# 11. Discretely and Continuously Compounded Rates of Return

## p. distinguish between discretely and continuously compounded rates of return and calculate and interpret a continuously compounded rate of return, given a specific holding period return;

What is a discretely compounded rate of return? A discretely compounded rate of return measures the rate of changes in the value of an asset over a period under the assumption that the number of compounding periods is countable. Most standard deposit and loan instruments are compounded at discrete and evenly spaced periods, such as annually or monthly.

How would you calculate the *quarterly* rate of return for a stock that compounds every quarter and has a yearly RoR = 50%?
\((1 - 0.5)^{1/4} - 1 = 10.67%\)

What is a continuously compounded rate of return?
- The continuously compounded rate of return measures the rate of change in the value of an asset associated with a holding period under the assumption of continuously compounding. It is the natural logarithm of 1 plus the holding period return, or equivalently, the natural logarithm of the ending price over the beginning price. From t to t + 1:
- \(r_{t,t+1} = ln \frac{S_{t+1}}{S_t} = ln(1 + R_{t,t+1})\)
- `S`

= stock price
- `R_{t,t+1}`

= the rate of return from `t`

to `t+1`

##### Source:

CFA

##### Graph:

- 113.021.02 Logarithms - e to 115.020.40.11 Reading 9 - 11. Discretely and Continuously Compounded Rates of Return
- 115.020.40 Reading 9 - Common Probability Distributions to 115.020.40.11 Reading 9 - 11. Discretely and Continuously Compounded Rates of Return
- 134.01.02 Newton - Law of acceleration to 115.020.40.11 Reading 9 - 11. Discretely and Continuously Compounded Rates of Return