Discounting money: A deep insight into the time value of money
Money today or tomorrow?
Which is better: 100 GE (monetary units, in our case EUR) today or 100 GE in a year's time? With 100 GE today you can make more next year thanks to a positive interest rate. Therefore, 100 GE today is preferable, and you would only have to choose between 100 GE today and 100+X GE next year. So money has a time value.
If you invest C GE today, you will receive an (annual) interest rate of r after t years. The future value (FV) is calculated as follows:
FV= C- (1+r)^t
The final value of C GE, called Future Value, is expressed as a percentage. For example, 13 % corresponds to a value of 0.13. At an interest rate of 7 %, EUR 100 results in a FV of EUR 107 after one year and EUR 114.49 after two years. The compound interest effect occurs in the second year, as the EUR 107 again earns 7% interest. With a higher interest rate and a longer term, the nominal return increases sharply.
The final value shows how much an initial value C has grown at the end of the investment term. The present value (PV) is calculated to determine how much you have to invest today in order to have a certain amount at your disposal in the future.
The present value is calculated as follows:
PV=C/(1+r)^t or PV=C-DF=1/(1+r)^t.
The discount factor (DF) measures the current value of a euro that will only be paid in t years. So if you want to have EUR 114.49 in two years' time, you have to invest EUR 100 today at an interest rate of 7%:
PV=114,49• 1/(1+0,07)^2=100,00.
At an interest rate of 10%, it would only be EUR 94.62. This means that with higher interest rates and longer terms, you have to invest less and less for the future.
Investment decisions
In an example investment, there are two payments (cash flows): Cv0 = -700,000 EUR is the cost of the facility, and Cv1 = 800,000 EUR is the selling price after one year. A discount rate is used to calculate the present value. In projects with the same risk and a return of 7% on the capital market, the opportunity costs correspond to this interest rate. The discount rate measures the "cost of capital" of the project and expresses the investment risk - a higher discount rate means higher risk.
The calculation for our example is
PV=Cv1/(1+r)^t = 800,000/(1.07)^1=747,664.
As the present value of EUR 747,664 is greater than the initial costs of EUR 700,000, it would be advisable to make the investment. The project results in an increase in value of EUR 47,664, the net present value (NPV). The NPV takes into account the initial costs and is calculated as follows:
Cv0+PV=Cv0+Cv1/(1+r)^t= -700,000+747,664=47,664>0 .
A positive NPV means that you are in the profit zone and are increasing your market value. If the investment were riskier, a higher interest rate would have to be applied and less value would be gained. At 12%, the NPV would only be EUR 14,286. You would still be in the black, but significantly less.
To calculate the return on investment (ROI) of the project, proceed as follows:
Return =Profit/Investment= 800,000-700,000/ 700,000=14.3%
At 14.3%, the return is higher than the opportunity cost of 12%, which means that the alternative investment would be worse than our project. The NPV rule states that investments should be made with a positive net value, and the return rule states that investments should be made with a return on investment (RoI) higher than the opportunity costs.
DCF formula (discounted cash flows)
In our example, we have assumed that there is only one payment, but how do you calculate multiple payments for projects?
PV= Cv1/(1+r)^1 + Cv2/(1+r)^2 +.....+Cvt/(1+r)^t
NPV= Cv0+Cv1/(1+r)^1 + Cv2/(1+r)^2 +.....+Cvt/(1+r)^t
=> The higher the discount rate r and the further into the future the payments C are, the lower the PV and NPV.
Internal rate of return
The internal rate of return of a payment series is the interest rate r for which the net present value of the payment series is zero, i.e.
NPV= Cv0+Cv1/(1+r)^1 + Cv2/(1+r)^2 +.....+Cvt/(1+r)^t=0
The internal rate of return is therefore the interest rate at which the present value of an investment cash flow corresponds to the initial investment. It shows what return an investment actually generates. It is usually calculated using a financial calculator or by trial and error.
Quiz
The opportunity cost of capital...
a) correspond to the interest rate at which the company can borrow capital.
b) are independent of the company's risk.
c) depend on the interest rate that the company's shareholders can expect if they invest their money in a company with identical risk.
d) occur when the company has the money for a project in a non-interest-bearing bank account.
CFO stands for:
a) Chief Executive Officer
b) Chief Operating Officer
c) Chief Technology (or Technical) Officer
d) None of the above
Source:
Brealey, R., S. Myers, F. Allen, A. Edmans (2022): Principles of Corporate Finance, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides
Sneak Peak: Special cash values
