Notes › MATH 5441: Financial Mathematics Lecture 1

Interest Theory Basics

calendar_today August 20, 2025 article 721 words access_time 6 min replay Modified August 21, 2025

Table of Contents

Meaning of Interest

Definition 01.1 (Interest).

Interest is an amount charged to a borrower for the use of the lender’s money over a time period.

The lender’s monetary investment is increasing with time due to interest being added
- This is why interest is also commonly referred to as the time value of money

Interest problems typically involve four quantities:
- The initial investment is the principal $(P)$
- The final amount is the amount value $(A)$
- The interest gained during the period of investment is $I = A-P$
- The interest rate is interest expressed as a percentage of the principal
  - May be calculated using the percent change formula: $\frac{FV-IV}{IV} \cdot 100\%$
Interest takes into account the risk of default (when the borrower cannot pay back the loan)
- Risk may be reduced if the borrower promises to release one of their assets in the event of their default called collateral
The unit with which the time of investment is measured is called the measurement period
The present value is the principal that must be invested to obtain a future amount value at a specified time

Types of Interest

Definition 01.2 (Simple Interest).

Simple interest occurs when interest is only calculated using the original principal; that is, interest earned from prior periods is not required for interest calculations for consecutive periods.

It is characterized by $P$, the interest rate per measurement period $(i)$ and number of measurement periods $(t)$
The amount value is linear in $t$

$$A(t)=P \cdot(1+it)$$

Definition 01.3 (Compound Interest).

Compound interest occurs when interest is accumulated on the principal in discrete subintervals of the measurement period.

It is characterized by $P$, the nominal interest rate $(r)$, the number of subdivisions for a single measurement period $(n)$, and the number of measurement periods $(t)$
The amount value is exponential in $t$ (assuming $n > 1$)

The accumulated amount for one measurement period is:

$$ A(n) = P\left(1+\frac{r}{n}\right)^n. $$

For multiple periods:

$$ A(n, t) = P\left(1+\frac{r}{n}\right)^{nt}. $$

The effective interest rate (ER) for one measurement period is

$$ ER(n) = \left[ \left(1+\frac{r}{n}\right)^n - 1 \right] \cdot 100\%. $$

For multiple periods:

$$ ER(n,t) = \left[\left(1+\frac{r}{n}\right)^{nt} - 1\right] \cdot 100\%. $$

Example: If $r=0.10$ is compounded semi-annually ($n=2$), then after one year ($t=1$):

$$ A(2, 1) = P\left(1+\frac{0.10}{2}\right)^2 = 1.1025P, $$

so:

$$ ER(2,1) = 1.1025 - 1 = 0.1025 = 10.25\%. $$

Practice

Example 01.4.

(Problem 1.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12])

$$\frac{3294.08-3200}{3200} \cdot 100\% = 2.94\%$$

Example 01.5.

(Problem 1.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12]) a. $12780-12000 = \$780$ b. $$\frac{12780-12000}{12000} \cdot 100\% = 6.5\%$$

Example 01.6.

(Problem 1.3 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12]) $1000 \cdot 1.09 = \$1090$

Example 01.7.

(Problem 1.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12]) $A(t) = 1 + 0.01t$ (in dollars)

Example 01.8.

(Problem 1.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12])

$$\frac{\frac{615}{500}-1}{2.5} \cdot 100\%=9.2\% \text{ annually}$$

Example 01.9.

(Problem 1.6 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12])

$$\frac{\frac{630}{500}-1}{\frac{7.8\%}{100\%}}=3.33 \text{ years}$$

Example 01.10.

(Problem 1.7 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 12])

$$10000\left(1 + \frac{5\%}{100\%} \right)^3 =\$ 11576.25$$

Example 01.11.

(Problem 1.8 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13]) $\$4971.7673529829$

Example 01.12.

(Problem 1.9 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13]) $6.961037572506901\%$

Example 01.13.

(Problem 1.10 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13]) $55.4781076387804 \text{ years}$

Example 01.14.

(Problem 1.11 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13]) Reject the offer, since $\$22500 < A(1, 10) = $25937.424601$.

Example 01.15.

(Problem 1.12 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13])

$$P\prod_{k=1}^{n}(1+i_{k})$$

Example 01.16.

(Problem 1.13 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 13]) $\$56.7426855718599$

Example 01.17.

(Problem 1.14 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 14]) a. $\$1004.006004001$ b. $$4.00600400099995$ c. $0.4006%$

Example 01.18.

(Problem 1.15 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 14]) $\$100$

Sources

Finan, A Basic Course in the Theory of Interest and Derivatives Markets

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