Discount Rates and Bill Discounting Principles

Core Variables and Equations for Discounting

When dealing with interest paid in advance, discount rates, or discounting, specific equations must be applied. These equations relate the discount rate, interest rate, and the three monetary values: discount amount ($D$), future value ($s$), and present value ($k$).

• Discount Rate for a Specific Period:     $\frac{d}{m} = \frac{D}{s}$

  • $d$ is the nominal discount rate.

  • $m$ is the number of periods per year.

  • If $m = 1$, the rate is for a full year.

• Effective Interest Rate per Period:     $\frac{i_m}{m} = \frac{D}{k}$

  • This represents the interest in a period over the loan amount.

  • In a standard interest scenario, the loan amount is the present value ($k$).

• Relationship Between Monetary Values:     $D = s - k$

  • Discount ($D$) or interest is the difference between the future value ($s$) and the present value ($k$).

• Conceptual Differences in Loan Amounts:

  • For normal interest, the loan amount is the present value ($k$).

  • For discount (prepaid interest), the loan amount is the future value ($s$).

Equivalent Rates and Conversion Strategies

Equivalent rates mean that starting with the same amount results in the same future value. When converting between interest rates and discount rates where no specific monetary values are given, an assumed loan amount (e.g., $100$) is used to facilitate calculations. The ratio between the calculated values remains constant regardless of the starting amount chosen.

Example: Finding Discount Rate ($d$) from Daily Interest ($i$)

• Given: Annual rate of interest $i = 15\%$, where $m = 1$.

• Objective: Find the annual rate of discount ($d$).

• Procedure:

  1. Assume a present value ($k$) of $100$.

  2. Calculate the interest amount: $15\%$ of $100 = 15$. Thus, $D = 15$.

  3. Calculate the future value ($s$): $s = k + D = 100 + 15 = 115$.

  4. Calculate the annual discount rate ($d$): $d = \frac{D}{s} = \frac{15}{115} = 0.13043$ (or $13.043\%$).

Example: Finding Interest Rate ($i$) from Discount Rate ($d$)

• Given: Annual rate of discount $d = 15\%$, where $m = 1$.

• Objective: Find the annual rate of interest ($i$).

• Procedure:

  1. Assume a future value ($s$) of $100$ (since the discount rate is given, $s$ is the natural loan amount reference).

  2. Calculate the discount amount: $15\%$ of $100 = 15$. Thus, $D = 15$.

  3. Calculate the present value ($k$): $k = s - D = 100 - 15 = 85$.

  4. Calculate the annual interest rate ($i$): $i = \frac{D}{k} = \frac{15}{85} = 0.17647$ (or $17.647\%$).

Example: Mismatched Time Units

• Given: Rate of discount of $4\%$ per quarter.

• Objective: Find the effective annual rate of interest ($i$ with $m = 1$).

• Rule: The information given (the quarterly rate) determines what can be calculated first. You must calculate the interest rate for the same period ($m = 4$) before converting it to an annual rate.

• Procedure:

  1. $m = 4$. The quarterly discount rate is $\frac{d}{4} = 4\%$.

  2. Assume $s = 100$. Therefore, $D = 4$ and $k = s - D = 96$.

  3. Calculate the quarterly interest rate: $\frac{i_4}{4} = \frac{4}{96} = 0.041666...$

  4. Convert to effective annual rate using the calculator's nominal-to-effective function:

     • Nominal rate ($i_4$) = $(\frac{4}{96}) \times 4 = 0.16666...$ (or $16.666\%$).

     • Input $16.666$ as Nominal, $4$ as periods per year, then solve for Effective ($17.36\%$).

Prepaid Rate of Interest Scenarios

Prepaid interest implies that the borrower pays interest immediately and receives the net amount (present value).

• Scenario: A borrower takes $R10,000$ for three months at a prepaid annual interest rate of $13\%$.

• Identification of Variables:

  • $s = R10,000$ (The amount to be paid back in the future).

  • $d = 13\%$ ($0.13$).

  • $m = 4$ (Three months corresponds to four periods per year).

• Calculations:

  1. Discount amount ($D$): $\frac{d}{m} \times s = \frac{0.13}{4} \times 10,000 = R325$.

  2. Amount received ($k$): $k = s - D = 10,000 - 325 = R9,675$.

  3. Effective quarterly interest rate: $\frac{i_4}{4} = \frac{D}{k} = \frac{325}{9,675} = 0.03359$ (or $3.359\%$).

  4. Effective annual interest rate: Convert the nominal rate ($i_4 = \frac{325}{9,675} \times 4 = 13.4367\%$) using $m = 4$ on a financial calculator to find the effective rate ($14.129\%$).

Discounting of Bills

A bill is a financial instrument indicating an amount to be paid on a specific future date (maturity date).

• Face Value ($s$): The amount shown on the bill to be paid at maturity.

• Discounted Value ($k$): The price paid for the bill today if sold before maturity.

• Discount ($D$): The difference between face value and discounted value.

Bill Discounted by Time Period

• Example 1: A bill with a face value of $R3,500$ is discounted for one month at a discount rate of $11\%$.

  • $s = 3,500$, $m = 12$, $d = 0.11$.

  • $D = \frac{0.11}{12} \times 3,500 = R32.08$.

  • $k = 3,500 - 32.08 = R3,467.92$.

  • Nominal interest rate ($i_{12}$): $(\frac{32.08}{3,467.92}) \times 12 = 11.10\%$.

Bill Discounted by Specific Dates

    If given dates (e.g., August 20 to October 15), use the calculator's Delta Days ($\Delta \text{DAYS}$) function to determine the duration. * For a duration of $56$ days: * $m = \frac{365}{56} \approx 6.51786$ * Finding Face Value ($s$) Given Discount ($D$): * Given $d = 13\%$ and $D = R53.60$. * Using $\frac{d}{m} = \frac{D}{s}$, then $\frac{0.13}{(\frac{365}{56})} = \frac{53.60}{s}$. * This can be rewritten as $0.13 \times (\frac{56}{365}) = \frac{53.60}{s}$. * $s = \frac{53.60 \times 365}{0.13 \times 56} = R2,689.18$.

Advanced Mathematical Solving for Two Unknowns

In scenarios where only the rate ($d$), the timing ($m$), and the discounted value ($k$) are known, the future value ($s$) and discount amount ($D$) must be found using simultaneous equations.

• Equations Used:

  1. $D = (\frac{d}{m}) \times s$

  2. $D = s - k$

• Example: $k = 2,635.54$, $d = 0.13$, $m = \frac{365}{56}$.

  • Equating the two: $(\frac{0.13}{\frac{365}{56}}) \times s = s - 2,635.54$

  • $0.019945 \times s = s - 2,635.54$

  • $2,635.54 = s - 0.019945s$

  • $2,635.54 = s(1 - 0.019945)$

  • $s = \frac{2,635.54}{0.980055} = R2,689.18$.

Conversion with Fractional Time Periods ($m$ as a Fraction)

Financial calculators typically error when a fractional value is entered for payments per year ($P/Y$ or $m$). In these cases, the analytical formula must be used to convert between nominal and effective rates.

• Conversion Formula:     $1 + i = (1 + \frac{i_m}{m})^m$

• Example: Nominal rate $i_m = 13.27\%$ with $m = \frac{365}{56}$.

  1. Find periodic rate: $\frac{i_m}{m} = \frac{0.1327}{(\frac{365}{56})}$.

  2. Calculate expression: $(1 + \frac{0.1327}{\frac{365}{56}})^{\frac{365}{56}}$.

  3. Subtract $1$ to find effective annual interest rate ($i$).

  4. Using numerical values ($m \approx 6.518$):         $i = (1 + \frac{0.1327}{6.518})^{6.518} - 1 = 0.1404$ (or $14.04\%$).