Question

Suppose you buy a machine that costs \(£ 10000\) today and rent it out. You earn a rent of \(£ 1500\) on that machine every year for four years. After four years the machine can be sold as scrap for \(£ 4000\). Assume that the interest rate is 10 per cent in all four years. What is the present value of the machine? What is the net present value of the machine? Is the investment worthwhile?

Short answer

The NPV is -£2514.38, making the investment not worthwhile.

Step-by-step solution

Understanding the Cash Flows

The initial cost of the machine is £10000. For the next four years, you will receive £1500 annually. After four years, the machine can be sold for £4000. So, there are four sets of cash flows to consider: initial investment of £10000 (outflow), annual rent receipts of £1500 for four years (inflows), and a scrap value of £4000 at the end of the fourth year (inflow).

Calculate Present Value of Rent Inflows

The present value (PV) of the inflows needs to be calculated by discounting each receipt back to today's value. The formula is \( PV = \frac{C}{(1 + r)^n} \), where \(C\) is the cash flow, \(r\) is the interest rate, and \(n\) is the number of periods.- Year 1 rent: \( PV = \frac{1500}{(1 + 0.10)^1} = \frac{1500}{1.10} \approx £1363.64 \)- Year 2 rent: \( PV = \frac{1500}{(1 + 0.10)^2} = \frac{1500}{1.21} \approx £1239.67 \)- Year 3 rent: \( PV = \frac{1500}{(1 + 0.10)^3} = \frac{1500}{1.331} \approx £1126.26 \)- Year 4 rent: \( PV = \frac{1500}{(1 + 0.10)^4} = \frac{1500}{1.4641} \approx £1023.29 \)

Calculate Present Value of Scrap Value

The present value of the scrap value is calculated using the same PV formula:- Scrap value year 4: \( PV = \frac{4000}{(1 + 0.10)^4} = \frac{4000}{1.4641} \approx £2732.76 \)

Calculate Total Present Value of Inflows

Sum up all the individual present values of the inflows to get the total present value: \[PV_{inflows} = £1363.64 + £1239.67 + £1126.26 + £1023.29 + £2732.76 \approx £7485.62\]

Calculate Net Present Value (NPV)

The net present value is the present value of the inflows minus the present value of the outflows:\[NPV = PV_{inflows} - PV_{outflows} = £7485.62 - £10000 = -£2514.38\]

Assess Investment Worthiness

With an NPV of -£2514.38, the investment is not worthwhile since the NPV is negative. This signals that the present value of the future cash flows is less than the initial investment cost.

Key concepts

Cash Flows

Cash flows represent the movement of money in and out resulting from an investment over time. In our machine investment example, there are several important cash flows to note. Initially, there is an outflow of £10,000 for purchasing the machine. This is the initial investment required to start the rental business.
Next, you experience cash inflows from the rent you earn, which is £1,500 annually for four years. These represent returns on investment during the operation period. Finally, at the end of the fourth year, there is another inflow from selling the machine as scrap for £4,000.
Understanding these cash flows is crucial as they determine the financial viability of an investment. It involves tracking inflows and outflows over the investment's life cycle.

Present Value

Present value (PV) is a key concept in evaluating investments. It refers to the worth of future cash flows in today's terms, taking into account the time value of money. The formula to calculate PV is: \[ PV = \frac{C}{(1 + r)^n} \]where:

• \(C\) is the future cash flow.
• \(r\) is the interest rate.
• \(n\) is the number of time periods.

To find the total present value of all inflows in this investment, we calculate the PV for each annual rent and the scrap value separately, then sum them up.
For example, the first year's rent inflow is discounted back to the present at the interest rate of 10%, giving its present value as approximately £1,363.64. This process is repeated for each yearly cash inflow and the scrap value. Adding them provides the total present value of future inflows, which is around £7,485.62 in this scenario.

Investment Worthiness

Investment worthiness is assessed through the net present value (NPV) of the project. NPV provides a metric to understand if the investment is likely to be profitable, essentially comparing the present value of cash inflows with the cash outflow at the beginning.
If the NPV is positive, it indicates that the projected earnings (adjusted for their present value) exceed the initial costs, making the investment favorable. Conversely, a negative NPV means that the investment is expected to bring in less money than it cost, thus considered unworthy.
In this machine purchase scenario, the NPV is -£2,514.38, which suggests the project is not worth pursuing. It raises a clear red flag by showing the present value of expected inflows does not cover the cost of investment, recommending the investor to reconsider or look for better opportunities.