2. Computing Present Values

Example 1

Suppose somebody offers you a payment of 1'100 in one year (C1). How valuable is this proposal to you today? Put differently, how much money are you willing to pay today in exchange for this proposal?

To find the answer, we have to take two steps:

  • Investment alternative: First, we need to understand what else we could do with our money. More specifically, we need to determine the rate of return that we could achieve by investing in an alternative investment proposal with identical risk.
     
    • Let's assume, for the moment, that this so-called opportunity rate of return (R) is 10%.
       
    • Put differently, instead of investing in the proposal outlined above, we could buy the alternative assets, which has identical risk and generates a return of 10%.
        
  • Required investment today: Second, we need to know how much money we have to invest today in the alternative investment proposal to end up with the exact same payment as the original investment proposal in one year (C1 = 1'100):
     
    • Since the alternative investment proposal generates a return (R) of 10%, the answer is 1'000: If we invest 1'000 today at a rate of return of 10%, we will end up with 1'100 in one year.

The Present Value of the investment proposal therefore is 1'000: If we invest 1'000 today at a rate of return of 10%, we can exactly replicate the future cash flows of the investment proposal.

 

Since this replication strategy has identical cash flows and identical risk as the investment proposal, its value also has to be identical.

  

More formally, we have seen that we can compute present values by reversing the calculation of future values. We have looked for the present value (PV0) that grows to 1'100 in one year (C1), if invested at the opportunity rate of return of 10% (R):

 

\( C_1 = PV_0 \times (1+R)^1 \)

 

When we solve this equation for PV0, we get:

 

\( PV_0 = \frac{C_1}{(1+R)^1} = \frac{1'100}{1.1} = 1'000 \)

 

More generally speaking, for any cash flow that occurs at time t (Ct), the present value at time 0 can be computed as follows:

 

\( \bf{PV_0 = \frac{C_t}{(1+R)^t}} \)
 

where R is the rate of return that we could earn on an alternative asset with identical risk.

  

Example 2

Suppose an investment proposal promises a payment of 25'000 in 5 years. An alternative investment with identical risk yields an annual return of 5%. What's the present value of this investment proposal?

To find the answer, we can use the above equation:

 

\( PV_0 = \frac{C_t}{(1+R)^t} = \frac{C_5}{(1+R)^5} = \frac{25'000}{1.05^5} = 19'588 \)

 

The current value of the investment proposal is 19'588. Why exactly 19'588? Because today, you could take the 19'588.15 and invest them at 5% for 5 years in the alternative asset. At the end of that investment period, you would have exactly 25'000:

 

\( 19'588 \times 1.05^5 = 25'000 \)

 

Consequently, the maximum you are willing to pay for the investment proposal is 19'558.

 

19'558 therefore is the fair value of the investment proposal.

  • If the counterparty asks for a higher price (e.g., 21'000), you are better off investing in the alternative asset.

  • In contrast, if the counterparty asks for a lower price (e.g., 18'000), the investment proposal is attractive, as it generates a higher return than the alternative asset (at the same risk).

 

As we will discuss at the end of this section, the actual price of an investment proposal should be close to its fair value, at least in competitive markets, where there are many potential buyers and sellers of such proposals.

   

Terminology

The crucial element of the equation is the "interest rate" R. This rate of return comes with many names, which we use as synonym. In particular:

  • Discount rate: The process of computing present values is generally referred to as discounting. The interest rate R is therefore also known as the discount rate. 
     
  • Opportunity rate: We have seen that R is defined as the rate of return which we can earn on an alternative investment with identical risk. The opportunity of our capital is therefore an investment at the rate R, hence the term opportunity rate.
     
  • Cost of capital: Similarly, by investing in a specific investment proposal, we forego investing the capital at the rate R. R is therefore the cost of the invested capital.