then the
arithmetic return over a single period t is defined as:
- i.e. the return on share j at time t = closing price + dividend - opening share price then divided by opening price
- d is dividend in the period
- Arithmetic return will always be greater than or equal to -1 (cannot be less than -1)
○ In a worst case scenario if you were to buy a share is that you don’t get a dividend, your share is valued at 0 at the end of the period, meaning
you would have (0 + 0 - x)/x --> your capital gain collapses to 0 so arithmetic returns can't be below -1
The capital gain is the difference between what it is worth at the end less what it was worth at the beginning of the period
EXAMPLE
Holding period return
Suppose an investment in shares j is valued at vj,(t-N) at time t-N, and valued at vj,t at time t; with dividends paid during the intervening period reinvested in
the share
- Then the holding period return over N periods is:
- Similar to arithmetic returns, it cannot have a value less than -1 as the worst case scenario would mean the final value falls to 0 = (0 - x)/x = -1
EXAMPLE
- In this case, it is important to keep track of how many shares are held as you need to calculate what proportion of shares can be bought with the
dividend value
- For the first year, holding return should be the same as arithmetic return
- When working out holding period return vj,(t-N) is the price of the share at the start of the period
Given dividends are re-invested immediately on receipt in the same share,:
Whenever dealing with rate of return, always use the decimal representation
- Always write rates as a decimal then find the percentage
Annualised holding period return
Finance Page 1
, Suppose an investment in shares of j is valued vj,(t-N) at time t-N, and at vj,t at time t; with dividends paid during the intervening years reinvested in the share
- Then the annualised holding period return over each of these N years is:
- R prime from the period (t-N) to t = the nth root of (1 + Holding period return on a particular share) less 1
- Again, the annualised holding period return can't be less than -1
○ Worst case scenario is where the holding period return is -1 meaning (1+H) = 0 and R' = -1
- 1 year holding return is equal to arithmetic return which is also equal to the annualised holding period return for year 1
EXAMPLE
Logarithmic return
If share j is priced at pj,t-1 per share at time t-1, and at pj,t per share at time t; and a dividend of dj,t per share is paid during period t
- Then logarithmic return over a single period is:
EXAMPLE
Collated returns from the continued example:
- In penultimate column, cumulative returns are the sum of all logarithmic returns up to and including that period
Difference between arithmetic and logarithmic returns
Finance Page 2
