Question

Find the present value of the income \(c\) (measured in dollars) over \(t_{1}\) years at the given annual inflation rate \(r\). $$ c=100,000+4000 t, r=5 \%, t_{1}=10 \text { years } $$

Short answer

The present value of the income is the sum of the present values for each year, calculated over \(t_1 = 10\) years at an annual inflation rate of \(r = 5\%\).

Step-by-step solution

Determine the general formula for present value

The general formula for present value is given by: \(PV = \frac{C}{(1+r)^n}\) where \(C\) is the cash flow at time \(n\), \(r\) is the interest rate, and \(n\) is the number of times the interest is compounded.

Apply the formula for each year for 10 years

Apply the formula to calculate the present value for each year, from \(t = 0\) to \(t = 10\). Since the income is growing over time, we need to find the present value for each year individually. Instead of \(C\), we substitute the formula for income growth, \(c = 100,000 + 4000t\), and \(r = 5\% = 0.05\). The calculation for each year is then added together to get the total present value.

Calculate the total present value over 10 years

The total present value over 10 years, \(PV_{TOTAL}\), is obtained by summing up the present values for each year: \(PV_{TOTAL} = PV_0 + PV_1 + PV_2 + ... + PV_{10}\), where each \(PV_t\) is calculated using the formula from Step 2.

Key concepts

Inflation Rate

Understanding inflation is crucial when calculating present value. Inflation refers to the rate at which the general level of prices for goods and services increases over time.It's expressed as a percentage and indicates a decrease in the purchasing power of money.
When dealing with long-term financial calculations, including present value computations, inflation must be considered to ensure that you are creating accurate and realistic projections.
In this problem, we're given an inflation rate (\( r \)) of 5%.This means that each year, the purchasing power of currency decreases by 5%.Hence, a cash flow received in the future will be worth less in today's terms.

• An inflation rate impacts the calculation of present value by reducing the value of future cash flows.
• In the formula \(PV = \frac{C}{(1+r)^n}\), the inflation rate is used as the discount rate \( r \), making future dollars less valuable.

By adjusting for inflation, we can determine how much current money is equivalent to in future terms.

Income Growth

Income growth plays a crucial role in the present value calculation in this exercise.
The problem specifies that the income grows by a constant amount each year, given by the expression \( c = 100,000 + 4000t \).Here, \( 100,000 \) represents the base income at year zero, while \( 4000 \) is the annual increment.
This formula reflects that the income doesn't remain static.It increases annually by a fixed dollar amount.By using this formula to substitute for each year into the present value calculation, we account for these incremental income increases.

• Income growth over time affects how much income one would expect to receive in future years.
• In present value calculations, each year's distinct income value needs its own adjustment for inflation.

This makes the calculation more comprehensive, as we account for how revenue increases can potentially neutralize some effects of inflation.

Cash Flow

Cash flow refers to the amount of money being transferred in and out of a business, or in this case, received as income over a period of 10 years.
In our scenario, cash flow starts at \(100,000 and increases by \)4,000 each year.This represents a steadily growing income stream, which is a common financial scenario.
Calculating the present value of cash flows helps in understanding the true value of these future receipts.We assess how much all these future cash inflows are worth in today's dollars.

• The method involves calculating the discounted value of future cash flows to today's value using the formula \( PV = \frac{C}{(1+r)^n} \) for each year.
• This allows an individual or business to understand their exact financial standing or potential profitability at present.
• Knowing the present value of future cash flows is crucial for financial planning and investment decisions.

Considering cash flow in present value calculations ensures the financial feasibility of income expectations and helps adapt plans as needed.