Numericals

Discrete vs. Continuously Compounded Returns

Here are a few problems illustrating the concepts of discrete and continuously compounded returns.

Problem 1: Basic Calculation

A stock's price changes from $50 to $52 in one day.

(a) Calculate the discrete return. (b) Calculate the continuously compounded return.

Solution:

(a) Discrete Return:

R = (P_t - P_{t-1}) / P_{t-1} = (52 - 50) / 50 = 2 / 50 = 0.04

The discrete return is 4%.

(b) Continuously Compounded Return:

r = ln(P_t / P_{t-1}) = ln(52 / 50) = ln(1.04) ≈ 0.0392

The continuously compounded return is approximately 3.92%.

Problem 2: Multi-Period Returns

An asset has the following prices over three days:

• Day 0: $100
• Day 1: $105
• Day 2: $112

(a) Calculate the discrete return for Day 1 and Day 2. (b) Calculate the continuously compounded return for Day 1 and Day 2. (c) Calculate the total discrete return over the two days (from Day 0 to Day 2). (d) Calculate the total continuously compounded return over the two days (from Day 0 to Day 2) by summing the individual daily returns. (e) Calculate the total continuously compounded return directly from Day 0 to Day 2 using the formula.

Solution:

(a) Discrete Returns:

• Day 1: R_1 = (105 - 100) / 100 = 0.05 = 5%
• Day 2: R_2 = (112 - 105) / 105 ≈ 0.0667 = 6.67%

(b) Continuously Compounded Returns:

• Day 1: r_1 = ln(105 / 100) = ln(1.05) ≈ 0.0488 = 4.88%
• Day 2: r_2 = ln(112 / 105) = ln(1.0667) ≈ 0.0645 = 6.45%

(c) Total Discrete Return:

1 + R_{0,2} = (1 + R_1) * (1 + R_2) = (1 + 0.05) * (1 + 0.0667) = 1.05 * 1.0667 ≈ 1.12 R_{0,2} = 1.12 - 1 = 0.12

The total discrete return is approximately 12%. Alternatively: R_{0,2} = (112-100)/100 = 12/100 = 0.12

(d) Total Continuously Compounded Return (Sum of Daily Returns):

r_{0,2} = r_1 + r_2 = 0.0488 + 0.0645 = 0.1133

The total continuously compounded return is approximately 11.33%.

(e) Total Continuously Compounded Return (Direct Calculation):

r_{0,2} = ln(112 / 100) = ln(1.12) ≈ 0.1133

The total continuously compounded return is approximately 11.33%. Notice how the results in (d) and (e) are the same, illustrating the additivity property.

Problem 3: Comparing Returns

An investment of $100 grows to $150 over a year.

(a) What is the discrete annual return? (b) What is the continuously compounded annual return? (c) If you reinvested the $150 and it grew to $225 the following year, what is the total continuously compounded return over the two years?

Solution:

(a) Discrete Annual Return:

R = (150 - 100) / 100 = 0.50

The discrete annual return is 50%.

(b) Continuously Compounded Annual Return:

r = ln(150 / 100) = ln(1.5) ≈ 0.4055

The continuously compounded annual return is approximately 40.55%.

(c) Total Continuously Compounded Return: Year 2 Price = 225 Continuous Return Year 2 = ln(225/150) = ln(1.5) = 0.4055 Total Continuous Return = 0.4055 + 0.4055 = 0.811

Notes on these Problems:

• Always pay attention to the time period (daily, monthly, annually).
• Be careful when calculating multi-period discrete returns; remember to compound the (1 + return) values.
• Continuously compounded returns are generally smaller in magnitude than discrete returns for the same price change, but the difference becomes smaller as the returns approach zero.