Section 1.5: Continuous Compounding and Force of Interest

Subject Matter: The focus is on section 1.5 of the textbook (page 25), specifically continuous compounding, which is also formally referred to as the force of interest.
Exam Importance: The force of interest typically accounts for approximately $2\%$ to $3\%$ of the final grade. In a standard assessment of 50 multiple-choice questions, one might expect roughly two simple interest questions and two to three force of interest questions.
Total Assessment Scope: Out of a total of 150 questions across three assessments, potentially six or seven questions will cover simple interest and force of interest combined.
Complexity Level: Questions regarding continuous compounding are generally kept simple. They typically do not involve advanced complications like changing interest rates or erratic cash flows (money moving in and out of accounts frequently).

Hypothetical Investment Scenario:
- Present Value ( $PV$ ): $-1000$
- Interest Rate ( $I$ ): $15\%$
- Time ( $n$ ): Exactly $1\,year$
Minimum Growth (Annual Compounding): If interest is calculated only once per year ( $M=1$ ), the interest earned is $15\%$ of $1000$ , which is $150$ . The final amount is $1150$ .
Increased Frequency Effects: As the frequency of compounding ( $M$ ) increases, the terminal value of the investment increases because the investor begins earning "interest on interest."
Comparative Growth Data ( $PV = 1000$ , $15\%$ interest, $1\,year$ ):
- Once a year ( $M=1$ ): $1150.00$
- Every six months ( $M=2$ ): $1155.63$ (An increase of $5.63$ )
- Three times a year (every 4 months, $M=3$ ): $1157.63$
- Four times a year (every 3 months, $M=4$ ): $1158.65$
- Six times a year (every 2 months, $M=6$ ): $1159.69$
- Monthly ( $M=12$ ): $1160.75$
- Weekly ( $M=52$ ): $1161.58$
- Daily ( $M=365$ ): $1161.798$ (rounded to $1161.80$ —earning $11.80$ more than annual compounding)
- 1,000 times a year (every few hours): $1161.82$
- 20,000 times a year: $1161.83$
The Concept of Frequent Reinvestment: Historically, if an investor were to manually close and reopen an account daily to capture interest, they would maximize their returns. Continuous compounding takes this to the mathematical extreme where the compounding frequency ( $M$ ) reaches infinity ( $\infty$ ).

Definition: Continuous compounding is the mathematical identity reached when the frequency of compounding intervals becomes infinite ( $M \rightarrow \infty$ ).
Mathematical Property: When $M$ approaches infinity in the standard compound interest equation $\left(1 + \frac{i}{m}\right)^{m}$ , the term involving the large number division and the large power simplifies into a function of the constant $e$ .
The Constant $e$ : Like $\pi$ , $e$ is a mathematical constant used as the base for natural logarithms. It is accessible on financial calculators via the $e^{x}$ function.
Force of Interest ( $r$ ): In the context of continuous compounding, the interest rate is denoted as $r$ rather than $i$ . The symbol $r$ is specifically reserved for interest that is added continuously to an account.
Maximum Yield: Continuous compounding represents the absolute maximum amount of money an investor can earn in a year for a given nominal rate, regardless of how frequently interest is applied.

Future Value Formula: To calculate the accumulated sum ( $S$ ) when interest is compounded continuously, use:
- $S = K \times e^{r \times n}$
- $S$ : Final amount (Future Value)
- $K$ : Principal amount (Capital/Present Value)
- $r$ : Force of interest per year
- $n$ : Number of years
Rate Conversion Formula: To convert between the force of interest ( $r$ ) and a normal effective compound interest rate ( $i$ ), use:
- $1 + i_{effective} = e^{r}$

Limitation of TVM Functions: The standard Time Value of Money (TVM) functions (the top five buttons on a financial calculator) cannot handle force of interest or continuous compounding directly.
Manual Calculation Requirements: Use the following specialized buttons:
- Second Function 1 ( $e^{x}$ ): Used for calculating the exponential growth factor.
- Second Function 2 ( $Ln$ ): Used when solving for the interest rate or time.
Walkthrough Example Steps:
1. Clear the calculator.
2. Input $1000$ as a negative value for Present Value ( $PV$ ).
3. Set Payment ( $PMT$ ) to zero to ensure no periodic payments interfere.
4. Input the interest rate (e.g., $15\%$ ).
5. Adjust the payments per year ( $P/Y$ ) using Second Function then I/Y and update $N$ (interest periods) accordingly to observe the changes in value across different compounding frequencies.

Solving for the Rate ( $r$ ): Because the rate $r$ is located in the exponent of the equation $e^{r \times n}$ , solving for it requires the use of logarithms.
Natural Logarithm ( $Ln$ ): The specific logarithm used is log base $e$ , which is written as $Ln$ .
Comparison to Standard Logs: $Ln$ follows the same rules as log base 10 (common logs used in school). Just as $10^{x}$ is solved with log base 10, $e^{x}$ is solved with log base $e$ .

There are four primary ways of calculating interest that are concentrated on in this course:

Simple Interest: Interest is calculated only on the principal amount.
Normal Compound Interest: Interest is calculated on the principal plus any accumulated interest at discrete intervals (e.g., monthly, quarterly).
Continuous Interest (Force of Interest): A special type of compound interest where the compounding frequency ( $M$ ) is infinity.
Discount: A separate method involving calculating interest from a future value back to the present (to be covered in future sections).