# Finance

The correct formula for the calculation of the future value of a lump sum is PV×(1+r)nPV×1+rn, where PV is the present value of the cash flow, r is the opportunity cost rate or the interest rate (return) paid or received each period, and n represents the number of periods for which interest is earned.

PMTrPMTr is the equation used to calculate the present value of a perpetuity.

PMT×(1−1(1+r)(n)r×(1+r))PMT×1−11+rnr×1+r is the formula used to calculate the present value of an annuity due.

FV(1+r)nFV1+rn is the equation used to calculate the present value of a lump sum.

A schedule or table that reports the amount of principal and the amount of interest that make up each payment made to repay a loan by the end of its regular termAmortization schedule   A process that involves calculating the current value of a future cash flow or series of cash flows based on a certain interest rateDiscounting   The concept that states that the timing of the receipt or payment of a cash flow will affect its value to the holder of the cash flowTime value of money   A 6% return that you could have earned if you had made a particular investmentOpportunity cost   A cash flow stream that is created by an investment or loan that requires its cash flows to take place on the last day of each quarter and requires that it last for 10 yearsOrdinary annuity

The correct formula for the calculation of the present value of a lump sum is FV(1+r)nFV1+rn, where FV is the future value of the cash flow, r is the opportunity cost rate or the interest rate (return) paid or received each period, and n represents the number of periods for which interest is earned.

PMT×(1−1(1+r)(n)r)PMT×1−11+rnr is the formula for the calculation of a present value of an ordinary annuity.

PMTrPMTr is the equation used to calculate the present value of a perpetuity.

PV×(1+r)nPV×1+rn is the formula for calculating the future value of a lump sum.

The correct formula for the calculation of the future value of an ordinary annuity is PMT×((1+r)(n)−1r)PMT×1+rn−1r, where PMT is the amount of the constant cash flow received or paid each period, r is the opportunity cost rate or the interest rate (return) paid or received each period, and n represents the number of periods for which interest is earned.

FV(1+r)nFV1+rn is the equation used to calculate the present value of a lump sum.

PMT×((1+r)(n)−1r×(1+r))PMT×1+rn−1r×1+r is the formula for calculating the future value of an annuity due.

PMT×(1−1(1+r)(n)r)PMT×1−11+rnr is the formula for the calculation of a present value of an ordinary annuity.

Kenji should invest in Investment A, since its future value of \$42,500 is greater than that offered by either Investments B (\$37,591) or C (\$40,145). The assumption that Kenji is an economically rational investor means he will want to maximize the return, or future value, generated by his investments. What makes this situation unique is that an investment earning only simple interest—albeit at a higher interest rate—is able to outperform an account earning compound interest.

The future value of Investment A, which earns simple interest, is calculated as:

 FVAFVA = PV+(PV×r×n)PV+PV×r×n = \$25,000 + (\$25,000 x 0.10 x 7) = \$42,500

In contrast, the future values of Investments B and C, both of which earn compound interest, are calculated as:

 FVBFVB = PV×(1+r)nPV×1+rn = \$25,000×(1+0.06)7\$25,000×1+0.067 = \$37,591\$37,591 FVCFVC = \$25,000×(1+0.07)7\$25,000×1+0.077 = \$40,145\$40,145

Since the future value of Investment A is greater than those of Investments B and C, Kenji should select it instead of either Investments B or C.

Since you are depositing the same amount in the bank at the end of each year you can treat this cash flow as an ordinary annuity. When solving for the future value (FV) of an ordinary annuity make sure that your calculator is set to END mode. It is an annuity of eight years (n = 8) that has annual payments of \$5,000 (PMT = \$5,000). The interest rate is 9% (r = 9), and there is no present value (PV = 0). Perform the following calculations to find the future value of this annuity:

 FVAnFVAn =  = PMT×(1+r)n−1rPMT×1+rn−1r

Therefore:

 FVAnFVAn =  = \$5,000×(1+0.09)8−10.09\$5,000×1+0.098−10.09 =  = \$55,142\$55,142

Or, with a financial calculator: END mode

 Input 8 9 0 -5000 Keystroke N I/Y PV PMT FV Output 55142

Alternately, a spreadsheet can be used:

 1 A B C D 2 FV = FV(rate, nper, pmt, [pv], [type]) 3 = FV(0.09,8,-5000,0,0) = \$55,142 4

Please note that the [type] element in the Excel spreadsheet is a value representing the timing of payment: payment at the beginning of the period = 1; payment at the end of the period = 0 or omitted.

You’ve decided to deposit your money in the bank at the beginning of the year instead of the end of the year, but now you are making payments of \$5,000 at an annual interest rate of 9%. How much money will you have available at the end of eight years—rounded to the nearest whole dollar?

\$55,142

\$60,105

\$42,074

\$84,147

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Explanation:

Now you are making annual deposits at the beginning of each year, so you will need to treat these cash flows as an annuity due. To solve for an annuity due you need to set your calculator to BEGIN mode. You are making payments of \$5,000 (PMT = \$5,000) over eight years (n = 8) at an interest rate of 9% (r = 9). First solve for the future value of the ordinary annuity. Then, using this value, solve for the future value of the annuity due. Perform the following calculations to find the future value of this annuity:

 FVA(DUE)nFVA(DUE)n =  = PMT×(1+r)n−1r×(1+r)PMT×1+rn−1r×1+r

Solve as follows:

 FVA(DUE)nFVA(DUE)n =  = FVAn×(1+r)FVAn×1+r =  = \$55,142×(1+0.09)\$55,142×1+0.09 =  = \$60,105\$60,105

Or, with a financial calculator: BEGIN mode

 Input 8 9 0 -5000 Keystroke N I/Y PV PMT FV Output 60105

You can also use a spreadsheet to solve for an annuity due. Note that “type”, which is usually 0, or left blank, should now have the value of 1. That tells the spreadsheet that the payment happens at the beginning of the period.

 1 A B C D 2 FV = FV(rate, nper, pmt, [pv], [type]) 3 = FV(0.09,8,-5000,0,1) = \$60,105 4

You are planning to put \$10,000 in the bank at the end of each year for the next seven years in hopes that you will have enough money for a down payment on a condo. If you are investing at an annual interest rate of 5%, how much money will you have at the end of seven years—rounded to the nearest whole dollar?

\$65,136

\$85,491

\$97,704

\$81,420

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Explanation:

Since you are depositing the same amount in the bank at the end of each year you can treat this cash flow as an ordinary annuity. When solving for the future value (FV) of an ordinary annuity make sure that your calculator is set to END mode. It is an annuity of seven years (n = 7) that has annual payments of \$10,000 (PMT = \$10,000). The interest rate is 5% (r = 5), and there is no present value (PV = 0). Perform the following calculations to find the future value of this annuity:

 FVAnFVAn =  = PMT×(1+r)n−1rPMT×1+rn−1r

Therefore:

 FVAnFVAn =  = \$10,000×(1+0.05)7−10.05\$10,000×1+0.057−10.05 =  = \$81,420\$81,420

Or, with a financial calculator: END mode

 Input 7 5 0 -10000 Keystroke N I/Y PV PMT FV Output 81420

Alternately, a spreadsheet can be used:

 1 A B C D 2 FV = FV(rate, nper, pmt, [pv], [type]) 3 = FV(0.05,7,-10000,0,0) = \$81,420 4

Please note that the [type] element in the Excel spreadsheet is a value representing the timing of payment: payment at the beginning of the period = 1; payment at the end of the period = 0 or omitted.

You’ve decided to deposit your money in the bank at the beginning of the year instead of the end of the year, but now you are making payments of \$10,000 at an annual interest rate of 5%. How much money will you have available at the end of seven years—rounded to the nearest whole dollar?

\$85,491

\$59,844

\$119,687

\$81,420

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Explanation:

Now you are making annual deposits at the beginning of each year, so you will need to treat these cash flows as an annuity due. To solve for an annuity due you need to set your calculator to BEGIN mode. You are making payments of \$10,000 (PMT = \$10,000) over seven years (n = 7) at an interest rate of 5% (r = 5). First solve for the future value of the ordinary annuity. Then, using this value, solve for the future value of the annuity due. Perform the following calculations to find the future value of this annuity:

 FVA(DUE)nFVA(DUE)n =  = PMT×(1+r)n−1r×(1+r)PMT×1+rn−1r×1+r

Solve as follows:

 FVA(DUE)nFVA(DUE)n =  = FVAn×(1+r)FVAn×1+r =  = \$81,420×(1+0.05)\$81,420×1+0.05 =  = \$85,491\$85,491

Or, with a financial calculator: BEGIN mode

 Input 7 5 0 -10000 Keystroke N I/Y PV PMT FV Output 85491

You can also use a spreadsheet to solve for an annuity due. Note that “type”, which is usually 0, or left blank, should now have the value of 1. That tells the spreadsheet that the payment happens at the beginning of the period.

 1 A B C D 2 FV = FV(rate, nper, pmt, [pv], [type]) 3 = FV(0.05,7,-10000,0,1) = \$85,491 4

You found out that now you are going to receive payments of \$7,000 for the next 14 years. You will receive these payments at the beginning of each year. The annual interest rate will remain constant at 14%.

What is the present value of these payments? (Note: Round your answer to the nearest whole dollar.)

\$64,661

\$42,015

\$47,897

\$38,318

1. Amortized loans

Imagine you are finally able to buy your first classic sports car. To do so, you have arranged to borrow \$72,500 from your local savings and loan association. The interest rate on the loan is 5.00%. To simplify the calculations, assume that you will repay your loan over the next seven years by making annual payments at the end of each year. According to the loan officer at the savings and loan association, you must answer the following questions before you can go pick up your new car.

How much is the annual payment on your new car loan?

\$11,652.38

\$12,529.44

\$13,782.38

\$15,661.80

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Explanation:

The annual payment for your new \$72,500 classic car is \$12,529.44, which is calculated as:

 PVAordinaryPVAordinary =  = PMT×1−1(1+r)nrPMT×1−11+rnr \$72,500\$72,500 =  = PMT×1−1(1+0.05)70.05PMT×1−11+0.0570.05 PMTPMT =  = \$12,529.44\$12,529.44

This payment, which is made at the end of each year for seven years, contains both principal and interest.

How much of your Year 2 payment will constitute interest on your loan?

\$2,957.20

\$3,179.78

\$3,497.76

\$3,974.73

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Explanation:

The amount of interest included in Year 2’s annual payment is \$3,179.78. To calculate the amount of interest paid in Year 2, it is necessary to either construct an amortization table or simply complete the same calculations. For example, to calculate the loan’s ending balance at the end of Year 1, it is necessary to (1) determine the amount of interest contained within Year 1’s payment, and (2) calculate and deduct the remaining payment from the loan’s beginning balance. Year 1’s ending balance is the same as Year 2’s beginning balance. With this data, you can calculate the amount of interest contained in Year 2’s payment.

 Interest1Interest1 =  = Beginning Balance1×Interest RateBeginning Balance1×Interest Rate =  = \$72,500×0.05\$72,500×0.05 =  = \$3,625.00\$3,625.00

Therefore, the amount of interest included in Year 1’s payment is \$3,625.00. Deducting the payment’s interest component (\$3,625.00) from the amount of the payment (\$12,529.44) leaves \$8,904.44 available to reduce Year 1’s beginning balance, which is equal to the amount of the loan (\$72,500). Therefore, Year 1’s ending balance is \$63,595.56.

Again, as Year 2’s beginning balance equals Year 1’s ending balance, Year 2’s interest expense is calculated as:

 Interest2Interest2 =  = Beginning Balance2×Interest RateBeginning Balance2×Interest Rate =  = \$63,595.56×0.05\$63,595.56×0.05 =  = \$3,179.78\$3,179.78

How much of your Year 3 payment will be used to repay principal on the loan?

\$9,129.94

\$9,817.14

\$10,798.85

\$12,271.43

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Explanation:

Year 3’s principal repayment component of \$9,817.14 is calculated as:

 Repayment2Repayment2 =  = Payment2−Interest2Payment2−Interest2 =  = \$12,529.44−\$3,179.78\$12,529.44−\$3,179.78 =  = \$9,349.66\$9,349.66 Ending Balance2Ending Balance2 =  = Ending Balance1−Repayment2Ending Balance1−Repayment2 =  = \$63,595.56−\$9,349.66\$63,595.56−\$9,349.66 =  = \$54,245.90\$54,245.90 Beginning Balance3Beginning Balance3 =  = \$54,245.90\$54,245.90 Interest3Interest3 =  = \$54,245.90×0.05\$54,245.90×0.05 =  = \$2,712.30\$2,712.30 Repayment3Repayment3 =  = \$12,529.44−\$2,712.30\$12,529.44−\$2,712.30 =  = \$9,817.14\$9,817.14

How much will you pay in total interest to finance the purchase of your \$72,500 car?

\$14,141.65

\$15,206.08

\$16,726.69

\$19,007.60

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Explanation:

The total amount of interest paid over the life of your automobile loan is \$15,206.08. The total amount of interest paid on an amortized loan can be calculated by either summing the interest component of each year’s payment or by the following calculation:

 Total Interest PaidTotal Interest Paid =  = (Payment×Number of Payments)−Amount of LoanPayment×Number of Payments−Amount of Loan =  = (\$12,529.44×7 years)−\$72,500\$12,529.44×7 years−\$72,500 =  = \$15,206.08\$15,206.08

1. Present value

To find the present value of a cash flow expected to be paid or received in the future, you willmultiply   the future value cash flow by (1+r)n1+rn.

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Explanation:

The process of calculating the present value of one or more cash flows is called discounting. Remember that the equation used to discount future cash flows is the inverse of the equation used to compound present cash flows to calculate a future value. That is:

 FVnFVn =  = PV×(1+r)nPV×1+rn

When this equation is rearranged to solve for the present value variable (PV), the following equation results:

 PVPV =  = FVn(1+r)nFVn(1+r)n

What is the value today of a \$42,000 cash flow expected to be received four years from now based on an annual interest rate of 8%?

\$47,850

\$24,697

\$30,871

\$38,589

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Explanation:

The present, or today’s, value of a \$42,000 cash flow expected to be received four years in the future, given the expectation of receiving an interest rate of 8%, is \$30,871.

You can use the following equation to solve for the present value of an individual, or single, cash flow:

 PVPV =  = FVn(1+r)nFVn1+rn =  = \$42,000(1+0.08)4\$42,0001+0.084 =  = \$30,871\$30,871

Alternatively, you can solve the problem using your financial calculator by inputting the following variables and solving for the present value (PV):

 Input 4 8 0 42000 Keystroke N I/Y PV PMT FV Output 30871

Excel is another tool that can be used to solve for present value. The inputs are shown below:

 A B C D 1 2 PV = PV(rate, nper, pmt, [fv], [type]) 3 = PV(8%,4,0,42000,0) =\$30,871 4 5

Please note that the [type] element in the Excel spreadsheet is a value representing the timing of payment: payment at the beginning of the period = 1; payment at the end of the period = 0 or omitted.

Your broker called earlier today and offered you the opportunity to invest in a security. As a friend, he suggested that you compare the current, or present value, cost of the security and the discounted value of its expected future cash flows before deciding whether or not to invest.

The decision rule that should be used to decide whether or not to invest should be:

everything else being equal, you should invest if the current cost of the security is greater than the present value of the security’s expected future cash flows.

everything else being equal, you should invest if the discounted value of the security’s expected future cash flows is greater than or equal to the current cost of the security.

everything else being equal, you should invest if the present value of the security’s expected future cash flows is less than the current cost of the security.

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Explanation:

The decision to purchase a security should involve a comparison of the security’s current cost and the discounted, or present, value of its expected future cash flows. Remember, a critical requirement of a time value of money analysis is that all cash inflows and outflows will be valued at one point on the timeline, or within the same time period. In the majority of situations, this time period will be today (time zero), and all cash flows will be expressed in present value terms.

This means that the appropriate decision rule by which to determine whether to invest in a given situation is: Invest in the security if the present value of the security’s expected future cash flows is greater than or equal to the present value, or current cost, of the security, everything else being equal. Doing so will increase your wealth. Spending more than the security is worth, expressed in discounted terms, will reduce your wealth.

Now that you’ve thought about the decision rule that should be applied to your decision, apply it to the following security offered by your broker:

Jing Associates, LLC, a large law firm in Denver, is building a new office complex. To pay for the construction, Jing Associates is selling a security that will pay the investor the lump sum of \$10,250 in four years. The current market price of the security is \$8,674.

Assuming that you can earn an annual return of 5.25% on your next most attractive investment, how much is the security worth to you today?

\$8,353

\$8,771

\$9,606

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From strictly a financial perspective, should you invest in the Jing security?

Yes

No

Points:

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Why or why not?

Because the discounted value of the security’s future cash flows is greater than the cost of the security.

Because the cost of the security is greater than the discounted value of the security’s future cash flows.

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Close Explanation

Explanation:

The security offered by Jing Associates has a current value of \$8,353 and a current market value of \$8,674. Given these two values, you should not purchase the security because the cost of the security (\$8,674) is greater than the discounted value of the security’s future cash flows (\$8,353). Purchasing the security under these circumstances will decrease your wealth.

The current (discounted or present) value of Jing’s security is calculated as:

 PVPV =  = FVn(1+r)nFVn1+rn =  = \$10,250(1+0.0525)4\$10,2501+0.05254 =  = \$8,353\$8,353

You can also use a financial calculator to solve for the present value as follows:

 Input 4 5.25 0 10250 Keystroke N I/Y PV PMT FV Output 8353

Excel is another tool that can be used to solve for present value. The inputs are shown below:

 A B C D 1 2 PV = PV(rate, nper, pmt, [fv], [type]) 3 = PV(5.25%,4,0,10250,0) =\$8,353 4 5

Please note that the [type] element in the Excel spreadsheet is a value representing the timing of payment: payment at the beginning of the period = 1; payment at the end of the period = 0 or omitted.

This means that if Jing’s security is worth \$8,353 to you today, given the opportunity to earn 5.25% on your next available investment, and the cost of the security is \$8,674, then investing in the security will decrease your wealth by \$321 (\$8,353 – \$8,674).

1. Perpetuities

Perpetuities are also called annuities with an extended, or unlimited, life.

Which of the following are characteristics of a perpetuity? Check all that apply.

The value of a perpetuity is equal to the sum of the present value of its expected future cash flows.

A perpetuity is a stream of regularly timed, equal cash flows that continues forever.

The value of a perpetuity cannot be determined.

The current value of a perpetuity is based more on the discounted value of its nearer (in time) cash flows and less by the discounted value of its more distant (in the future) cash flows.

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Explanation:

perpetuity is a stream of equal payments that occur at regular and fixed intervals and that are expected to continue forever. A perpetuity’s present value is calculated as the sum of the discounted, or present value (PV), of each of its future cash flows. That is:

 PVPPVP =  = Payment per periodInterest rate per periodPayment per periodInterest rate per period

When performing this calculation, it is critical to match the timing of the cash flows with the compounding frequency of the interest rate. That is, a monthly cash flow should be discounted using a monthly interest rate, an annual cash flow should be discounted by an annual interest rate, and so on. It may be necessary to adjust the timing of the interest rate to make it consistent with the timing of the perpetuity’s cash flows.

A perpetuity’s returns may take the form of dividends (in the case of preferred stock) or interest payments (as in the case of perpetual bonds, such as consol bonds issued by the British government during the last several centuries). Although there are an infinite number of cash flows in a perpetuity, the present value of the nearer cash flows contributes greatly to its current value, while the more distant cash flows make a smaller—and ever decreasing—contribution to its current value. Nevertheless, a perpetuity’s current, or present, value reflects the sum of its discounted expected future cash flows.

Your grandfather wants to establish a scholarship in his father’s name at a local university and has stipulated that you will administer it. As you’ve committed to fund a \$25,000 scholarship every year beginning one year from tomorrow, you’ll want to set aside the money for the scholarship immediately. At tomorrow’s meeting with your grandfather and the bank’s representative, you will need to deposit\$363,636    (rounded to the nearest whole dollar) so that you can fund the scholarship forever—assuming that the account will earn 5.50% per annum every year.

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Oops! The bank representative just reported that he misquoted the available interest rate on the scholarship’s account. Your account should earn 3.50%. The amount of your required deposit should be revised to\$785,715   .

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This suggests there isa direct   relationship between the interest rate earned on the account and the present value of the perpetuity.

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Explanation:

Your grandfather’s scholarship is an example of a perpetuity, since it involves a series of regularly timed (annual), constant (\$25,000) payments to an unending number of student recipients. To ensure that sufficient money is available to meet this commitment, you must deposit \$454,545 into an account that is expected to earn 5.50% per year every year forever.

When the bank adjusts the interest rate to 3.50%, the present value of the scholarship is revised to \$714,286. These calculations indicate that there is an inverse relationship between the interest rate and the present value of a perpetuity.

Remember, the present value of the sum of the scholarship’s future payments is the amount that must be deposited today to ensure there is sufficient capital available when it is needed to make the unlimited number of annual scholarship payments. When performing this calculation, it is necessary to discount the annual scholarship payment using an annual interest rate. The required deposit, \$454,545, should be calculated as:

 PVP(Scholarship)PVP(Scholarship) =  = PaymentInterest ratePaymentInterest rate =  = \$25,0005.50%\$25,0005.50% =  = \$454,545\$454,545

When you learn that the bank misquoted the interest rate, and the actual rate applied to your account is 3.50%, the amount of the required deposit will be adjusted to \$714,286.

Since the interest rate variable is in the denominator, an increase in the interest rate will decrease the present value of the perpetuity or the required deposit, and a rate decrease will cause an increase in the present value variable. Assume you want to maintain a fixed level of withdrawals (equal to the amount of the annual scholarship) from a money pool created by only one deposit into the pool. If the interest rate causing the money pool to grow is small, then it requires a larger deposit in order to support those fixed scholarship payments. In contrast, a higher interest rate will allow greater growth in the money pool and permit a smaller initial deposit into the pool.

To calculate the present value of this perpetuity when the revised interest rate is 3.50%, solve the following equation:

 PVP(Scholarship)PVP(Scholarship) =  = PaymentInterest ratePaymentInterest rate =  = \$25,0003.50%\$25,0003.50% =  = \$714,286\$714,286

When the interest rate is 5.50%, the present value of the scholarship perpetuity is \$454,545. When the interest rate is revised to 3.50%, the adjusted present value of the perpetuity is \$714,286. These values indicate that there is an inverse relationship between the interest rate and the discounted value of the scholarship fund.

The future value and present value equations also help in finding the interest rate and the number of years that correspond to present and future value calculations.

If a security of \$8,000 will be worth \$12,029.04 seven years in the future, assuming that no additional deposits or withdrawals are made, what is the implied interest rate the investor will earn on the security?

4.50%

4.80%

6.00%

7.20%

If an investment of \$40,000 is earning an interest rate of 6.50% compounded annually, it will take     for this investment to grow to a value of \$54,803.47—assuming that no additional deposits or withdrawals are made during this time.

Which of the following statements is true, assuming that no additional deposits or withdrawals are made?

It takes 10.5 years for \$500 to double if invested at an annual rate of 5%.

It takes 14.2 years for \$500 to double if invested at an annual rate of 5%.