MindTap: Chapter 4: Time Value of Money
Present value of annuities
1. You got into a car accident and settled out of court for equal payments of $1,000 at the end of each year for the next seven years. If the annual interest rate stays constant at 5%, what is the value of these payments in today's dollars? (Note: Round your answer to the nearest whole dollar.) 1000 x (1- 1 divided by (1+0.05)^7) all divided by 0.05 = 5786 ANSWER 5786 ------------------------ 2. You found out that now you are going to receive payments of $7,500 for the next 16 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.) PVAn= 7500 x (1- 1 divided by (1+0.14)^16) all divided by 0.14 =46988 PVA(DUE)n= 46988 x (1+0.14)= 53566 ANSWER 53566
Calculate annuity cash flows
1. Your goal is to have $7,500 in your bank account by the end of eight years. If the interest rate remains constant at 9% and you want to make annual identical deposits, what amount will you have to deposit into your account at the end of each year to reach your goal? 7500= PMT x ((1+0.09)^8 -1 divided by 0.09) ANSWER: PMT= 680.06 per year ------------------------ 2. If your deposits were made at the beginning of each year rather than an at the end, what is the amount your deposit would change by if you still wanted to reach your financial goal by the end of eight years? 7500= PMT x ((1+0.09)^8 -1 divided by 0.09) multiply by (1+0.09) PMT= 623.90 per year 623.90 - 680.06 = -56.16 ANSWER 56.16 ------------------------ FORMULA: difference in deposits= deopsit (annuity due) - deposit (ordinary annuity)
Present value
1. To find the present value of a cash flow expected to be paid or received in the future, you will DIVIDE the future value cash flow by (1+r)^n. ------------------------ 2. What is the value today of a $12,000 cash flow expected to be received nine years from now based on an annual interest rate of 8%? 12000 divided by (1+0.08)^9 = 6002.99 ANSWER: 6003 ------------------------ 3. 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 discounted value of the security's expected future cash flows is greater than or equal to the current cost of the security. ------------------------ 4. 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? 10250 divided by (1+0.0525)^4 = 8352.86 ANSWER: 8353 ------------------------ 5. From strictly a financial perspective, should you invest in the Jing security? NO ------------------------ 6. Why or why not? Because the discounted value of the security's future cash flows is greater than the cost of the security.
Future value of annuities II
1. Which of the following statements about annuities are true? Check all that apply. a. When equal payments are made at the end of each period for a certain time period, they are treated as an annuity due. FALSE b. A perpetuity is a series of equal payments made at fixed intervals that continue infinitely and can be thought of as an infinite annuity. TRUE c. When equal payments are made at the end of each period for a certain time period, they are treated as ordinary annuities. TRUE d. An ordinary annuity of equal time earns less interest than an annuity due. TRUE Explination: Annuities are defined as a series of equal payments at regular intervals either made, received, or both, for a certain number of periods. When a payment is made at the end of each period, it is treated as an ordinary annuity; when a payment is made at the beginning of each period, it is treated as an annuity due. A perpetuity is a series of payments made at fixed intervals that continue infinitely and can be thought of as an infinite annuity. Annuity due payments are made one period earlier than ordinary annuity payments, so they will earn interest for an additional period. Therefore, the value of an annuity due will be greater than the value of a similar ordinary annuity. To take this additional period of interest into account, multiply the value of the ordinary annuity by 1 plus the interest rate (1 + r) to find the value of the annuity due. ------------------------ 2. Which of the following is an example of an annuity? a. A lump-sum payment made to a life insurance company that promises to make a series of equal payments later for some period of time. b. An investment in a certificate of deposit (CD) ANSWER: a Explanation: You pay the life insurance company a single lump-sum amount as a payment toward your life insurance. The insurance company will, in return, pay interest on the lump-sum amount and will make regular payments for a stated time period. Thus, a life insurance policy is an annuity product. An investment in a CD would return a lump-sum payment, which includes the investment amount and the interest earned. ------------------------ 3. Ana had a high monthly food bill before she decided to cook at home every day in order to reduce her expenses. She starts to save $750 every year and plans to renovate her kitchen. She deposits the money in her savings account at the end of each year and earns 8% annual interest. Ana's savings are an example of an annuity. If Ana decides to renovate her kitchen, how much would she have in her savings account at the end of 8 years, rounded to the nearest whole dollar? 750 x [(1+0.08)^8 -1] over 0.08 = 7977.47 ANSWER: 7977 ------------------------ 4. If Ana deposits the money at the beginning of every year and everything else remains the same, she will save___ at the end of 8 years? 7977 x (1+0.08) = 8615.16 ANSWER 8615
Uneven cash flows
1. A series, or stream, of cash flows may not always necessarily be an annuity. Cash flows can also be uneven and nonconstant, but the concept of the time value of money applies to uneven cash flows as well. Consider the following case: Swanky Beverage Co. expects the following cash flows from its manufacturing plant in Palau over the next 5 years: year:annual cash flows 1:3,300,000 2:3,350,000 3:4,250,000 4:4,000,000 5:5,150,000 The CFO of the company believes that an appropriate annual interest rate on this investment is 8.5%. What is the present value of this uneven cash flow stream (rounded to the nearest whole dollar)? NVP5= NVP(0.0850,3300000, 3350000, 4250000, 4000000,5150000)= ANSWER: 15,525,790 ------------------------ 2. Identify whether the situations described in the following table are examples of uneven cash flows or annuity payments. a. Antonio has been donating 10% of his salary at the end of every year to charity for the last three years. His salary increased by 15% every year in the last three years. UNEVEN CASH FLOW b. You deposit a certain equal amount of money every year into your pension fund. ANNUITY PAYMENTS c. Charles receives quarterly dividends from his investment in a high-dividend yield, index exchange-traded fund. UNEVEN CASH FLOW d. Gilberto borrowed some money from his friend to start a new business. He promises to pay his friend $2,650 every year for the next five years to pay off his loan along with interest. ANNUITY PAYMENTS
Semiannual and other compounding periods
Semiannual compounding occurs when interest is compounded twice a year which would be the same thing as saying interest is compounded every six months. Quarterly compounding occurs when interest is compounded four times a year which would be the same thing as saying interest is compounded every three months. Monthly compounding occurs when interest is compounded 12 times a year which would be the same thing as saying interest is compounded every month. ------------------------ 1. Monthly compounding implies that interest is compounded (# times) per year. ANSWER: 12 times ------------------------ 2. You have deposited $10,800 into an account that will earn an interest rate of 8% compounded semiannually. How much will you have in this account at the end of fourteen years? STEP 1: 0.08/2 = 0.04 STEP 2: 14 x 2= 28 STEP 3: 10800 x (1+0.04)^28 = 32386 ANSWER: 32386
Finding the interest rate and the number of years
1. Interest Rate= RATE(nper, pmt, pv, [fv], [type], [guess]) 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 $6,000 will be worth $7,558.27 three 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? USE EXCEL =RATE(3,0,-6000,7558.27,0) ANSWER: 8% ------------------------ 2. N= NPER(rate, pmt, pv, [fv], [type]) If an investment of $50,000 is earning an interest rate of 8.50% compounded annually, it will take (# YEARS) for this investment to grow to a value of $96,030.22—assuming that no additional deposits or withdrawals are made during this time. USE EXCEL =NPER(0.085,0,-50000,96030.22,0) ANSEWR: 8 YEARS ------------------------ 3. FV= FV(rate, nper, pmt, [pv], [type]) Which of the following statements is true, assuming that no additional deposits or withdrawals are made? A. If you invest $5 today at 15% annual compound interest for 82.3753 years, you'll end up with approximately $100,000. B. If you invest $1 today at 15% annual compound interest for 82.3753 years, you'll end up with approximately $100,000. USE EXCEL A.=FV(0.15,82.3753,0,-5,0) = 500,003.23 B.=FV(0.15,82.3753,0,-1,0)= 100,000.65 ANSWER: B
Non annual compounding period
1. The number of compounding periods in one year is called compounding frequency. The compounding frequency affects both the present and future values of cash flows. An investor can invest money with a particular bank and earn a stated interest rate of 6.60%; however, interest will be compounded quarterly. Nominal Rate= 6.60% Periodic Rate= 1.65% Effective Annual Rate= 6.77% ------------------------ Periodic Interest Rate: 6.60% per year / 4 quarters per year= 1.65% per quarter ------------------------ Effective Annual Rate: rEAR= (1+ 0.0660 OVER 4) ^4 -1 =6.77% ------------------------ 2. You want to invest $15,000 and are looking for safe investment options. Your bank is offering a certificate of deposit that pays a nominal rate of 6.00% that is compounded quarterly. Your effective rate of return on this investment is... 6.14% ------------------------ Effective Rate: (1+0.0600 over 4)^4 -1 =0.0614 = 6.14% ------------------------ 3. Another bank is also offering favorable terms, so Felix decides to take a loan of $15,000 from this bank. He signs the loan contract at 7.80% compounded daily for nine months. Based on a 365-day year, what is the total amount that Felix owes the bank at the end of the loan's term? (Hint: To calculate the number of days, divide the number of months by 12 and multiply by 365.) $15904.89 ------------------------ Periodic Rate 0.0780/ 365= 0.000214 Ending Loan Amount 15000 x (1+0.000214)^273.7500 = 15904.89 NOTE: to get the 273.7500 you compute # months given /12 months per year) x 365 days per year so it was 9 months/12 months per year) x 365 days per year= 273.7500
Future value of annuities I
1. To compute the value of an annuity due, multiply the value of the ordinary annuity by (1+I) Explanation: Annuity due payments are made one period earlier than ordinary annuity payments, so they will earn interest for an additional period. Therefore, the value of an annuity due will be greater than the value of a similar ordinary annuity. To take this additional period of interest into account you will multiply the value of the ordinary annuity by one plus the interest rate (1 + r) to find the value of the annuity due. ------------------------ 2. You are planning to put $8,000 in the bank at the end of each year for the next eight years in hopes that you will have enough money for a new boat. If you are investing at an annual interest rate of 6%, how much money will you have at the end of eight years—rounded to the nearest whole dollar? FVAn= 8000 x [(1+0.06)^8 - 1] over 0.06 FVAn= 79179.74 Answer: $79,180 Explination: 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 $8,000 (PMT = $8,000). The interest rate is 6% (r = 6), and there is no present value (PV = 0). ------------------------ 3. 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 $8,000 at an annual interest rate of 6%. How much money will you have available at the end of eight years—rounded to the nearest whole dollar? FVA(DUE)n= 79180 x (1+0.06) FVA(DUE)n= 83930.9 ANSWER: $83,931 Explination: 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 $8,000 (PMT = $8,000) over eight years (n = 8) at an interest rate of 6% (r = 6). First solve for the future value of the ordinary annuity. Then, using this value, solve for the future value of the annuity due.