Introduction
In financial mathematics, the concept of discounting is used to find the present value of future money. When the discounting is done continuously, it means that the money loses its value every instant in time rather than at specific intervals. This question is asking for the effective discount rate when the nominal rate of discount is 10% compounded continuously. Let us understand what this means and how to calculate it.
Understanding Discount Rates
Nominal Rate of Discount (d): This is the stated rate at which money is discounted over a period, not accounting for compounding.
Effective Discount Rate (deff): This is the actual discount rate which, when applied once, gives the same effect as the nominal rate applied continuously over the period.
Formula for Effective Discount Rate (Continuous Compounding)
When the nominal discount rate is compounded continuously, the effective discount rate can be calculated using the formula:
deff = 1 – e-d
Where:
- d = nominal rate of discount (as a decimal)
- e = Euler’s number, approximately equal to 2.71828
Given Data
- Nominal rate of discount = 10% = 0.10
Substitute in Formula
deff = 1 – e-0.10
Step-by-step Calculation
We need to calculate e-0.10:
e-0.10 ≈ 2.71828-0.10 ≈ 0.904837
deff = 1 – 0.904837 = 0.095163
Convert to Percentage
Effective Discount Rate ≈ 9.5163%
Conclusion
When a nominal discount rate of 10% is compounded continuously, the actual or effective discount rate becomes approximately:
Effective Discount Rate = 9.52% (rounded to two decimal places)
This concept is useful in finance, banking, and investment analysis. It helps compare different discounting scenarios fairly and shows how compounding affects the real value of a financial transaction over time. Continuous compounding leads to a lower effective discount rate compared to nominal values because the discount is applied at every moment, reducing the present value less aggressively than periodic discounting.