# The Time Value of Money & Net Present Value

Many companies seek to invest in new or existing projects in order to generate income. They can invest in real assets which can provide a new revenue stream and diversify their income. These can be tangible assets such as new machinery, office space, the expansion of store locations etc. They can also choose to invest in intangible assets like research & development, trademarks/patents etc.

There are two types of decisions that a financial manager must make. One is the investment decision, where they must weigh up the costs as well as the benefits of the project/investment. The costs include the purchase of the asset and the cost of managing it. A manager must also be aware of the risk of an investment as it may not be as profitable as expected if at all. The second decision is how to raise capital to finance the investment, in which there are a number of ways to do so and these methods can also be combined.

- The sale of existing assets
- Raiding funds from new investors by through issuance of new shares
- Simply borrowing the money (can take on debt by issuing bonds or bank loans)
- Reinvesting profits of the firm

The main aim of a financial manager when choosing which projects to invest in is to ensure it will increase the value of the firm. The project must be expected to be profitable and the manager will want to see a great return on investment (ROI) in order to satisfy shareholders. This is crucial since shareholders will have essentially given up money in the present if profits are reinvested (giving up present dividends) or new shares are sold (dilution of their shares) to fund the investment. They will be expecting to be compensated by larger future profits otherwise they would object the investment decision.

Below is the process of how funds flow between the investors and the company

- Funds raised from investors or profits reinvested
- Invested into new projects by the firm
- Revenue generated by the operations of the project
- Investors are compensated with ideally larger profits than if they did not invest

Investors can also again give up the profits generated the operations of the investment and instead reinvest in further projects.

### The Time Value of Money

Net Present Value (NPV) is a common method of evaluating an investment, it allows one to calculate the future costs and benefits in the present day and analyse the profitability of the investment [1]. In order to better grasp this concept, it is important to first understand the time value of money.

Consider the following example, you can either receive £100 at this moment, or £100 one year from now. The value of the £100 cash flow in the present is worth more than the future £100 cash flow because you could choose to invest that money. Depending on the interest rate of your investment you can end up with more than £100 after one year. If the interest rate was 5% you would receive £105. We can conclude is that the present value of a cash flow larger than in the present value of a future cash flow of the same amount.

### Discounted Cash Flow

Suppose you have £100 to invest at an interest rate of 5% per year. The amount of money you will have after one year is:

£100 * 1.05 = £105 (Future value after one year)

The investment grows at a compound rate, because for the following years you earn interest on the initial £100 and the previous year’s interest. To calculate the value of the investment after a given number of years (t) the calculation becomes:

£100 * (1.05)^{t} = Future value after n years

The formula for calculating the future value (FV) is simply: (r = interest rate, t = number of years, & PV = present value)

PV * (1 + r)^{t} = FV

Many companies seek to invest in new or existing projects in order to generate income. They can invest in real assets which can provide a new revenue stream and diversify their income. These can be tangible assets such as new machinery, office space, the expansion of store locations etc. They can also choose to invest in intangible assets like research & development, trademarks/patents etc.

There are two types of decisions that a financial manager must make. One is the investment decision, where they must weigh up the costs as well as the benefits of the project/investment. The costs include the purchase of the asset and the cost of managing it. A manager must also be aware of the risk of an investment as it may not be as profitable as expected if at all. The second decision is how to raise capital to finance the investment, in which there are a number of ways to do so and these methods can also be combined.

- The sale of existing assets
- Raiding funds from new investors by through issuance of new shares
- Simply borrowing the money (can take on debt by issuing bonds or bank loans)
- Reinvesting profits of the firm

The main aim of a financial manager when choosing which projects to invest in is to ensure it will increase the value of the firm. The project must be expected to be profitable and the manager will want to see a great return on investment (ROI) in order to satisfy shareholders. This is crucial since shareholders will have essentially given up money in the present if profits are reinvested (giving up present dividends) or new shares are sold (dilution of their shares) to fund the investment. They will be expecting to be compensated by larger future profits otherwise they would object the investment decision.

Below is the process of how funds flow between the investors and the company

- Funds raised from investors or profits reinvested
- Invested into new projects by the firm
- Revenue generated by the operations of the project
- Investors are compensated with ideally larger profits than if they did not invest

Investors can also again give up the profits generated the operations of the investment and instead reinvest in further projects.

### The Time Value of Money

Net Present Value (NPV) is a common method of evaluating an investment, it allows one to calculate the future costs and benefits in the present day and analyse the profitability of the investment [1]. In order to better grasp this concept, it is important to first understand the time value of money.

Consider the following example, you can either receive £100 at this moment, or £100 one year from now. The value of the £100 cash flow in the present is worth more than the future £100 cash flow because you could choose to invest that money. Depending on the interest rate of your investment you can end up with more than £100 after one year. If the interest rate was 5% you would receive £105. We can conclude is that the present value of a cash flow larger than in the present value of a future cash flow of the same amount.

### Discounted Cash Flow

Suppose you have £100 to invest at an interest rate of 5% per year. The amount of money you will have after one year is:

£100 * 1.05 = £105 (Future value after one year)

The investment grows at a compound rate, because for the following years you earn interest on the initial £100 and the previous year’s interest. To calculate the value of the investment after a given number of years (t) the calculation becomes:

£100 * (1.05)^{t} = Future value after n years

The formula for calculating the future value (FV) is simply: (r = interest rate, t = number of years, & PV = present value)

PV * (1 + r)^{t} = FV

Therefore, we can just rearrange this equation to obtain the present value formula which will allow us to calculate the how much a future cash flow (CF) is worth today. The future cash flow is multiplied by the discount rate 1 / (1+r)^{t} to give the present value.

PV = CF/[(1+r)^{t}]

The future cash flow can be either positive (revenue generated from the operations of the firm’s project) or negative (the costs associated with the maintenance of the project).

Present value is a useful technique that can be used to analyse cash flows for not just projects but also financial instruments such as stocks and bonds.

### Net Present Value (NPV)

NPV is the sum of the present value of cash inflows and outflows over time. A project will be profitable if the NPV is positive, but it will make a loss if the NPV is negative. Ideally investors would like to undertake projects that have the highest NPV, as they would like to get the best return on their investment [2].

The discount rate (r) is the opportunity cost of capital. This is the rate of return that one would have received by alternatively investing their money elsewhere that has the same level of risk as the project [3]. This can be in the form of Treasury bulls, or shares in the stock market etc.

Here is a simple example of how NPV can be calculated.

A construction company contemplates undertaking a new project of building an office block which will cost a total of £800,000. They estimate that they can sell the office block for £1,200,000 after two years. The opportunity cost of capital is 8% (the returns offered by the market from instruments of similar risk).

The cost of constructing the building is an upfront cost and does not need to be discounted, unlike the £1,000,000 which is received after one year.

**Present value of the payoff £1,000,000 / (1+0.08) ^{2} = £857,338.82**

**NPV = -£800,000 + £857,338.82**

**NPV = £57,338.82**

According to the NPV valuation principle, they should build the office block because the NPV is positive and the project is profitable, this will increase the value of the company.

The NPV approach while useful has its limitations. It relies heavily on the assumptions in terms of the costs of investment, the estimated returns and the discount rate [4]. Projects can run into unexpected additional costs which would decrease the NPV. Also, it can be seen from the present value formula that any delays in the project will reduce the present value of future cash flows. As the number of years (n) increases PV decreases. There are also assumptions about the discount rate. The project may be riskier than assumed. There the opportunity cost of capital would be higher, since investing in riskier capital market securities will give higher returns. Once again looking at the PV formula it can be seen that increasing r will decrease PV, so the NPV would be lower than expected. The NPV approach does not give the return on investment in percentage terms, which is helpful for comparing projects.

The previous example was just a simple one. However, for large scale projects with more complex/multiple cash flows that span over a number of years, a discounted cash flow model (DCF) can be constructed in Excel [5]. This can look at NPV as well as other evaluation methods such as Internal Rate of Return (IRR) and Return on Investment (IRR) which can help to compare projects and decide on which to invest in. One should be cautious of just using NPV to assess a project’s profitability as the various assumptions can lead to errors. Using other metrics in conjunction with NPV would better analyse an investment opportunity.