Discounted Payback Period Calculator

1. What is the Discounted Payback Period Calculator?

Definition: The Discounted Payback Period Calculator computes the time required for a project to recover its initial investment in present value terms, assuming uniform annual cash inflows and using a logarithmic formula to account for the time value of money.

Purpose: It helps investors and managers assess the risk and profitability of a project by determining how quickly the investment can be recovered, considering a discount rate and constant cash inflows.

2. How Does the Calculator Work?

The calculator uses the following formula, as shown in the image above:

$ \text{DPP} = \frac{-\ln(1 - I \times R / C)}{\ln(1 + R)} $

Where:

$ \text{DPP} $: Discounted Payback Period (in years);
$ I $: Total Sum Invested (initial investment);
$ R $: Discount Rate (as a decimal);
$ C $: Annual Cash Inflow (uniform yearly cash flow).

Steps:

Enter the total sum invested ($ I $), discount rate ($ R $, as a percentage), and annual cash inflow ($ C $).
Convert the discount rate from percentage to decimal form.
Calculate the DPP using the formula above, ensuring the logarithm arguments are valid.
Display the result, formatted with 4 decimal places or in scientific notation if less than 0.001.

3. Importance of Discounted Payback Period Calculation

Calculating the discounted payback period is essential for:

Risk Assessment: It measures how quickly an investment can be recovered in present value terms, highlighting the project’s risk exposure.
Investment Decisions: Helps compare projects by considering the time value of money, favoring projects with shorter payback periods.
Financial Planning: Assists in budgeting by estimating the recovery timeline for investments with steady cash inflows.

4. Using the Calculator

Example 1: Calculate the discounted payback period for a project with an initial investment of $10,000, a discount rate of 10%, and annual cash inflows of $3,000:

Total Sum Invested ($ I $): $10,000;
Discount Rate ($ R $): 10% (0.10);
Annual Cash Inflow ($ C $): $3,000;
Check condition: $ I \times R / C = 10000 \times 0.10 / 3000 = 0.3333 < 1 $;
Argument: $ 1 - I \times R / C = 1 - 0.3333 = 0.6667 $;
DPP: $ \frac{-\ln(0.6667)}{\ln(1 + 0.10)} = \frac{-\ln(0.6667)}{\ln(1.10)} = \frac{-(-0.4055)}{0.0953} = 4.2550 $ years.

Example 2: Calculate the discounted payback period for a project with an initial investment of $20,000, a discount rate of 5%, and annual cash inflows of $6,000:

Total Sum Invested ($ I $): $20,000;
Discount Rate ($ R $): 5% (0.05);
Annual Cash Inflow ($ C $): $6,000;
Check condition: $ I \times R / C = 20000 \times 0.05 / 6000 = 0.1667 < 1 $;
Argument: $ 1 - I \times R / C = 1 - 0.1667 = 0.8333 $;
DPP: $ \frac{-\ln(0.8333)}{\ln(1 + 0.05)} = \frac{-\ln(0.8333)}{\ln(1.05)} = \frac{-(-0.1823)}{0.0488} = 3.7377 $ years.

5. Frequently Asked Questions (FAQ)

Q: What does the discounted payback period tell us?
A: It indicates the time needed to recover the initial investment in present value terms, accounting for the time value of money, assuming uniform cash inflows.

Q: Why does the formula require $ I \times R / C < 1 $?
A: This condition ensures the project can recover its investment. If $ I \times R / C \geq 1 $, the discounted cash inflows may never cover the initial investment, making the DPP undefined or infinite.

Q: How does this DPP calculation differ from other methods?
A: This method assumes constant annual cash inflows and uses a logarithmic formula for continuous discounting, while other methods may calculate DPP by summing discounted cash flows year by year for variable cash flows.