Subject 11. Discretely and continuously compounded rates of return

CFA Exam: Dec. 2015 Level I > Study Session 3. Quantitative Methods: Application > Reading 9. Common Probability Distributions > LOS 9.p.

A discretely compounded rate of return measures the rate of changes in the value of 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. For example, suppose that the holding period return on a stock over a year is 50%. If the rate of return is compounded on a quarterly basis, the compounded quarterly rate of return on the stock is ( 1 + 0.5)1/4 - 1 = 10.67%.

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:

S: stock price. Rt, t + 1: the rate of return from t to t + 1.

Example 1

S0 = $30, S1 = $34.50. ==> Rt, t + 1 = $34.50/$30 - 1 = 0.15, and r0, 1 = 0.139762. The continuously compounded return is smaller than the associated holding period return.

Example 2

Assume that a particular stock has a price of $200 at the start of a period, and a price of $250 at the end of that period. That is, S0 = 200, and S1 = 250.

Hence, R, the holding period return, is: R = [S1/ S0] - 1 = [250/200] - 1 = 25%.

Using the formula for the continuously compounded rate of return gives: ln(1+R) = ln(S1/ S0) = ln(1.25) = 0.223 or 22.3%.

In order to see why the latter is preferable, consider the following:

Suppose that the stock now falls from $250 to $200. Then, R = [200/250] - 1 = -0.2 or -20%. So, effectively, the stock has returned to its original price, but combining the two rates of return and averaging them gives: [(0.25+ (-0.2)] / 2 = 0.05 / 2 = 0.025 or 2.5%, which is misleading, as in actual fact the stock has returned to its original price, and hence the return is effectively 0%.

But, ln(1+R) = ln[(1 + (-0.2)] = ln(0.8) = -0.223 or -22.3%, which is exactly the negative of the original return. So, averaging out these two rates gives: [(0.223 + (-0.223)] / 2 = 0 / 2 = 0 or 0%, which is the true rate of return for the period.

Thus, a continuously compounded return gives a more accurate account of the true picture, as far as the rate of return over a period is concerned.

User Contributed Comments 8 add a comment

surob: Perfect explanation why continuosly compounded return is better than HPY. Good job!!!
achu: That is a good explanation of why Cnts compouding is better. I wonder if it's used in the real world, though.
aakash1108: nice.
jpducros: Is Continuously coumpounding yield always < HPY ?

More generally, can we always write :
Cont Compound Y < HPY < MMY < BEY < EAY

And is BEY = BDiscountYield ?
So many Yields it becomes complicated to remember everything, and the logic behind.
pbielstein: You should keep in mind though that in this example the arithmetic average for the discretely compounded returns is taken.

If you take the geometric average then you obtain sqrt{(1-0.2)*(1+0.25)} - 1 = 0
johntan1979: Shouldn't the discretely compounded quarterly rate of return = (1 + HPR/4)^4 ? 1, not (1 + HPR)^1/4 - 1?
johntan1979: How can the quarterly compounding be less than the annual HPR?
fanDango: The quarterly compounding rate is not 4*X = annual HPR because you are compounding the principle and interest each quarter.