Reading: Lesson 3 - Finding the Interest Rate/Number of Years & Perpetuities

4.3.A - Finding the Interest Rate/Number of Years & Perpetuities 

1. Finding the Interest Rate, I

1. We have used equations to find future and present values. Those equations have four variables, and if we know three of them, then we (or our calculator or Excel) can solve for the fourth. Thus, if we know PV, I, and N, we can solve for FV, or if we know FV, I, and N, we can solve to find PV. That’s what we did in the preceding two sections. Now suppose we know PV, FV, and N, and we want to find I. For example, suppose we know that a given security has a cost of $100 and that it will return $150 after 10 years. Thus, we know PV, FV, and N, and we want to find the rate of return we will earn if we buy the security. Here’s the solution:
   

       2. Finding the interest rate by solving the formula takes a little time and thought, but financial calculators and spreadsheets find the answer almost instantly. Here’s the calculator setup:    

       3. Enter N = 10, PV = −100, PMT = 0 (because there are no payments until the security matures), and FV = 150. Then, when you press the I/YR key, the calculator gives the answer, 4.14%. Notice that the PV is a negative value because it is a cash outflow (an investment) and the FV is positive because it is a cash inflow (a return of the investment). If you enter both PV and FV as positive numbers (or both as negative numbers), you will get an error message rather than the answer. 

       4. In Excel, the RATE function can be used to find the interest rate: =RATE(N,PMT,PV,FV). For this example, the interest rate is found as =RATE(10,0,−100,150) = 0.0414 = 4.14%.

2. Finding the Number of Years, N

1. We sometimes need to know how long it will take to accumulate a specific sum of money, given our beginning funds and the rate we will earn. For example, suppose we now have $500,000 and the interest rate is 4.5%. How long will it be before we have $1 million?
   

2. We need to solve for N, and we can use three procedures: a financial calculator, Excel (or some other spreadsheet), or by working with natural logs. As you might expect, the calculator and spreadsheet approaches are easier. Here’s the calculator setup:

3. Enter I/YR = 4.5, PV = −500000, PMT = 0, and FV = 1000000. We press the N key to get the answer, 15.7473 years. In Excel, we would use the NPER function: =NPER(I,PMT,PV,FV). Inserting data, we have =NPER(0.045,0,−500000,1000000) = 15.7473.

3. Perpetuities 

1. The previous sections examined the relationship between the present value and future value of a single payment at a fixed point in time. However, some securities promise to make payments forever. For example, preferred stock promises to pay a dividend forever. Another “forever” security originated in the mid-1700s when the British government issued some bonds that never matured and whose proceeds were used to pay off other British bonds. Because this action consolidated the government’s debt, the new bonds were called “consols.” The term stuck, and now any bond that promises to pay interest perpetually is called a consol, or a perpetuity. The interest rate on the consols was 2.5%, so a consol with a face value of ₤1,000 would pay ₤25 per year in perpetuity (₤ is the currency symbol for a British pound.
   
2. A consol, or perpetuity, is simply an annuity whose promised payments extend out forever. Since the payments go on forever, you can’t apply the step-by-step approach. However, it’s easy to find the PV of a perpetuity with the following formula:
   

3. This examples demonstrate an important point: When interest rates change, the prices of outstanding bonds also change, but inversely to the change in rates. Thus, bond prices decline if rates rise, and prices increase if rates fall. This holds for all bonds, both consols and those with finite maturities.