Reading: Lesson 2 - Present Values

4.2.A - Present Values

1. Discuss Present Values

1. Suppose you have some extra money and want to make an investment. A broker offers to sell you a bond that will pay a guaranteed $115.76 in 3 years. Banks are currently offering a guaranteed 5% interest on 3-year certificates of deposit (CDs), and if you don’t buy the bond you will buy a CD. The 5% rate paid on the CD is defined as your opportunity cost, or the rate of return you would earn on an alternative investment of similar risk if you don’t invest in the security under consideration. Given these conditions, what’s the most you should pay for the bond?
   
2. First, recall from the future value example in the last section that if you invested $100 at 5% in a CD, it would grow to $115.76 in 3 years. You would also have $115.76 after 3 years if you bought the bond. Therefore, the most you should pay for the bond is $100—this is its “fair price,” which is also its intrinsic, or fundamental, value. If you could buy the bond for less than $100, then you should buy it rather than invest in the CD. Conversely, if its price were more than $100, you should buy the CD. If the bond’s price were exactly $100, you should be indifferent between the bond and the CD.
   
3. The $100 is defined as the present value, or PV, of $115.76 due in 3 years when the appropriate interest rate is 5%. In general, the present value of a cash flow due N years in the future is the amount which, if it were on hand today, would grow to equal the given future amount. Because $100 would grow to $115.76 in 3 years at a 5% interest rate, $100 is the present value of $115.76 due in 3 years at a 5% rate. Finding present values is called discounting, and as previously noted, it is the reverse of compounding: If you know the PV, you can compound it to find the FV, or if you know the FV, you can discount it to find the PV. We will solve the formula for the future value, for the PV to produce the present value equation as follows.
   

4. The top section of the figure below shows inputs and a time line for finding the present value of $115.76 discounted back for 3 years. We first calculate the PV using the step- by-step approach. When we found the FV in the previous section, we worked from left to right, multiplying the initial amount and each subsequent amount by (1 + I). To find present values, we work backward, or from right to left, dividing the future value and each subsequent amount by (1 + I), with the present value of $100 shown in Cell D105. The step-by-step procedure shows exactly what’s happening, and that can be quite useful when you are working complex problems or trying to explain a model to others. However, it’s inefficient, especially if you are dealing with more than a year or two. A more efficient procedure is to use the formula approach by simply dividing the future value by (1 + I)N. This gives the same result in Cell G107. This is actually programmed into financial calculators by entering values for N = 3, I/YR = 5, PMT = 0, and FV = 115.76, and then pressing the PV key to get −100.

5. Excel also has a function that solves for the PV function, and it is written as =PV(I,N,0,FV).4 Cell E113 shows the inputs to this function. Next, Cell E114 shows the Excel function with fixed numbers as inputs, with the actual function and the resulting −$100 in Cell G114. Cell E115 shows the Excel function using cell references, with the actual function and the resulting −$100 in Cell G115.

6. As with the future value calculation, students often wonder why the result of the present value calculation is sometimes positive and sometimes negative. In the algebraic calculations in Rows 105 and 107, the result is +$100, while the result of the calculation using a financial calculator or Excel’s function in Rows 111 and 114 is –$100. Again, the answer is in the signs of a correctly constructed time line. Outflows are negative and inflows are positive. The PV function for Excel and a financial calculator answer the question “How much must I invest today in order to have available to me a certain amount of money in the future?” If you want to have $115.76 available in 3 years (an inflow to you, and therefore positive), then you must invest $100 today (an outflow, and therefore negative). If you use the algebraic functions as in Rows 105 and 107, you must keep track of whether the results of your calculations are inflows or outflows.

7. The fundamental goal of financial management is to maximize the firm’s intrinsic value, and the intrinsic value of a business (or any asset, including stocks and bonds) is the present value of its expected future cash flows.