Unit 3: Interest

1. Simple Interest

Simple interest is a method of calculating interest charges on a principal amount. The interest is calculated only on the original principal and does not include interest accrued in previous periods. It is typically used for short-term loans or investments, usually for a period of one year or less.

a. Key Formulas

• Interest (I): The total amount of interest paid or earned.
  
  TEXT

      I = Prt
      

• Future Value / Amount (A): The total amount to be repaid or the total value of the investment after time t. It is the sum of the principal and the interest.
  
  TEXT

      A = P + I
      A = P + (Prt)
      A = P(1 + rt)
      

b. Variables

• P (Principal): The initial amount of money borrowed or invested. Also known as the Present Value.
• r (Rate): The annual interest rate, expressed as a decimal. To convert a percentage to a decimal, divide by 100 (e.g., 5% = 0.05).
• t (Time): The duration of the loan or investment, expressed in years.
  • If time is given in months: t = (Number of months) / 12
  • If time is given in days:
    • Exact Interest: t = (Number of days) / 365 (or 366 in a leap year). Used by governments and federal banks.
    • Ordinary Interest (Banker's Rule): t = (Number of days) / 360. This method is more common in commercial transactions as it yields slightly more interest.

c. Examples

Example 1: Calculating Interest and Future Value
A company takes a short-term loan of $20,000 for 9 months at a simple interest rate of 8% per annum. Calculate the interest paid and the total amount to be repaid.

• Given: P = $20,000, r = 8% = 0.08, t = 9/12 = 0.75 years.
• Interest (I):
  I = Prt
I = 20,000 * 0.08 * 0.75
I = $1,200
• Future Value (A):
  A = P + I
A = 20,000 + 1,200
A = $21,200
  Alternatively:
  A = P(1 + rt)
A = 20,000(1 + 0.08 * 0.75)
A = 20,000(1.06)
A = $21,200

Example 2: Calculating Principal (Present Value)
How much money must you invest today at 6% simple interest to have $15,000 in 2 years?

• Given: A = $15,000, r = 6% = 0.06, t = 2 years.
• Formula: A = P(1 + rt)
• Rearrange to solve for P: P = A / (1 + rt)
• Calculation:
  P = 15,000 / (1 + 0.06 * 2)
P = 15,000 / 1.12
P = $13,392.86
  You must invest $13,392.86 today.

2. Compound Interest

Compound interest is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods. It is often referred to as "interest on interest" and is the standard for most savings accounts, loans, and investments.

a. Key Formulas

• Future Value / Amount (A):
  
  TEXT

      A = P(1 + i)^n
      

• Present Value (P):
  
  TEXT

      P = A / (1 + i)^n   or   P = A(1 + i)^-n
      

b. Variables

• P (Principal): The initial amount of money.
• A (Amount): The future value of the investment/loan, including interest.
• r (Nominal Annual Rate): The stated annual interest rate.
• m (Compounding Frequency): The number of times that interest is compounded per year.
• i (Periodic Interest Rate): The interest rate per compounding period.
  i = r / m
• t (Time): The number of years the money is invested or borrowed for.
• n (Total Compounding Periods): The total number of times interest is calculated over the entire term.
  n = m * t

c. Compounding Frequencies (m)

d. Examples

Example 1: Calculating Future Value
An investment of $10,000 is made in an account that pays 6% interest per year, compounded quarterly. What will be the balance in the account after 5 years?

• Given: P = $10,000, r = 6% = 0.06, t = 5 years, m = 4 (quarterly).
• Calculate i and n:
  i = r / m = 0.06 / 4 = 0.015
n = m * t = 4 * 5 = 20
• Calculate Future Value (A):
  A = P(1 + i)^n
A = 10,000(1 + 0.015)^20
A = 10,000(1.015)^20
A = 10,000 * 1.346855
A = $13,468.55

Example 2: Calculating Present Value
A business needs to have $50,000 available in 4 years to replace a machine. How much must be invested today in an account that earns 8% interest, compounded semi-annually, to meet this goal?

• Given: A = $50,000, r = 8% = 0.08, t = 4 years, m = 2 (semi-annually).
• Calculate i and n:
  i = r / m = 0.08 / 2 = 0.04
n = m * t = 2 * 4 = 8
• Calculate Present Value (P):
  P = A / (1 + i)^n
P = 50,000 / (1 + 0.04)^8
P = 50,000 / (1.04)^8
P = 50,000 / 1.368569
P = $36,534.51

3. Interest Compounded Continuously

Continuous compounding is the mathematical limit that compound interest can reach. It is an extreme case where the compounding period is infinitesimally small, meaning interest is being calculated and added to the principal at every moment in time.

a. Key Formulas

The formula involves Euler's number, e, an irrational constant approximately equal to 2.71828.

• Future Value (A):
  
  TEXT

      A = Pe^(rt)
      

• Present Value (P):
  
  TEXT

      P = A / e^(rt)   or   P = Ae^(-rt)
      

b. Variables

• P (Principal): The initial amount.
• A (Amount): The future value.
• e: Euler's number (constant).
• r (Rate): The nominal annual interest rate, in decimal form.
• t (Time): The duration in years.

c. Example

Example: Calculating Future Value with Continuous Compounding
If $10,000 is invested at an annual rate of 6% compounded continuously, what is the value of the investment after 5 years?

• Given: P = $10,000, r = 6% = 0.06, t = 5 years.
• Calculate Future Value (A):
  A = Pe^(rt)
A = 10,000 * e^(0.06 * 5)
A = 10,000 * e^0.3
A = 10,000 * 1.3498588
A = $13,498.59

Comparison: Notice this amount (13,468.55). Continuous compounding yields the highest return for a given nominal rate.

4. Nominal vs. Effective Rate of Interest

When comparing investment or loan options, the stated annual rate (nominal rate) can be misleading if the compounding frequencies are different. The effective rate provides a way to make a true comparison.

• Nominal Rate of Interest (r): The advertised or stated annual interest rate. Also known as the Annual Percentage Rate (APR).
• Effective Rate of Interest (E or APY): The actual annual rate of interest earned or paid after accounting for the effect of compounding. Also known as the Effective Annual Rate (EAR) or Annual Percentage Yield (APY). The effective rate is the simple interest rate that would produce the same amount of interest in one year.

a. Formulas

• For discrete compounding (annually, quarterly, monthly, etc.):
  

  TEXT

      E = (1 + r/m)^m - 1
      

  
  where r is the nominal rate and m is the number of compounding periods per year.

• For continuous compounding:
  

  TEXT

      E = e^r - 1
      

b. Examples

Example 1: Calculating the Effective Rate
What is the effective rate of interest for a loan with a nominal rate of 12% per year, compounded monthly?

• Given: r = 12% = 0.12, m = 12 (monthly).
• Calculate Effective Rate (E):
  E = (1 + r/m)^m - 1
E = (1 + 0.12/12)^12 - 1
E = (1 + 0.01)^12 - 1
E = (1.01)^12 - 1
E = 1.126825 - 1
E = 0.126825 or 12.68%
  This means that a 12% nominal rate compounded monthly is equivalent to a 12.68% simple annual interest rate.

Example 2: Comparing Investment Options
Which investment is better?

• Option A: 7% compounded semi-annually.
• Option B: 6.9% compounded daily.

To compare, we calculate the effective rate for each option.

• Effective Rate for Option A:
  r = 0.07, m = 2
E_A = (1 + 0.07/2)^2 - 1
E_A = (1.035)^2 - 1
E_A = 1.071225 - 1 = 0.071225 or 7.12%

• Effective Rate for Option B:
  r = 0.069, m = 365
E_B = (1 + 0.069/365)^365 - 1
E_B = (1.000189)^365 - 1
E_B = 1.0714 - 1 = 0.0714 or 7.14%

Conclusion: Option B is slightly better because its effective annual rate (7.14%) is higher than Option A's (7.12%).