### Continuous Compounding

Continuous compounding applies when either the frequency with which 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 differs 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 under discrete compounding, the effective annual return increases as the frequency of compounding increases.

### Example 1: Discrete Compounding

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 given a stated annual rate of Rcc. the formula used is:

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

### Example 2: Continuous Compounding

Given a stated rate of 10%, calculate the effective rate based on continuous compounding.

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 formula used is:

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

Where S1 = end of period value and S0 is the value at the beginning of the period

### Example 3: Continuous compounding given the beginning and ending values

An investor purchases a stock for $1000 and sells it for$1080 after a period of one year. Compute the annual rate of return on the stock on a continuously compounded basis.

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

### Example 4: Continuous compounding given the HPR

A stock has a holding period return of 20%. Calculate its continuously compounded rate of return.

$$\text{Continuously compounded rate} = ln(1 + 0.20) = 18.2\%$$

Note that we can also calculate the holding period return given the continuously compounded rate, Rcc. 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 for $50 and sells it for$56 after a year. The continuously compounded rate of return is closest to:

A. 11.3%

B. 10%

C. 12%

Solution

$$\text{The continuously compounded rate of return} = ln \left(\cfrac {56}{50} \right) = 11.3\%$$

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.

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