# CH 5, 7, 8, and 9 HW

Heather deposited $1,700 at her local credit union in a savings account at the rate of 9.8% paid as simple interest. She will earn interest once a year for the next 13 years. If she were to make no additional deposits or withdrawals, how much money would the credit union owe Heather in 13 years?

$3,865.80 Heather earns simple interest on her savings account, which means that she will earn interest once every period on only the account's initial investment. Interest is not earned on any previously-earned interest. This means that Heather's $1,700 initial investment will earn $166.60 of interest (9.8% x $1,700) each year for the next 13 years. Calculate the future value (FV) of the investment in 13 years as follows: I=Interest Earned in One Year x Number of Years =(9.8% x $1,700.00) per year x 13 years =$2,165.80 FVn=Initial Investment (PV0) + Total Interest Earned (I) FV13=$1,700.00 + $2,165.80 =$3,865.80

Now, assume that Heather's credit union pays a compound interest rate of 9.8% compounded annually. All other things being equal, how much will Heather have in her account after 13 years?

$5,731.65 In this case, Heather earns compound interest paid annually on the money she deposited at the credit union, which means that interest is computed each period on the principal invested and the interest earned in previous periods. She deposits $1,700 initially, which means the beginning balance of her account for the first year is $1,700. She will earn $1,700 x 9.8% = $166.60 in interest in the first year. The balance in her account at the beginning of the second year will become $1,866.60 ($1,700 + $166.60). Heather will now earn interest for the second year based on the balance at the beginning of the second year. Thus, she will earn $182.93 ($1,866.60 x 9.8%) in interest in the second year. The interest earned in the second year, $182.93, is more than the interest earned in the first year, $166.60. This is due to the effect of compounding. This process continues for each period, and interest earned on interest is called compound interest. This process will continue for 13 years. Because the beginning balance each year increases, the interest earned each year will continue to increase. Therefore, the future value of the deposits will be the sum of the principal deposited for a certain time period plus the interest that is paid by the credit union and added at the end of each year to the principal amount. You can calculate the future value (FV) using this equation: FV(subscript)N=PV x (1 + I)^N That is: FV13=$1,700 x (1 + 0.0980)^13 => =$1,700 x (1.098)^13 =$5,731.65 When using a financial calculator, the initial investment is $1,700 (PV = 1,700), the interest rate is 9.8% (I = 9.8), and money is invested for 13 years (N = 13). Note that the answer will show a negative sign; this merely represents whether money is coming in or going out. If you input the present value (PV) as a negative number, then the future value (FV) will become a positive number, and vice versa. You can perform the calculation as follows: Input: N=13, I/YR=9.8, PV=-1,7000, PMT=0 Output: FV => 5,731.65 Notes: •Calculators should be set at one payment each year; that is, P/Y = 1. Some calculators are preset assuming 12 payments are made; that is, P/Y = 12. If a problem requires you to use another P/Y setting, you will be prompted specifically to use that setting, but you should change back to P/Y = 1 immediately once you are finished. •Always keep your calculator in END mode. If a problem requires you to use BEGIN mode, you will be prompted specifically to use that mode, but change back to END mode immediately once you are finished. Excel may also be used to solve the problem, and the Excel inputs are as follows: FV= FV(rate, nper, pmt, [pv], [type]) = FV(9.8%, 13, 0, -1,700, 0) = $5,731.65

Before deciding to deposit her money at the credit union, Heather checked the interest rates at her local bank as well. The bank was paying a nominal interest rate of 9.8% compounded quarterly. If Heather had deposited $1,700 at her local bank, how much would she have had in her account after 13 years?

$5,985.09 You can solve this problem in two ways: 1.Break the investment period down to quarterly periods and use the periodic interest rate and use it to calculate the future value (FV), or 2.Compute the account's effective annual rate ('EFF'%) (also called the equivalent annual rate (EAR)) and use it to calculate the account's future value (FV). In this case, the compounding period changed from annually to quarterly. Using the first technique, in 13 years, there are 52 quarters (13 years x 4 quarters per year), which represents the number of interest-earning periods (N = 52) for this problem. The periodic interest rate, or the interest rate paid per interest-earning period, is 2.4500% (9.8% / 4), and the initial investment is still $1,700 (PV = -1,700). Remember, there is no intermediate payment (PMT = 0). Note: The precision of your calculation will depend on the number of decimal places set in your calculator. It is standard practice to set your calculator to four decimal places. Input: P/Y=1, N=52, I=2.4500, PV=-1,700, PMT=0 Output: FV => 5,985.09 Therefore, Heather would have a total of $5,985.09 in her account at the end of 52 quarters (13 years). Using the second solution method, the interest rate quoted in the question is called the nominal interest rate, also called the annual percentage rate (APR). But when the compounding period changes from yearly to something else, the actual interest rate to be used in the computation also needs to change—and is called the effective annual rate (EFF%), or the equivalent annual rate (EAR). Use the following formula to calculate the EAR: EFF%=(1 + I(sub)NOM / M)^M - 1 In this formula, M is the number of compounding periods per year. Therefore, the EAR or %eff can be calculated as follows: EFF%=(1 + I(sub)NOM / 4)^4 - 1 =(1 + 0.0980 / 4)^4 - 1 =0.10166068, or 10.1661% Then, you can use the EAR as the interest rate in the FV formula and solve as follows: FV(sub)N = = PV × (1 + I)^N FV13 = $1,700×(1+0.10166068)13 = $1,700×(1.101661)^13 = $5,985.09 You can also use the EAR as the interest rate in your financial calculator and solve as follows: Input: P/Y=1, N=13, I=10.1661, PV=-1,700, PMT=0 Output: FV => 5,985.09 Notice that regardless of the method used, the solution is same.

opportunity cost of funds

A 6% return that you could have earned if you had made a particular investment. The interest rate that represents the return on an investor's best available alternative investment of comparable (equal) risk is the investor's opportunity cost of funds.

annuity due

A cash flow stream that is created by a lease that requires the payment to be paid on the first of each month and a lease period of three years. An annuity due is the name given to a series of equal cash flows that occur at the beginning of each of the equally spaced intervals (such as daily, monthly, annually, and so on).

ordinary annuity

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 years. A series of equal cash flows that are paid or received at regular intervals, such as a day or a month, is called an annuity. When the cash flows occur at the end of each of the regular intervals, the series is called an ordinary annuity. An example of an ordinary annuity is the 60 monthly payments of $676.65 made at the end of each month to repay a $35,000 loan that charges 6% interest and is to be repaid over five years. If the cash flow were to occur at the beginning of each of the regular intervals, then the annuity would be called an annuity due.

perpetuity

A cash flow stream that is generated by a share of preferred stock that is expected to pay dividends every quarter indefinitely. A perpetuity is a series of equal cash flows that are expected to continue forever. A perpetuity can be considered to be a special type of annuity. While both a perpetuity and an annuity exhibit constant periodic cash flows, the annuity has a definite end date, and the perpetuity does not. Instead, a perpetuity's cash flows are expected to continue indefinitely.

amortized loan

A loan in which the payments include interest as well as loan principal. An amortized loan is one that is repaid with payments that are composed of both the interest owed on the loan and a portion of the loan's principal. In contrast, a zero-interest loan is one on which interest is not charged and the payments made to repay the loan will consist only of principal.

discounting

A process that involves calculating the current value of a future cash flow or series of cash flows based on a certain interest rate. Discounting is the process of calculating the present value of a cash flow to be received or paid in the future. Compounding, which is the process of determining the future, or terminal, value of a current cash flow, is the opposite of discounting.

amortization schedule

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 term. An amortization schedule or table 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 term. Remember, the term amortization has two meanings. One meaning refers to the process of decreasing the principal outstanding on a loan via payments containing both interest and principal. The second meaning refers to the depreciation of the intangible assets owned by a firm.

annual percentage rate

A value that represents the interest paid by borrowers or earned by lenders, expressed as a percentage of the amount borrowed or invested over a 12-month period. The annual percentage rate (APR) is the cost of borrowed funds as quoted by lenders and paid by borrowers, in which the interest required is expressed as a percentage of the principal borrowed. This rate does not reflect the effects of compounding if interest is earned more than once per year.

Nicholai is willing to invest $35,000 for six years, and is an economically rational investor. He has identified three investment alternatives (A, B, and C) that vary in their method of calculating interest and in the annual interest rate offered. Since he can only make one investment during the six-year investment period, complete the following table and indicate whether Nicholai should invest in each of the investments. Note: When calculating each investment's future value, assume that all interest is earned annually. The final value should be rounded to the nearest whole dollar. Investment A Interest Rate and Method: 9% simple interest Expected Future Value: ? Make this investment? Y/N Investment B Interest Rate and Method: 4% compound interest Expected Future Value: ? Make this investment? Y/N Investment C Interest Rate and Method: 6% compound interest Expected Future Value: ? Make this investment? Y/N

A: $53,900; Yes B: $44,286; No C: $49,648; No Nicholai should invest in Investment A, since its future value of $53,900 is greater than that offered by either Investments B ($44,286) or C ($49,648). The assumption that Nicholai 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: FV(sub)A = PV+(PV × I × N) = $35,000+($35,000 × 0.09 × 6years) = $35,000+$18,900 = $53,900 In contrast, the future values of Investments B and C, both of which earn compound interest, are calculated as: FV(sub)B = = PV × (1+I)^N = $35,000 × (1+0.04)^6 = $35,000 × 1.2653 = $44,286 Since the future value of Investment A is greater than those of Investments B and C, Nicholai should select it instead of either Investments B or C.

Eric wants to invest in government securities that promise to pay $1,000 at maturity. The opportunity cost (interest rate) of holding the security is 5.40%. Assuming that both investments have equal risk and Eric's investment time horizon is flexible, which of the following investment options will exhibit the lower price?

An investment that matures in seven years To find the lower-priced investment option, find the value of the investments today. Using the PV equation, calculate the present value of both investments as follows: PV6Years= $1,000/(1+0.054)^6 = $729.38 PV7Years= $1,000/(1+0.054)^7 = $692.02 Using a financial calculator, enter the following data: future value as $1,000 (FV = -1,000), interest rate as 5.40% (I = 5.40), and investment period as six years (N = 6). Perform the calculation as follows: Input: P/Y=1, N=6, I=5.40, FV=-1,000, PMT=0 Output: PV => 729.38 Now, keep all other values the same and change the time to seven years: Input: P/Y=1, N=7, I=5.40, FV=-1,000, PMT=0 Output: PV => 692.02 You can also solve this using a spreadsheet: PV6Years= PV(rate, nper, pmt, [fv], [type]) = PV(0.054, 6, 0, -1000) = $729.38 PV5.40%= PV(0.054, 7, 0, -1000) = $692.02 Because both investments have equal risk and Eric has a flexible time horizon, an investment with a longer maturity will have a lower present value.

Assume that the variables I, N, and PV represent the interest rate, investment or deposit period, and present value of the amount deposited or invested, respectively. Which equation best represents the calculation of a future value (FV) using: Compound interest? Simple interest?

Compound interest: FV = PV x (1 + I)^N Simple interest: FV = PV + (PV x I x N) In the United States, there are two frequently encountered methods for calculating interest: simple interest and compound interest. Both methods employ the same three variables—the amount of principal, the interest rate, and the investment or deposit period—to the amount deposited or invested in order to compute the amount of interest, but they differ in their relationships between the variables. For example, the equation that best represents the calculation of a future value (FV) using compound interest is FV = PV x (1 + I)^N. Notice the use of an exponent. This allows for the earning of interest on previously earned interest. This is not the case when calculating interest using the simple interest method. In contrast, the best equation for the calculation of a future value using simple interest is FV = PV + (PV x I x N). Notice this equation does not allow for the earning of interest on previously earned interest.

time value of money - basic concepts

Finance, or financial management, requires the knowledge and precise use of the language of the field.

time value of money - simple versus compound interest

Financial contracts involving investments, mortgages, loans, and so on are based on either a fixed or a variable interest rate. Assume that fixed interest rates are used throughout this question.

time value of money - present value

Finding a present value is the reverse of finding a future value.

future value

One of the four major time value of money terms; the amount to which an individual cash flow or series of cash payments or receipts will grow over a period of time when earning interest at a given rate of interest. A future value represents the amount to which a current (present) value will grow over a given period of time when compounded at a given rate of interest. Mathematically, a future value is calculated as FV = PV x (1 + r)^n.

Which of the following is true about present value calculations?

Other things remaining equal, the present value of a future cash flow decreases if the investment time period increases. You know that present value of future cash flows is calculated using the following equation: PV = FV(sub)N/(1 + I)^N As the investment time increases, the numerator stays the same, but the denominator increases, leading to a smaller value for PV. A protracted investment time period means that you will have to wait longer to receive the cash flow. The future value gets discounted more and more as the period of discounting increases, thus leading to a lower present value of the cash flow.

Time value of money calculations can be solved using a mathematical equation, a financial calculator, or a spreadsheet. Which of the following equations can be used to solve for the present value of a perpetuity?

PMT/r The correct formula for the calculation of the present value of a perpetuity is PMT/r, where PMT is the amount of the constant cash flow received or paid each period, and r is the opportunity cost or the interest rate (return) paid or received each period.

time value of money

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 flow. The financial concept that maintains that the timing of a receipt or payment of a cash flow will affect its value is called the time value of money (TVM). The time value of money illustrates that, due to its capacity to earn interest, a cash flow received today is worth more than an identical cash flow to be received on a future date. The exact current value of a future cash flow is a function of the magnitude of the future cash flow, the return required by the owner (recipient) of the cash flow, and when in the future the cash flow will occur.

time value of money - future value

The principal of the time value of money is probably the single most important concept in financial management. One of the most frequently encountered applications involves the calculation of a future value.

Which of the following investments that pay will $9,500 in 14 years will have a lower price today?

The security that earns an interest rate of 8.25%. To find the current value of the investment, calculate the present value of the security that pays $9,500 in 14 years at a given rate of return. The equation used to compute the present value (PV) of cash flows is: PV = FV(sub)N/(1 + I)^N Using the equation just given, calculate the present values of both investments: PV5.50%= $9,500/(1+0.06)^14 = $4,489.41 PV8.25%= $9,500/(1+0.08)^14 = $3,131.36 Or, using a financial calculator, enter the following data: future value as $9,500 (FV = -9,500), interest rate as 5.50% (I = 5.50), and investment period as 14 years (N = 14). Perform the calculation as follows: Input: P/Y=1, N=14, I=5.50, FV=-9,500, PMT=0 Output: PV => 4,489.41 Now, keep all other values the same and change the interest rate to 8.25%: Input: P/Y=1, N=14, I=8.25, FV=-9,500, PMT=0 Output: PV => 3,131.36 You can also solve this using a spreadsheet: PV5.50%= PV(rate, nper, pmt, [fv], [type]) = PV(0.06, 14, 0, -9500) = 4,489.41 PV8.25%= PV(0.08, 14, 0, -9500) = 3,131.36 The [type] element represents the timing of the payment in the spreadsheet. Leave the [type] element blank or use the values 0 or 1 based on the timing of the payment made. If the payment is made at the beginning of the period, [type] = 1; if the payment is made at the end of the period, [type] =0, or leave the [type] element blank. Because you are dealing with simple future values here, leave the [type] element blank.

Time Value of Money Concept Application: Does it matter whether I deposit my savings at the beginning of each month or at the end?

annuity due

Time Value of Money Concept Application: I want to make sure I don't run out of pocket money. How much should I save each month to have at least $500 at the beginning of next year?

annuity payments

The process for converting present values into future values is called _____. This process requires knowledge of the values of three of four time-value-of-money variables. Which of the following is not one of these variables?

compounding The inflation rate indicating the change in average prices Compounding is the term used to describe the process of computing a future value. It requires knowledge of three of the four time-value-of-money variables: •The present value (PV) of the amount invested •The interest rate (I) that could be earned by invested funds •The duration of the investment (N) The inflation rate indicating the change in average prices is not directly relevant to the calculation of a future value.

_____ is the process of calculating the present value of a cash flow or a series of cash flows to be received in the future.

discounting Discounting is the process of computing the present value of one or more future cash flows. Consider the time value of money: A dollar in hand today is worth more than a dollar tomorrow—if you have the opportunity to earn interest on your current funds. In contrast, compounding is the process of computing the future value of one or more cash flows.

Time Value of Money Concept Application: Do I prefer to use a credit card or a debit card for purchases? Does it matter which one I use?

effective interest rates

After the end of the second year and all other factors remaining equal, a future value based on compound interest will never exceed the future value based on simple interest. T/F

false After the end of the second year and all other factors remaining equal, a future value based on compound interest will never exceed the future value based on simple interest. One way to see that this statement is incorrect is to consider an example. Assume that an investor is considering depositing $25,000 into one of two accounts for a period of five years. Both accounts pay 7% interest (compounded annually) and differ only in that one calculates its interest using the simple-interest method and the other pays compounded interest. At the end of year 1, both accounts generate a future value of $26,750 ([$25,000 x (1.07)^1] for the compound-interest investment, and [$25,000 + ($25,000 x 0.07 x 1) for the simple-interest investment). By the end of year 2, however, the simple-interest investment will have a balance of $28,500 ($26,750 + ($25,000 x 0.07 x 1)), while the compound-interest investment will have a balance of $28,623 ($25,000 x (1.07)^2). The $123 difference is due to the interest earned on year 1's accrued interest ([$26,750 - 25,000) x 0.07] = $1,750 x 0.07).

Time Value of Money Concept Application: How much student debt will I bear by the time I graduate? What will my student loan payments be?

future value and payments

FV/(1 + r)^n

is the equation used to calculate the present value of a lump sum.

PV x (1 + r)^n

is the formula for calculating the future value of a lump sum.

PMT x ({1 - [1/(1 + r)^n]}/r)

is the present value of an ordinary annuity.

Time Value of Money Concept Application: I think I need a car to drive for my internship. If I took out a car loan, what would my loan payments be?

loan payments

Time Value of Money Concept Application: How much money do I need in an interest-bearing account today so that I have at least $1,000 in savings in 3 years?

present value

Investments and loans base their interest calculations on one of two possible methods: the _____ interest and the _____ interest methods. Both methods apply three variables—the amount of principal, the interest rate, and the investment or deposit period—to the amount deposited or invested in order to compute the amount of interest. However, the two methods differ in their relationship between the variables.

simple compound

All other factors being equal, both the simple interest and the compound interest methods will accrue the same amount of earned interest by the end of the first year. T/F

true All other factors being equal, both the simple interest and the compound interest methods will accrue the same amount of earned interest by the end of the first year. Because neither investment has accumulated interest by the end of the first year—because the simple interest investment can't accumulate this interest and the compound interest investment hasn't yet accumulated this interest—both investments will exhibit the same future value by the end of the first year. However, if both investments continue to earn interest into subsequent years, the future values earned by the two investments will diverge, assuming that the two investments are identical except in the method used to compute any earned interest.

The process of earning compound interest allows a depositor or investor to earn interest on any interest earned in prior periods. T/F

true The process of earning compound interest allows a depositor or investor to earn interest on any interest earned in prior periods. This is the definition of compound interest. The earning of interest on previously earned interest is the single biggest difference between the compound and simple methods of computing interest.