Reading: Present Values

This section has shown how to compute the present value of an individual future cash flow as well as the present value of a string of future cash flows. The ability to compute present values  is an extremely valuable tool in the toolbox of any financial manager and investor.

The reason why it is so important to understand discounting is very simple: Investment decisions generally involve payouts today in exchange for future expected cash flows. To determine whether these future expected cash flows justify today's investment, we need to incorporate the time value of money and thereby make the cash flows comparable over time. That's what discounting does.

It is important to note that the computation of present values hinges of some rather strict assumptions. In particular:
 

• First, we need to estimate the discount rate.
   
  • The discount rate is the rate of return the investor can expect to earn if she puts her money in an alternative investment with identical risk.
     
  • Consequently, we need to be able to understand the riskiness of the investment project in question, and we need to understand how risk and return are related. These topics are covered extensively in other modules.

       

• Second, we have to assume well-functioning capital markets, on which investment opportunities can be traded (bought and sold) freely. Why is that?
   
  • Remember that, to determine the present value of an investment opportunity, we have constructed a replicating portfolio that generates identical future cash flows with identical risk.
     
  • If the investment proposal and the replicating portfolio have identical cash flows and identical riskiness, we have argued that their values must also be identical.
     
  • Otherwise, investors would immediately start trading to exploit the price difference. And they would keep trading until the difference between the fair value and the actual price disappears. The following example illustrates this behavior.

Example 8

Suppose an investment promises a cash flow of 11'000 in one year. For simplicity, let us assume that this cash flow is guaranteed (risk free) and that the risk free rate of return is 10%. From our considerations before, we know that the present value of the investment project is 10'000:

\( PV_0 = \frac{11'000}{1.1} = 10'000 \)

This is the fair value of the project. Therefore, we would expect the project to trade at 10'000. But what if its actual price differs? Let's consider two cases.

Case 1: The actual price is lower than the fair value

If the actual price is lower than the fair value, investors would buy the investment project. For the purpose of illustration, let us assume that the price is 9'000. Investors would benefit from the mispricing in the following way:

• First, they borrow 9'000 from a bank at the risk free rate of return of 10%
   
• Then they buy the project for 9'000 today and wait for one year.
   
• In one year, the project pays off a cash flow of 11'000, according to our assumptions.
   
• At the same time, investors have to repay the borrowed money. Given that they borrowed 9'000 at an interest rate of 10%, the amount due is 9'900.
   
• Consequently, in one year, the trading strategy generates a risk free profit of 11'000 - 9'900 = 1'100.

Note that the trading strategy is very clever. The investor does not need any money of her own! She borrows the full amount and then locks in a risk-free gain of 1'100. Such a trading strategy that involves other people's money and guarantees a risk-free profit is called Arbitrage.

Clearly, everybody would jump the bandwagon and start trading along the lines outlined above. As a result of the increased demand for the project, the price would go up until it reaches its fair value of 10'000. With that price movement, the mispricing disappears and so does the arbitrage opportunity.

Such Arbitrage considerations are a key pillar of finance theory because very often, asset prices are determined via replicating portfolios. Whenever we use such replicating portfolios, we factually make an arbitrage argument to support the determined price.

Case 2: The actual price is higher than the fair value

But what if the price of the investment proposal is lager than the fair value of 10'000? To illustrate, let us assume that the price is 11'000. How could investors benefit from this situation? Let us take a look:

• First, investors sell the proposal. Put differently, they guarantee the buying third party to make a payment of 11'000 in one year in exchange for a payment of 11'000 today. The technical term for selling an asset (or investment proposal) that the investor does not possess is a "Short sale."
   
• The investors would then take the proceeds from the sale (11'000 today) and invest them for one year at the risk free rate of return. 
   
• In 1 year, the risk free asset will grow to 11'000 × 1.1 = 12'100.
   
• With this money, the investors make the promised payment of 11'000 to the original buyer of the investment proposal and then pocket the difference of 1'100 [= 12'100 - 11'000] as an arbitrage profit.

Again, the trading strategy has produced a risk free profit that does not involve any of the investor's own money. As in the preceding case, such an arbitrage opportunity would attract other investors who would short sell the proposal and thereby exert downward pressure on its price.

This example illustrates how important the assumption of free tradability is for the computation of present values. Only if markets function well can we assume that the observed prices correspond to their fair value. If tradability is limited or inexistent, observed prices can and often do deviate significantly from their fair values.