Notes › MATH 5441: Financial Mathematics Lecture 2

Accumulation and Amount Functions

calendar_today August 21, 2025 article 753 words access_time 9 min replay Modified August 27, 2025

Table of Contents

Amount Function

Characterizes the amount value $A$ with time $t$
$A(0) = P$

Accumulation Function

Characterizes how much the principal has accumulated with time

$$a(t) = \frac{A(t)}{A(0)}$$

A simple ratio of the amount function and the principal
- Represents the factor by which the principal has grown
- This means $A(t)$ is just a constant multiple of $a(t)$; specifically, the proportionality constant is $A(0)=P$

Example 02.1.

(Example 2.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 15])

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Properties

$a(0) = 1$ by definition
Always monotonically increasing
- Decreasing $a(t)$ implies negative interest
- $a(t) = C$ for all $t$ implies zero interest
May be continuous or discontinuous
- If $a(t)$ is continuous, then interest accrues for fractional values of $t$
- If $a(t)$ is discontinuous, then interest does not accrue between payment dates; the function will stay constant between measurement periods and then jump when interest is added as a step function

Since $A(t)$ is a positive multiple of $a(t)$ and $a(t)$ is increasing, properties 2/3 must hold for $A(t)$ as well

Example 02.2.

(Example 2.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 16])

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Accumulation Factor

Suppose we want to deposit a principal $k$ at a non-zero time $s$
- Then, the accumulated value of $k$ for any $t > s$ is $k \frac{a(t)}{a(s)}$
- This ratio is referred to as the accumulation factor, and it represents the factor at which the principal grew from time $s$ to $t$

Example 02.3.

(Example 2.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 17])

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Time Intervals

The $n^{\text{th}}$ time period is $\{t\ |\ n-1 \le t \le n\}$
- The interest earned in this time period is $I_n = A(n) - A(n-1)$
- “Final amount value minus initial amount value”

$$I_{[s, t]} = A(t) - A(s) = k(a(t)-a(s))$$

Example 02.4.

(Example 2.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 19])

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Example 02.5.

(Example 2.6 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 19])

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Example 02.6.

(Example 2.7 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 19])

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Practice

Example 02.7.

(Problem 2.1 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.8.

(Problem 2.2 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.9.

(Problem 2.3 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.10.

(Problem 2.4 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.11.

(Problem 2.5 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.12.

(Problem 2.6 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.13.

(Problem 2.7 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 21])

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Example 02.14.

(Problem 2.8 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22])

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Example 02.15.

(Problem 2.9 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22] and Problem 2.10 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22])

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Example 02.16.

(Problem 2.11 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22])

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Example 02.17.

(Problem 2.12 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22])

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Example 02.18.

(Problem 2.13 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 22])

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Example 02.19.

(Problem 2.14 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23])

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Example 02.20.

(Problem 2.15 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23])

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Example 02.21.

(Problem 2.16 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23]) (

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)

Example 02.22.

(Problem 2.17 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23])

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Example 02.23.

(Problem 2.18 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23])

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Example 02.24.

(Problem 2.19 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 23])

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Example 02.25.

(Problem 2.20 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 24])

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Example 02.26.

(Problem 2.21 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 24])

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Example 02.27.

(Problem 2.22 [Finan, A Basic Course in the Theory of Interest and Derivatives Markets, p. 24])

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Sources

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

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