Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The management of Gibraltar Brokerage Services anticipates a capital expenditure of $$\$ 20,000$$ in 3 yr for the purchase of new computers and has decided to set up a sinking fund to finance this purchase. If the fund earns interest at the rate of \(10 \% /\) year compounded quarterly, determine the size of each (equal) quarterly installment that should be deposited in the fund.

Step-by-Step Solution

Answer

The size of each equal quarterly installment that should be deposited in the sinking fund is approximately $$\$ 1,582.09$$. Gibraltar Brokerage Services should deposit this amount every quarter for 3 years to reach their goal of $$\$ 20,000$$ for purchasing new computers.

Step 1: Identify the variables

The problem asks us to find out the size of each equal quarterly installment. Therefore, the variables we need to identify are: - Future Value (FV) = $$\$ 20,000$$ (The goal amount after 3 years) - Annual Interest Rate (r) = 10% per year - Compounding Frequency (n) = Quarterly, so 4 times per year - Time Period (t) = 3 years

Step 2: Convert the annual interest rate to a quarterly rate

The interest rate is given in annual terms, so we need to convert it to a quarterly rate since installments are deposited quarterly. Quarterly interest rate (i) = Annual interest rate (r) divided by the number of compounding periods in a year (n) \(i = \frac{r}{n}\)

Step 3: Calculate the number of quarterly payments

We need to determine the total number of quarterly payments that will be made over the 3-year period. Number of quarterly payments (q) = Number of years (t) times compounding frequency (n) \(q = t * n\)

Step 4: Use the future value of an ordinary annuity formula

We will use the future value formula for an ordinary annuity to solve for the size of each quarterly installment (PMT) needed to reach the goal of $20,000 after 3 years. Future value (FV) = PMT * [(((1+i)^q)-1)/i] Rearrange the formula to solve for PMT: PMT = FV / [(((1+i)^q)-1)/i]

Step 5: Plug in values and calculate the quarterly installment

From previous steps, we have: FV = $$\$ 20,000$$ i = \(\frac{10\%}{4}\) = \(0.1 / 4\) = 0.025 q = \(3 * 4\) = 12 Now, plug these values into the rearranged formula: \(PMT = \frac{20,000}{(((1+0.025)^{12})-1)/0.025}\) Calculate the value of PMT: \(PMT \approx \$ 1,582.09\)

Step 6: Interpret the result

The size of each equal quarterly installment that should be deposited in the sinking fund is approximately $1,582.09. This means Gibraltar Brokerage Services should deposit $$\$ 1,582.09$$ into the fund every quarter for 3 years to reach their goal of $$\$ 20,000$$ for purchasing new computers.

Key Concepts

Future Value of An Annuity

The future value of an annuity is the total amount that a series of equal payments will be worth at a specific point in the future when invested at a particular interest rate. Annuities are used by individuals and entities to save for future expenses, like retirement or, as in our exercise, the purchase of equipment.

An annuity can be classified as ordinary (payments made at the end of each period) or as an annuity due (payments made at the beginning of each period). The sinking fund created by Gibraltar Brokerage Services is an example of an ordinary annuity, where funds are deposited regularly and interest is compounded periodically.

To calculate the future value of an ordinary annuity, we use the formula:
\[ FV = PMT \times \left(\frac{\left(1+i\right)^q - 1}{i}\right) \]
where:

• PMT is the amount of each payment,
• i is the periodic interest rate, and
• q is the total number of payments.

The exercise provides a practical application for this formula, assisting Gibraltar Brokerage Services in planning for their future capital expenditure needs.

Time Value of Money

The time value of money is a core principle of finance that reflects the idea that money available now is worth more than the same amount in the future due to its potential earning capacity. This concept underpins the rationale for the interest that accrues on investments or savings.

The principle encompasses the preference for money to be received sooner rather than later and is the foundation for concepts like interest rates, investment appraisal, and loan calculations. In our exercise, Gibraltar Brokerage Services is taking into account the time value of money by setting up a sinking fund to meet a future expense rather than saving the needed amount all at once later.

Understanding the time value of money enables individuals and businesses to make informed decisions about investments, savings, loans, and other financial transactions, as each decision can affect the current and future financial states.

Compound Interest

Compound interest is the addition of interest to the principal sum of a loan or deposit, where the added interest also earns interest from that moment on. This creates a snowball effect where the amount of interest grows exponentially over time.

In the Gibraltar Brokerage Services scenario, the fund compounds quarterly, which means the interest is calculated and added to the principal amount every three months. The formula to calculate compound interest on an annuity is integrated into the future value formula mentioned earlier:
\[ FV = PMT \times \left(\frac{\left(1+i\right)^q - 1}{i}\right) \]
This formula assumes that each installment is compounded at the same frequency as the payments are made.

Compound interest can significantly increase the growth of savings and investments compared to simple interest, which is calculated only on the initial principal.

Financial Mathematics

Financial mathematics involves applying mathematical methods to solve problems in finance. It includes concepts like the time value of money, future value, present value, annuities, and more. Understanding these concepts allows individuals and businesses to plan for future financial needs, evaluate investment opportunities, and effectively manage debt.

The sinking fund calculation from our exercise is a classic example of applying financial mathematics to determine the size of each payment needed to achieve a financial goal in the future. It employs formulas derived from the principles of financial mathematics to compute the future value of annuity payments considering compound interest.

By mastering financial mathematics, individuals can better understand and navigate the financial decisions required for personal and business finance, leading to more prudent and fruitful financial management.