# 1. Introduction This problem deals with the concepts of **discount rate** and **effective interest rate**, particularly focusing on a bill purchased at a discount before its maturity. The **nominal rate of discount** refers to the interest rate calculated using the face value of a bill, rather than the amount actually invested, and is often quoted on a convertible basis (e.g., bimonthly). The **effective rate of interest** refers to the actual rate earned over a year, accounting for the effect of compounding. When an investor purchases a bill at a price lower than its face value, the difference represents the **discount** earned. The calculation of the nominal rate of discount and the effective rate of interest involves understanding the relationship between the present value, the face value, time, and the compounding frequency. **Explanation:** Understanding the above concepts is crucial to determine both the nominal rate of discount and the effective annual interest rate. The nominal rate helps standardize interest earned over a period, while the effective rate provides a true measure of earnings accounting for compounding, which is essential for comparing investment alternatives. --- # 2. Presentation of Relevant Formulas Required To Solve The Question ### **a) Nominal Rate of Discount Convertible Bimonthly (\(d^{(6)}\)):** The nominal rate of discount \(d^{(m)}\) convertible \(m\) times per year is defined as: \[ d^{(m)} = m \left[1 - \left(\frac{P}{F}\right)^{1/n}\right] \] where: - \(P\) = Purchase price (present value) - \(F\) = Face value (future value) - \(n\) = Number of periods in the investment (relative to the compounding period) - \(m\) = Number of compounding periods per year ### **b) Annual Effective Rate of Interest (\(i_{eff}\)):** The annual effective rate of interest is calculated as: \[ i_{eff} = \left( \frac{F}{P} \right)^{k} - 1 \] where: - \(k\) = Number of investment periods per year **Explanation:** The nominal rate of discount formula relates the periodic discount earned to the face value, allowing conversion to standard compounding intervals. The effective annual interest formula captures the total return on an investment over a year, factoring in compounding, thus providing a better basis for comparing investments. --- # 3. Detailed Step-by-Step Solution ## **Given Data:** - Face value of bill, \(F = \$200\) - Purchase price, \(P = \$193\) - Time before maturity = 3 months - Compounding period for bimonthly = 2 months per period (\(m = 6\) per year) --- ### **Step 1: Find the Discount Rate per Period** The time period between purchase and maturity is 3 months. Each bimonthly period is 2 months. Number of bimonthly periods in 3 months: \[ \text{Number of periods} = \frac{3}{2} = 1.5 \] **Explanation:** The time from purchase to maturity must be expressed in terms of the compounding period to apply the discount rate formula correctly. --- ### **Step 2: Find the Discount Rate per Bimonthly Period** The formula for the present value at a discount rate \(d\) for \(n\) periods is: \[ P = F \cdot (1 - d)^n \] Solving for \(d\): \[ \frac{P}{F} = (1 - d)^n \implies 1 - d = \left(\frac{P}{F}\right)^{1/n} \implies d = 1 - \left(\frac{P}{F}\right)^{1/n} \] Plug in the values: - \(P = 193\) - \(F = 200\) - \(n = 1.5\) \[ d = 1 - \left(\frac{193}{200}\right)^{1/1.5} \] Calculate: \[ \frac{193}{200} = 0.965 \] \[ 0.965^{1/1.5} = 0.965^{0.6667} \] Calculate the exponent: \[ \ln(0.965) = -0.0357 \] \[ -0.0357 \times 0.6667 = -0.0238 \] \[ e^{-0.0238} = 0.9765 \] So, \[ d = 1 - 0.9765 = 0.0235 \] **Explanation:** This step finds the periodic discount rate, which is required to determine the nominal rate convertible bimonthly. --- ### **Step 3: Find the Nominal Rate of Discount Convertible Bimonthly** The nominal rate is: \[ d^{(6)} = 6 \times d = 6 \times 0.0235 = 0.1410 = 14.10\% \] **Explanation:** The nominal rate is annualized by multiplying the periodic rate by the number of bimonthly periods in a year, providing a standardized comparison rate. --- ### **Step 4: Find the Annual Effective Rate of Interest** The effective annual rate: \[ i_{eff} = \left(\frac{F}{P}\right)^{k} - 1 \] Here, \(k = \frac{12}{3} = 4\) (since four periods of 3 months in a year). \[ i_{eff} = \left(\frac{200}{193}\right)^4 - 1 \] \[ \frac{200}{193} = 1.03627 \] Calculate \(1.03627^4\): \[ 1.03627^4 = (1.03627 \times 1.03627 \times 1.03627 \times 1.03627) \approx 1.1531 \] \[ i_{eff} = 1.1531 - 1 = 0.1531 = 15.31\% \] **Explanation:** The effective annual interest rate incorporates compounding, providing a true measure of the annual return on the investment, which is essential for comparing to other rates. --- # Conclusion The nominal rate of discount convertible bimonthly earned by the purchaser is **14.10%**, and the annual effective rate of interest earned by the purchaser is **15.31%**. These results present a complete assessment of the rates applicable to this discounted bill transaction, facilitating accurate comparisons and financial decision-making. --- ## **Summary Table** | Quantity | Value | |------------------------------------------------|------------| | Nominal rate of discount (bimonthly, \(d^{(6)}\)) | 14.10% | | Annual effective rate of interest (\(i_{eff}\)) | 15.31% |