Time Value of Money Examples
Suppose the interest rate on a bank savings account is only 3%, half the rate in our earlier example. How much money will be in the account in five years, and how much growth does that represent on the initial investment?
By using a spreadsheet or financial calculator to solve Equation 3.1 using the inputs given here, we can quickly determine that the future value of a $100 investment for five years is $115.93. FV = $100 × (1 + 0.03)5 = $115.93 In this case, the investment grew by $15.93. Notice that while the interest rate used in this calculation was half the rate we used before (3% rather than 6%), the growth in the account's value was less than half of what it was before ($15.93 versus $33.82). This is the effect of compound interest. When the interest rate on an investment increases, the value of the investment rises at an increasing rate.
You wish to save money on a regular basis to finance an exotic vacation in five years. You are confident that, with sacrifice and discipline, you can force yourself to deposit $1,000 annually, at the end of each of the next five years, into a savings account paying 7% annual interest. This situation is depicted graphically at the top of Figure 3.8.
Compute the future value (FV) of this ordinary annuity, using Equation 3.3. Use the assumed interest rate (r) of 7% and plug in the known values of each of the five yearly (n = 5) cash flows (CF1to CF5), as follows: FV = CF1 × (1 + r)n−1 + CF2 × (1 + r)n−2 + ... + CFn × (1 + r)n−n FV = CF1 × (1 + r)5−1 + CF2 × (1 + r)5−2 + ... + CFn × (1 + r)5−5 = $1,000(1.07)4 + $1,000(1.07)3 + $1,000(1.07)2 + $1,000(1.07)1 + $1,000 = $1,310.80 + $1,225.04 + $1,144.90 + $1,070 + $1,000 = $5,750.74 The year 1 cash flow of $1,000 earns 7% interest for four years, the year 2 cash flow earns 7% interest for three years, and so on. The future value of the ordinary annuity is $5,750.74.
determine the rate of return on a particular investment or to calculate the rate of growth over time in a firm's sales or profits. Google, Inc., became a public company when it conducted an initial public offering (IPO) of common stock in August 2004. Originally priced at $85 per share, Google stock soared after the IPO. By August 2011, Google shares stood at $600. What annual rate of return did the investors who bought Google shares at the IPO and held them through August 2011 earn? Once again, start with Equation 3.1 FV = PV(1 + r)n
In this case we know FV = $600, PV = $85, and n = 7 years. Plug those values into Equation 4.1 and solve for r. $600 = $85(1 + r)7 $600 ÷ $85 = (1 + r)7 ($600 ÷ $85)(1÷7) = (1 + r) 1.322 = 1 + r r = 0.322 = 32.2%
How many years will it take you to triple your money? You saved $1,000, and you plan to put it into an investment earning 8% interest. To solve this problem, start with Equation 3.1: FV = PV(1 + r)n
In this case, we know PV = $1,000 and r = 0.08. We also know that FV = $3,000 because the goal is to triple the initial $1,000 investment. The unknown quantity is n, the number of years needed for $1,000 to grow to $3,000 if the interest rate is 8%. Therefore, we have: (If using a calculator PV and FV must have opposite signs) $3,000 = $1,000(1 + 0.08)n Dividing both sides by $1,000 leaves a simplified equation 3.0 = (1.08)n To solve this last expression algebraically, we take the natural logarithm of each side and then simplify terms as follows ln(3.0) = ln(1.08)n ln(3.0) = ln(1.08) ln(3.0) ÷ ln(1.08) = n 1.0986 ÷ 0.0770 = n 14.3 years = n
suppose an investment will pay you $300 one year from now. How much would you be willing to spend today to acquire this investment if you can earn 6% on an alternative investment of equal risk? To answer this question, you must determine how many dollars you would have to invest at 6% today in order to have $300 one year from now.
PV × (1 + 0.06) = $300 Solving this equation for PV gives us PV = __________ $300 -------------------------- (1+0.06) = $283.02 The present value of $300 one year from today is $283.02 in today's dollars. That is, $283.02 invested today at a 6% interest rate would grow to $300 at the end of one year. Therefore, today you would be willing to pay no more than $283.02 for an investment that pays you $300 in one year.
As in the previous example, you plan to save $1,000 at the end of each of the next five years to accumulate money for a vacation, and you expect to earn 7% on the money that you save. In addition, you just received a bonus at work, which gives you another $5,000 to invest immediately. How much can you accumulate in five years if you invest your bonus in addition to the $1,000 per year that you originally intended to save? Previously, we found that the future value of a $1,000 annuity invested over five years at 7% was $5,750.74. To that total, we now want to add the future value of a $5,000 lump sum invested immediately. Using Equation 3.1 we have:
$5,000(1 + 0.07)5 = $7,012.76 Adding that to the future value of the annuity gives you a total of $12,763.50 ($7,012.76 + $5,750.74) for your vacation. A quick way to solve this is to enter into Excel =fv(0.07,5,−1000,−5000,0). Notice here that you enter both the $1,000 deposits and the initial $5,000 lump sum as negative numbers. These inputs have the same sign because they both represent money flowing in the same direction (out of your wallet into your savings account).
You wish to save money on a regular basis to finance an exotic vacation in five years. You are confident that, with sacrifice and discipline, you can force yourself to deposit $1,000 annually, at the beginning of each of the next five years, into a savings account paying 7% annual interest. This situation is depicted graphically at the top of Figure 3.8.
Equation 3.5 demonstrates that the future value of an annuity due always exceeds the future value of a similar ordinary annuity (for any positive interest rate) by a factor of 1 plus the interest rate. We can check this by comparing the results from the two different five-year vacation savings plans presented earlier. We determined that, given a 7% interest rate, after five years the value of the ordinary annuity was $5,750.74, and that of the annuity due was $6,153.29. Multiplying the future value of the ordinary annuity by 1 plus the interest rate yields the future value of the annuity due: FV (annuity due) = $5,750.74 × (1.07) = $6,153.29
Equation 3.1 to make a wise decision when confronted with different options for borrowing money to purchase a consumer durable good. You are thinking of treating yourself to a nice graduation present, the latest and greatest laptop designed specifically for gamers. It has a huge screen with ultra fast graphics capabilities. Fully equipped, the laptop costs $3,000. The problem is that you do not have any money today. In a few months, you will be working full time, and by the end of the year you believe you will have enough money to purchase the laptop. Of course, you want it now. As a promotion, a local electronics retailer is offering a special deal. Consumers can either buy the laptop at a discount, paying just $2,700, or they can take the laptop home today and wait one year before paying the full asking price, $3,000. Though you don't have $2,700 in cash on hand, you could charge that amount to your credit card and pay 10% interest. Which is the better option—buying the laptop today for $2,700 and paying interest to the credit card company, or paying nothing today and writing a check to the retailer for $3,000 a year later? Once again, let's write down Equation 3.1 and plug in values that we know. You can spend $2,700 today and pay 10% interest for a year. In this case, we could write Equation 3.1 as follows:
FV = $2,700(1 + 0.10)1 = $2,970 Borrowing $2,700 today on your credit card will cost you $2,970 in one year. The second option is to pay nothing today and pay the retailer $3,000 at the end of the year. You save $30 by using your credit card and repaying the credit card company $2,970 next year rather than paying the retailer $3,000. Another way to frame this problem is to determine the implicit interest rate that the retailer is charging if you accept the offer to pay $3,000 in one year. The retailer is essentially lending you $2,700 today (the amount that you would be charged if you paid up front), but you have to pay the full price at the end of one year. In this case, Equation 3.1 looks like this: $3,000 = $2,700(1 + r)1 ($3,000 ÷ $2,700) − 1 = r 0.1111 = 11.11% = r Solving for r, the implicit interest rate charged by the retailer, we obtain a rate of 11.11%. If you can borrow at a rate of 10% using your credit card, then that is preferable to accepting the retailer's loan that carries a rate of 11.11%.
how to use Equation 3.1 to develop a simple measure of how Google performed as a company from 2004-2011. In 2004, the year of its IPO, Google generated total revenue of about $3.2 billion. Seven years later, the firm reported 2011 revenues of about $37 billion. What was the annual growth rate in Google's revenues during this period? Again we apply Equation 3.1, substituting the values that we know as follows
FV = PV(1 + r)n $37 = $3.2(1 + r)7 ($37 ÷ $3.2)(1÷7) = (1 + r) 1.419 = 1 + r r = 0.419 = 41.9% Notice here that we are still solving for r, just as we did in the previous example. In this case, the interpretation of r is a little different. It is not the rate of return (or the rate of interest) on some investment, but rather the compound annual growth rate between Google's 2004 and 2011 revenues. It is a simple measure of how fast the company was growing during this period
Jim and Mary Cummings just had their first child, and they want to begin saving for college expenses. They estimate they will need a college fund worth $100,000 to pay for four years of college costs. They plan to set aside $2,700 at the end of each of the next 18 years, investing the money to earn 8% interest. Given that plan, how long will it take Jim and Mary to accumulate the money they need?
Plug in $100,0000 for FV, 0.08 for r, and $2,700 for PMT and you obtain: 17.9 years
Equipment Rental, Inc., leases packaging machines to manufacturing firms. Their leases run for eight years, which corresponds to the useful life of the equipment. Part of their standard lease agreement dictates that at the end of the lease, Equipment Rental must remove the old equipment, and fulfilling that requirement costs Equipment Rental about $1,700 per machine. Managers at Equipment Rental want to know the present value of this cost so that they can add it to the list of up-front fees that they charge when they sign a new lease with a client. Assuming that the relevant discount rate is 8%, what is the present value of a $1,700 payment that occurs eight years in the future?
Substituting FV = $1,700, n = 8, and r = 0.08 into Equation 3.2 yields the following: PV = $1,700/(1 + 0.08)8 = $1,700/1.85093 = $918.46
You have an opportunity to invest $100 cash in a bank savings account that pays 6% annual interest. You would like to know how much money you will have at the end of five years.
Substituting PV = $100, r = 0.06, and n = 5 into Equation 3.1 gives the future value at the end of year 5: FV = $100 × (1 + 0.06)5 = $100 × (1.3382) = $133.82 Your account will have a balance of $133.82 at the end of the fifth year, so your investment grew by $33.82.
