finance first test chapter 6
You have just won the lottery and will receive $1,500,000 in one year. You will receive payments for 30 years, and the payments will increase by 2.7 percent per year. If the appropriate discount rate is 6.8 percent, what is the present value of your winnings?
We can use the present value of a growing annuity equation to find the value of your winnings today. Doing so, we find: PV = C{[1/(r - g)] - [1/(r - g)] × [(1 + g)/(1 + r)]t} PV = $1,500,000{[1/(.068 - .027)] - [1/(.068 - .027)] × [(1 + .027)/(1 + .068)]30} PV = $25,280,038.83
Find the EAR in each of the following cases: Stated Rate Number of Times Effective Rate (APR) Compounded (EAR) 9% quarterly 16 monthly 12 daily 11 infinite
For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR/m)]m - 1 EAR = [1 + (.09/4)]4 - 1 = .0931, or 9.31% EAR = [1 + (.16/12)]12 - 1 = .1723, or 17.23% EAR = [1 + (.12/365)]365 - 1 = .1275, or 12.75% To find the EAR with continuous compounding, we use the equation: EAR = eq - 1 EAR = e.11 - 1 EAR = .1163, or 11.63%
First National Bank charges 13.1 percent compounded monthly on its business loans. First United Bank charges 13.4 percent compounded semiannually. As a potential borrower, which bank would you go to for a new loan?
For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR/m)]m - 1 So, for each bank, the EAR is: First National: EAR = [1 + (.1310/12)]12 - 1 = .1392, or 13.92% First United: EAR = [1 + (.1340/2)]2 - 1 = .1385, or 13.85% Notice that the higher APR does not necessarily result in the higher EAR. The number of compounding periods within a year will also affect the EAR.
You want to be a millionaire when you retire in 40 years. How much do you have to save each month if you can earn an annual return of 9.7 percent? How much do you have to save each month if you wait 10 years before you begin your deposits? 20 years?
Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is an APR of 9.7 percent, with monthly deposits. We must make sure to use the number of months in the equation. So, using the FVA equation: Starting today: 0 1 ... 480 I-----I-----I-----I-----I--- ----I----I------I----I------I $1,000,000 C C C C C C C C C FVA = C[{[1 + (.097/12) ]480 - 1}/(.097/12)] C = $1,000,000/5,774.1984 C = $173.18 Starting in 10 years: 0 1 ... 120 121 ... 480 I-----I--- I-----I-----I----I--- ----I----I------I $1,000,000 C C C C C C C FVA = C[{[1 + (.097/12) ]360 - 1}/(.097/12)] C = $1,000,000/2,120.8215 C = $471.52 Starting in 20 years: 0 1 ... 240 241 ... 480 I----I----I---- ----I------I----I-----I--- ---I------I $1,000,000 C C C C C FVA = C[{[1 + (.097/12) ]240 - 1}/(.097/12)] C = $1,000,000/730.4773 C = $1,368.97 Notice that a deposit for half the length of time, i.e., 20 years versus 40 years, does not mean that the annuity payment is doubled. In this example, by reducing the savings period by one-half, the deposit necessary to achieve the same ending value is about 8 times as large.
You want to have $60,000 in your savings account 12 years from now, and you're prepared to make equal annual deposits into the account at the end of each year. If the account pays 6.4 percent interest, what amount must you deposit each year?
The time line is: 0 1 ... 12 I-----I-----I-----I-----I-- ----I-----I-----I-----I----I $60,000 C C C C C C C C C Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the FVA equation: FVA = C{[(1 + r)t - 1]/r} $60,000 = $C[(1.06412 - 1)/.064] We can now solve this equation for the annuity payment. Doing so, we get: C = $60,000/17.269218 C = $3,474.39
A local finance company quotes an interest rate of 17.1 percent on one-year loans. So, if you borrow $20,000, the interest for the year will be $3,420. Because you must repay a total of $23,420 in one year, the finance company requires you to pay $23,420/12, or $1,951.67, per month over the next 12 months. Is the interest rate on this loan 17.1 percent? What rate would legally have to be quoted? What is the effective annual rate?
The time line is: 0 1 ... 12 I------I-----I-----I-----I----- ------I-----I-----I------I-----I------I $20, 1,951. 1,951. 1,951. 1,951 1,951. 1,951. 1,951. 1,951. 1,951. 1,951. 000 67 67 67 67 67 67 67 67 67 67 To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest rate quoted in the problem is only relevant to determine the total interest under the terms given. The interest rate for the cash flows of the loan is: PVA = $20,000 = $1,951.67{(1 - [1/(1 + r)12])/r} Again, we cannot solve this equation for r, so we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find: r = 2.516% per month So the APR that would legally have to be quoted is: APR = 12(2.516%) APR = 30.20% And the EAR is: EAR = (1.02516)12 - 1 EAR = .3475, or 34.75%
If you put up $41,000 today in exchange for a 5.1 percent, 15-year annuity, what will the annual cash flow be?
The time line is: 0 1 ... 15 I-----I-----I-----I-----I-- --I-----I-----I-----I----I $41,000 C C C C C C C C C Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity payment. Using the PVA equation: PVA = C({1 - [1/(1 + r) t]}/r) PVA = $41,000 = $C{[1 - (1/1.05115)]/.051} We can now solve this equation for the annuity payment. Doing so, we get: C = $41,000/10.30985 C = $3,976.78
You need a 30-year, fixed-rate mortgage to buy a new home for $235,000. Your mortgage bank will lend you the money at an APR of 5.35 percent for this 360-month loan. However, you can afford monthly payments of only $925, so you offer to pay off any remaining loan balance at the end of the loan in the form of a single balloon payment. How large will this balloon payment have to be for you to keep your monthly payments at $925?
The time line is: 0 1 ... 360 I------I-------I------I----I--- ----I------I------I-----I------I PV $925 $925 $925 $925 $925 $925 $925 $925 $925 The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present value of the $925 monthly payments is: PVA = $925[(1 - {1/[1 + (.0535/12)]360})/(.0535/12)] PVA = $165,647.80 The monthly payments of $925 will amount to a principal payment of $165,647.80. The amount of principal you will still owe is: $235,000 - 165,647.80 = $69,352.20 0 1 ... 360 I-----I-----I-----I-----I--- ---I-----I-----I-----I-------I $69,352.20 FV This remaining principal amount will increase at the interest rate on the loan until the end of the loan period. So the balloon payment in 30 years, which is the FV of the remaining principal will be: Balloon payment = $69,352.20[1 + (.0535/12)]360 Balloon payment = $343,996.22
You want to buy a new sports car from Muscle Motors for $57,500. The contract is in the form of a 60-month annuity due at an APR of 5.9 percent. What will your monthly payment be?
The time line is: 0 1 ... 59 60 I-----I------I-----I----I--- ----I-----I-----I-----I-----I -$57,500 C C C C C C C C C We need to use the PVA due equation, that is: PVAdue = (1 + r)PVA Using this equation: PVAdue = $57,500 = [1 + (.059/12)] × C[{1 - 1/[1 + (.059/12)]60}/(.059/12) $57,218.67 = $C{1 - [1/(1 + .059/12)60]}/(.059/12) C = $1,103.54 Notice, to find the payment for the PVA due we compound the payment for an ordinary annuity forward one period.
You want to buy a new sports coupe for $84,500, and the finance office at the dealership has quoted you an APR of 5.2 percent for a 60-month loan to buy the car. What will your monthly payments be? What is the effective annual rate on this loan?
The time line is: 0 1 ... 60 I-----I-----I-----I-----I--- ----I-----I-----I-----I----I $84,500 C C C C C C C C C We first need to find the annuity payment. We have the PVA, the length of the annuity, and the interest rate. Using the PVA equation: PVA = C({1 - [1/(1 + r)t] }/r) $84,500 = $C[1 - {1/[1 + (.052/12)]60}/(.052/12)] Solving for the payment, we get: C = $84,500/52.7343 C = $1,602.37 To find the EAR, we use the EAR equation: EAR = [1 + (APR/m)]m - 1 EAR = [1 + (.052/12)]12 - 1 EAR = .0533, or 5.33%
Given an interest rate of 5.3 percent per year, what is the value at date t=7 of a perpetual stream of $6,400 payments that begins at date t=15?
The time line is: 0 1 ... 7 ... 14 15 ... ∞ I-----I--- ----I---- ---I----I---------I---- -----I---------I PV $6,400 $6,400 $6,400 $6,400 To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using this equation we find: PV = $6,400/.053 PV = $120,754.72 0 1 ... 7 ... 14 I-----I----I---- ----I-----I----I----I---- -----I-----I PV $120,754.72 Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as: PV = $120,754.72/1.0537 PV = $84,121.21
One of your customers is delinquent on his accounts payable balance. You've mutually agreed to a repayment schedule of $450 per month. You will charge 1.3 percent per month interest on the overdue balance. If the current balance is $18,000, how long will it take for the account to be paid off?
The time line is: 0 1 ... ? I------I------I-----I----I-- -------I-----I-----I-----I-----I -$18,000$450$450$450$450 $450$450$450$450$450 Here we need to find the length of an annuity. We know the interest rate, the PVA, and the payments. Using the PVA equation: PVA = C({1 - [1/(1 + r)t] }/r) $18,000 = $450{[1 - (1/1.013)t]/.013} Now we solve for t: 1/1.013t = 1 - ($18,000/$450)(.013) 1/1.013t = .48 1.013t = 1/.48 = 2.083 t = ln 2.083/ln 1.013 t = 56.83 months
Live Forever Life Insurance Co. is selling a perpetuity contract that pays $1,250 monthly. The contract currently sells for $245,000. What is the monthly return on this investment vehicle? What is the APR? The effective annual return?
The time line is: 0 1 ... ∞ I-----I----------I------I-----I-- ---I------I------I------I------I -$245,000$1,250$1,250$1,250$1,250$1,250$1,250$1,250$1,250$1,250 Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash flows. Using the PV of a perpetuity equation: PV = C/r $245,000 = $1,250/r We can now solve for the interest rate as follows: r = $1,250/$245,000 r = .0051, or .51% per month The interest rate is .51% per month. To find the APR, we multiply this rate by the number of months in a year, so: APR = (12).51% = 6.12% And using the equation to find an EAR: EAR = [1 + (APR/m)]m - 1 EAR = [1 + .0051]12 - 1 EAR = 6.30%
What is the value today of $4,400 per year, at a discount rate of 8.3 percent, if the first payment is received 6 years from today and the last payment is received 20 years from today?
The time line is: 0 1 2 3 4 5 6 ... 20 I-----I-----I-----I-----I----I-----I------I--- -------I---------I PV $4,400 $4,400 $4,400 4,400 We want to find the value of the cash flows today, so we will find the PV of the annuity, and then bring the lump sum PV back to today. The annuity has 15 payments, so the PV of the annuity is: PVA = $4,400{[1 - (1/1.08315)]/.083} PVA = $36,981.52 Since this is an ordinary annuity equation, this is the PV one period before the first payment, so this is the PV at t = 5. To find the value today, we find the PV of this lump sum. The value today is: PV = $36,981.52/1.0835 PV = $24,822.33
Consider a firm with a contract to sell an asset for $145,000 four years from now. The asset costs $91,700 to produce today. Given a relevant discount rate of 11 percent per year, will the firm make a profit on this asset? At what rate does the firm just break even?
The time line is: 0 4 I----I----I------I-----I PV $145,000 The profit the firm earns is just the PV of the sales price minus the cost to produce the asset. We find the PV of the sales price as the PV of a lump sum: PV = $145,000/1.114 PV = $95,515.99 And the firm's profit is: Profit = $95,515.99 - 91,700 Profit = $3,815.99 To find the interest rate at which the firm will break even, we need to find the interest rate using the PV (or FV) of a lump sum. Using the PV equation for a lump sum, we get: 0 4 I-----I-----I-----I-----I -$91,700 $145,000 $91,700 = $145,000/(1 + r)4 r = ($145,000/$91,700)1/4 - 1 r = .1214, or 12.14%
Suppose an investment offers to triple your money in 12 months (don't believe it). What rate of return per quarter are you being offered?
The time line is: 0 4 I-------------------------------------------------------I $1 $3 Since we are looking to triple our money, the PV and FV are irrelevant as long as the FV is three times as large as the PV. The number of periods is four, the number of quarters per year. So: FV = $3 = $1(1 + r)(12/3) r = .3161, or 31.61%
After deciding to buy a new car, you can either lease the car or purchase it on a three-year loan. The car you wish to buy costs $43,000. The dealer has a special leasing arrangement where you pay $4,300 today and $505 per month for the next three years. If you purchase the car, you will pay it off in monthly payments over the next three years at an APR of 6 percent. You believe you will be able to sell the car for $28,000 in three years. Should you buy or lease the car? What break-even resale price in three years would make you indifferent between buying and leasing?
To answer this question, we should find the PV of both options, and compare them. Since we are purchasing the car, the lowest PV is the best option. The PV of leasing is the PV of the lease payments, plus the $4,300. The interest rate we would use for the leasing option is the same as the interest rate of the loan. The PV of leasing is: 0 1 ... 36 I------I-----I------I-----I----- ---------I-----I------I------I----I $4,300 $505 $505 $505 $505 $505 $505 $505 $505 505 PV = $4,300 + $505{1 - [1/(1 + .06/12)12(3)]}/(.06/12) PV = $20,899.86 The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV of the resale price is: 0 1 ... 36 I-----I-----I-----I-----I---- ----I----I-----I----I-------I $43,000 -$28,000 PV = $28,000/[1 + (.06/12)]12(3) PV = $23,398.06 The PV of the decision to purchase is: $43,000 - 23,398.06 = $19,601.94 In this case, it is cheaper to buy the car than leasing it since the PV of the purchase cash flows is lower. To find the break-even resale price, we need to find the resale price that makes the PV of the two options the same. In other words, the PV of the decision to buy should be: $43,000 - PV of resale price = $20,899.86 PV of resale price = $22,100.14 The break-even resale price is the FV of this value, so: Break-even resale price = $22,100.14[1 + (.06/12)]12(3) Break-even resale price = $26,446.80
Suppose you are going to receive $13,500 per year for five years. The appropriate interest rate is 6.8 percent. a. What is the present value of the payments if they are in the form of an ordinary annuity? What is the present value if the payments are an annuity due? b. Suppose you plan to invest the payments for five years. What is the future value if the payments are an ordinary annuity? What if the payments are an annuity due? c. Which has the highest present value, the ordinary annuity or the annuity due? Which has the highest future value? Will this always be true?
a. If the payments are in the form of an ordinary annuity, the present value will be: 0 1 2 3 4 5 I---------I----------I----------I----------I------------I $13,500 $13,500 $13,500 $13,500 $13,500 PVA = C({1 - [1/(1 + r)t]}/r)) PVA = $13,500[{1 - [1/(1 + .068)5]}/ .068] PVA = $55,650.35 If the payments are an annuity due, the present value will be: 0 1 2 3 4 5 I-------------I-----------I----------I-----------I------------I $13,500 $13,500 $13,500 $13,500 $13,500 PVAdue = (1 + r)PVA PVAdue = (1 + .068)$55,650.35 PVAdue = $59,434.57 b. We can find the future value of the ordinary annuity as: FVA = C{[(1 + r)t - 1]/r} FVA = $13,500{[(1 + .068)5 - 1]/.068} FVA = $77,325.75 If the payments are an annuity due, the future value will be: FVAdue = (1 + r) FVA FVAdue = (1 + .068)$77,325.75 FVAdue = $82,583.90 c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an annuity due will always be higher than the future value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding.
