Reading: Present Values

In the last step of this section, we show how to compute the present value of a string of future cash flows. Again, the basic logic is the same as in the case of future values. Using the principle of value additivity, we compute the present value of a cash flow string as the sum of the present value of each individual cash flows.

From before, we know that the present value of any future cash flow (C_t) is:

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

Consequently, the sum of the present value of each individual cash flow between time 0 and T is:

\( \bf{PV_0 = C_0 + \frac{C_1}{(1+R)}+\frac{C_2}{(1+R)^2}+\frac{C_3}{(1+R)^3}+...+\frac{C_T}{(1+R)^T}} \)

Using the symbol \( \sum \) to indicate a sum of a sequence of cash flows, we can write:

\( \bf{PV_0 = \sum_{t=0}^{T} \frac{C_t}{(1+R)^t}} \)

Example 6

An investment promises a cash flow of 5'000 in 2 years and a cash flow of 10'000 in 5 years. An alternative asset with identical risk generates an annual return of 8%. Based on this information, what is the present value of the investment proposal?

The following graph illustrates the investment proposal and it's valuation: 

In words:

• The first cash flow of 5'000 arrives in 2 years (C₂). Given the discount rate of 8%, it's present value is 4'286.69.
   
  
• Similarly, the second cash flow of 10'000 in year 5 (C₅) has a present value of 6'805.83.
   
  
• Using the principle of value additivity, the overall value of the investment proposal therefore is 11'092.52.

Why exactly is the value of the investment proposal 11'092.52? Because that is the amount of money we need to invest at the opportunity rate of 8% to exactly replicate the cash flows of the investment proposal (with identical risk).

Put differently, instead of taking the investment proposal, we could invest 11'092.52 in the alternative asset and thereby generate a cash flow of 5'000 in 2 years and another cash flow of 10'000 in 5 years. The following table illustrates this replication strategy (see also the accompanying Excel file):

Similar to what we did in the section on future values, the table shows the year-to-year development of an investment of 11'092.52 today at a rate of return of 8% and the relevant cash outflows:

• As per the investment proposal above, we withdraw 5'000 at the end of year 2 and 10'000 at the end of year 5.
   
• As the table shows, the balance of the account will be exactly 0 after the second withdrawal.
   
• Put differently, with the investment of 11'092.52 in the alternative asset, we can exactly replicate the cash flows of the investment proposal.

Since original investment proposal and the replicating portfolio promise the same cash flows and exhibit the same risk (as per our assumptions), their values must be identical. We would therefore recommend to go ahead with the investment proposal if its price is smaller than 11'092.52. Otherwise, we are better off investing in the alternative asset.

Example 7

An investment proposal promises a cash flow of 10'000 in 6 months and another cash flow of 10'000 in 1 year. The cost of capital is 10%. Based on this information, what is the present value of the project's future cash flows? 

The special element of this example is that the first cash flow occurs during the year, not at the end of it. However, that does not pose any valuation challenge: If the first cash flow already occurs after 6 months, we simply discount that cash flow over 6 months (= 0.5 years) instead of a full year. Consequently:

\( PV_0 = \frac{C_{0.5}}{(1+R)^{0.5}}+\frac{C_1}{(1+R)}=\frac{10'000}{1.1^{0.5}}+\frac{10'000}{1.1}= 18'625.53 \)

The present value of the project's future cash flows is 18'625.53.