Problem 19 - Applied Calculus

Question

A person who deposits money in a bank account starts a long process described by the reserve-deposit ratio, \(r\). For every dollar deposited, the bank keeps \(r\) dollars and lends \(1-r\) dollars to someone else, who deposits the loan in a bank account. The same fraction of the second deposit is loaned out, to be deposited in turn, and so on. If the initial deposit is \(N\) dollars, find the total value of the bank accounts generated by this deposit: (a) After the second deposit (b) After the third deposit (c) If the process continues forever

Step-by-Step Solution

Answer

(a) \( N(2-r) \), (b) \( N[2-r + (1-r)^2] \), (c) \( \frac{N}{r} \)."

Step 1: Initial Setup

Let's start by understanding the problem. When a deposit is made, the bank keeps a reserve of \( r \) times the deposit and lends out the rest. This process repeats itself with each subsequent re-deposit.

Step 2: Calculate After First Deposit

The initial deposit is \( N \) dollars. After the first deposit, the bank keeps \( r \times N \) dollars, and loans out \( (1-r) \times N \) dollars. The first deposit thus counts as \( N \) dollars in the accounts.

Step 3: Calculate Second Deposit

The loaned amount from the first deposit, \( (1-r) \times N \), is deposited in another account. The bank keeps \( r \times (1-r) \times N \) dollars from this new deposit and lends out \( (1-r)^2 \times N \) dollars. The second deposit thus adds \( (1-r) \times N \) to the accounts.

Step 4: Total After Second Deposit

After the second deposit, the total value in the bank accounts is \( N + (1-r) \times N = N \times (1 + (1-r)) = N + N - rN = N(2-r) \).

Step 5: Calculate Third Deposit

The amount loaned from the second deposit is \( (1-r)^2 \times N \), which is deposited again. The bank keeps \( r \times (1-r)^2 \times N \), and a new loan of \( (1-r)^3 \times N \) is made. The third deposit adds \( (1-r)^2 \times N \) to the accounts.

Step 6: Total After Third Deposit

After the third deposit, the total value in the bank accounts is \( N (2-r) + (1-r)^2 N = N [2-r + (1-r)^2] \).

Step 7: Process Continuing Forever

Each new deposit is a geometric series of the form \( N (1-r)^n \) which continues forever. The sum of an infinite geometric series \( a + ar + ar^2 + ar^3 + \, \ldots \) is \( \frac{a}{1-r} \) for \(|r| < 1\). Here, \( a = N \) and the ratio is \( 1-r \), so the total value of deposits is \[ \frac{N}{1 - (1-r)} = \frac{N}{r} \].

Key Concepts

Reserve-Deposit Ratio

The reserve-deposit ratio is a critical concept in banking mathematics. It represents the fraction of each deposit a bank must keep in reserve and not loan out.
In our scenario, for every dollar deposited, the bank retains an amount defined by this ratio, denoted as \( r \).
Here's how it works:

• The bank retains \( r \) dollars for every dollar deposited.
• The remaining \( 1-r \) dollars are available for loans.

These loans enhance the liquidity of the market but also come with the responsibility for banks to ensure they have enough reserves.
This reserve-deposit ratio is regulated to prevent banks from running out of liquid cash.
Essentially, it controls how much money can be multiplied and circulate in the economy through successions of deposits and loans.

Infinite Series

An infinite series is a sum of an endless sequence of numbers. When it comes to geometric series, they follow a specific rule: each term is a constant multiple of the previous term.
In the banking context, deposits and loans follow this pattern.After every deposit:

• The initial deposit amount \( N \) is reduced by the bank's reserve, and the rest continues the cycle.
• This process keeps repeating and forms a geometric series with the term \( N(1-r)^n \).

For deposits, the infinite series is written as \( N + N(1-r) + N(1-r)^2 + \ldots \)
The sum of an infinite geometric series is calculated by \( \frac{a}{1-r} \), where \( a \) is the first term and \( r \) the common ratio. This formula applies when \( |r| < 1 \).
In our case, \( a \) is \( N \) and the sum becomes \( \frac{N}{r} \). This tells us the total accumulated in bank accounts over an infinite number of redistributions.

Banking Mathematics

Banking mathematics involves principles and formulas to manage financial operations. It spans beyond basic arithmetic to the use of geometric series, like in our reserve-deposit exercise.
Key elements include:

• Determining the reserve required from a deposit, aiding in liquidity management.
• Calculating iterative deposits and loans using series to project growth.

This branch of mathematics ensures banks can maintain capital adequacy.
It helps predict outcomes of continual reinvestments of deposits, valuable for strategic planning.
Banks apply these mathematical concepts to assess the overall effect of their deposit and lending activities in the long run.
The logical structure allows evaluation of risks and opportunities inherent in the banking models.