Continuously Compounded Rate of Return Given Holding Period Return

Continuous compounding applies either when the frequency with at we calculate interest is infinitely large or the time interval is infinitely small. Put quite simply, under continuous compounding, time is viewed as continuous. This is a departure from discrete compounding, where we deal with finite time intervals.

We have previously seen how discrete compounding works, given a finite compounding period such as a month or a year. It is important to recall that the effective annual return increases concurrently with the compounding frequency under discrete compounding.

Example: Discrete Compounding #1

For a stated rate of 20%, semiannual compounding gives an effective rate of:

$$ \left(1 + \frac {0.20}{2} \right)^2 – 1 = 21\% $$

And monthly compounding gives an effective rate of:

$$ \left(1 + \frac {0.20}{12} \right)^{12} – 1 = 21.94\% $$

Daily or hourly compounding will produce even larger effective rates.

We can calculate the effective annual rate based on continuous compounding if we are given a stated annual rate of \(R_{cc}\). The formula used is:

$$ \text{Effective annual rate} = \text e^{R_{cc}} – 1 $$

Example: Continuous Compounding #2

Given a stated rate of 10%, the effective rate based on continuous compounding is closest to:

Solution

Applying the formula above,

$$ \text{Effective rate} = e^{0.10} – 1 = 10.52\% $$

Continuous Compounding Given the Holding Period Return (HPR)

We can calculate the continuous compound rate of return if we have the holding period return. The following is the formula used in the calculation:

$$ \text{Continuous rate} = \text{ln}(1 + \text{HPR}) = \text{ln} \left(\cfrac {S_1}{S_0} \right) $$

Where:

\(S_1\) = Value at the end of the period.

\(S_0\) = Value at the beginning of the period.

Example: Continuous Compounding Given the Beginning and Ending Values

An investor purchases a stock at $1,000 and sells it for $1,080 after a period of one year. The annual rate of return on the stock on a continuously compounded basis is closest to:

Solution

$$ \text{Continuously compounded rate} = \text{ln} \left( \cfrac {1,080}{1,000} \right) = 7.7\% $$

Example: Continuous Compounding Given a Holding Period Return (HPR)

A stock has a holding period return of 20%. Its continuously compounded rate of return is closest to:

Solution

$$ R_{cc} = \text{ln}(1 + 0.20) = 18.2\% $$

Note to candidates: We can also calculate the holding period return given the continuously compounded rate, \(R_{cc}\). In general, to determine the HPR after \(t\) years:

$$ \text{HPR}_{\text t} = e^{\text{Rcc} *{\text t}} – 1 $$

Question

A portfolio manager buys a stock at $50 and sells it for $56 after a year. The continuously compounded rate of return is closest to:

A. 10.0%.

B. 11.3%.

C. 12.0%.

Solution

The correct answer is B.

$$ R_{cc} = \text{ln} \left(\cfrac {56}{50} \right) = 11.3\% $$