Question

What is the present value of a perpetual cash flow of: (a) \(\$ 1,450\) per year, discounted at \(r=5 \% ?\) (b) \(\$ 2,460\) per year, discounted at \(r=8 \% ?\)

Short answer

(a) The present value is $29,000. (b) The present value is $30,750.

Step-by-step solution

Understand the Concept of Perpetuity

A perpetuity is a type of annuity that provides an infinite series of cash flows. The present value of a perpetuity can be calculated using the formula: \( PV = \frac{C}{r} \), where \( C \) is the annual cash flow and \( r \) is the discount rate.

Identify the Variables for Each Part

For part (a), \( C = 1450 \) and \( r = 0.05 \). For part (b), \( C = 2460 \) and \( r = 0.08 \). We will use these values to calculate the present value for each cash flow.

Calculate Present Value for Part (a)

Substitute the given values into the formula \( PV = \frac{C}{r} \) for part (a):\[ PV = \frac{1450}{0.05} \]Calculate this to get:\[ PV = 29000 \]

Calculate Present Value for Part (b)

Substitute the given values into the formula \( PV = \frac{C}{r} \) for part (b):\[ PV = \frac{2460}{0.08} \]Calculate this to get:\[ PV = 30750 \]

Conclusion

For part (a), the present value of the perpetuity is \\(29,000. For part (b), the present value of the perpetuity is \\)30,750. Both values represent the present worth of receiving these annual cash flows indefinitely, given the respective discount rates.

Key concepts

Present Value

When we talk about the present value, we're essentially discussing the current worth of a future sum of money or stream of cash flows, discounted at a specific rate. Understanding the present value helps us compare the value of money now versus its value in the future because of the time value of money. The formula used for calculating the present value of a perpetuity is a simplified version, \( PV = \frac{C}{r} \), where \( C \) is the annual cash flow and \( r \) is the discount rate. This equation assumes that cash flows continue indefinitely and at a constant rate, making it significantly useful for estimating the worth of perpetual financial arrangements. The calculated present value enables investors and analysts to assess whether receiving the series of cash flows is worth the initial investment today.

Discount Rate

The discount rate is a key concept when calculating present value and serves as a measure of time value of money. It's the interest rate used to discount future cash flows back to their present value, reflecting the decreasing worth of money over time due to factors like inflation and opportunity costs. The choice of discount rate can greatly impact the calculated present value. For example, a higher discount rate decreases the present value since money in the future is worth less today. Conversely, a lower discount rate suggests that future cash flows are more valuable in today's terms. The discount rate is often dictated by the market interest rate or the required rate of return, depending on the type of investment and associated risks.

Cash Flows

Cash flows refer to the net amount of cash being transferred in and out of a business or investment. In the context of perpetuities, it specifically refers to the regular, unending stream of income received every year. Knowing the cash flow amount is vital because it directly influences the calculation of the present value. In our exercise, we have two different cash flows: \(1,450\) and \(2,460\) annually. These amounts are used in conjunction with their respective discount rates to find the present value of the perpetual income stream. The consistency and predictability of cash flows in perpetuities make them a valuable consideration for investors looking for steady income.

Annuity

An annuity is a series of periodic payments, usually made at equal intervals. While annuities often have a finite end, a perpetuity is a special type of annuity that continues indefinitely. In the realm of finance, understanding annuities can help decipher more complex financial products that provide regular payments over time. Perpetuities are particularly beneficial for those seeking everlasting income from an investment. For practical understanding, think of it as a never-ending stream of cash flows, just like an annuity but with no termination date. This unique characteristic of perpetuities means that the calculation of present value assumes these payments will continue forever, contributing to its intrinsic financial appeal.