A

The

S

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, S

Hence, R, the holding period return, is: R = [S

Using the formula for the continuously compounded rate of return gives: ln(1+R) = ln(S

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%.

However, ln(1+R) = ln[(1 + (-0.2)] = ln(0.8) = -0.223 or -22.3%, which is exactly the negative of the original return. Averaging 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 of the rate of return over a period.

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. |