MGMT 310 Final Exam Review

Your coin collection contains 53 1957 silver dollars. If your grandparents purchased them for their face value when they were new, how much will your collection be worth when you retire in 2048, assuming they appreciate at an annual rate of 12 percent?

$1,596,305.22 To find the FV of a lump sum, we use: FV = PV(1 + r)t FV = $53(1.12)91 FV = $1,596,305.22

If you put up $30,000 today in exchange for a 8.75 percent, 17-year annuity, what will the annual cash flow be?

$3,455.18 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 = $30,000 = $C{[1 − (1/1.0875)17 ]/.0875} We can now solve this equation for the annuity payment. Doing so, we get: C = $30,000/8.68261C = $3,455.18

Prescott Bank offers you a $22,000, 6-year term loan at 12 percent annual interest. What will your annual loan payment be?

$5,350.97 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)$22,000 = C{[1 − (1/1.12^6) ]/.12} We can now solve this equation for the annuity payment. Doing so, we get: C = $22,000/4.11141C = $5,350.97

What is the IRR of the following set of cash flows? YearCash Flow 0 −$ 11,164 1 6,800 2 4,000 3 6,800

26.94% The IRR is the interest rate that makes the NPV of the project equal to zero. So, the equation that defines the IRR for this project is: 0 = −$11,164 + $6,800/(1+IRR) + $4,000/(1+IRR)^2 + $6,800/(1+IRR)^3 Using a spreadsheet, financial calculator, or trial and error to find the root of the equation, we find that: IRR = 26.94%

Say you own an asset that had a total return last year of 11 percent. If the inflation rate last year was 2.5 percent, what was your real return?

8.29 The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation is: (1 + R) = (1 + r)(1 + h)r = [(1 + 0.11)/(1.025)] − 1 r = 0.0829, or 8.29%

Both Bond Sam and Bond Dave have 6 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 4 years to maturity, whereas Bond Dave has 10 years to maturity.

If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Sam? If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Dave? If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Sam be then? If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Dave be then? Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial YTM on both bonds is the coupon rate, 6 percent. If the YTM suddenly rises to 8 percent: PSam = $30(PVIFA4%,8) + $1,000(PVIF4%,8) PSam = $932.67 PDave = $30(PVIFA4%,20) + $1,000(PVIF4%,20) PDave = $864.10 The percentage change in price is calculated as: Percentage change in price = (New price − Original price)/Original price ΔPSam% = ($932.67 − 1,000)/$1,000 ΔPSam% = -.0673, or -6.73% ΔPDave% = ($864.10 − 1,000)/$1,000 ΔPDave% = -.1359, or -13.59% If the YTM suddenly falls to 4 percent: PSam = $30(PVIFA2%,8) + $1,000(PVIF2%,8) PSam = $1,073.25 PDave = $30(PVIFA2%,20) + $1,000(PVIF2%,20) PDave = $1,163.51 ΔPSam% = ($1,073.25 − 1,000)/$1,000 ΔPSam% = .0733, or 7.33% ΔPDave% = ($1,163.51 − 1,000)/$1,000 ΔPDave% = .1635, or 16.35% All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in interest rates.

Imprudential, Incorporated, has an unfunded pension liability of $900 million that must be paid in 22 years. To assess the value of the firm's stock, financial analysts want to discount this liability back to the present. If the relevant discount rate is 6.0 percent, what is the present value of this liability?

To find the PV of a lump sum, we use:PV = FV / (1 + r)tPV = $900,000,000 / (1.060)22PV = $249,754,587

Workman Software has 11.2 percent coupon bonds on the market with 14 years to maturity. The bonds make semiannual payments and currently sell for 139.3 percent of par.

What is the current yield on the bonds? The YTM? The effective annual yield? a. The current yield is: Current yield = Annual coupon payment/Price Current yield = $112/$1,393Current yield = .0804, or 8.04% b. The bond price equation for this bond is: P0 = $1,393 = $56(PVIFAR%,28) + $1,000(PVIFR%,28) Using a spreadsheet, financial calculator, or trial and error we find: R = 3.400% This is the semiannual interest rate, so the YTM is: YTM = 2 × 3.400% YTM = 6.80% c. The effective annual yield is the same as the EAR, so using the EAR equation: Effective annual yield = (1 + .03400)^2 − 1 Effective annual yield = .0692, or 6.92%

Red, Incorporated, Yellow Corporation, and Blue Company each will pay a dividend of $1.50 next year. The growth rate in dividends for all three companies is 7 percent. The required return for each company's stock is 8.60 percent, 11.30 percent, and 14.80 percent, respectively.

What is the stock price for Red, Incorporated? What is the stock price for Yellow Corporation? What is the stock price for Blue Company? We can use the constant dividend growth model, which is:Pt = Dt × (1 + g)/(R − g)So the price of each company's stock today is: Red stock price = $1.50/(.086 − .07) Red stock price = $93.75 Yellow stock price = $1.50/(.113 − .07) Yellow stock price = $34.88 Blue stock price = $1.50/(.148 − .07) Blue stock price = $19.23

You have just received notification that you have won the $1.5 million first prize in the Centennial Lottery. However, the prize will be awarded on your 100th birthday (assuming you're around to collect), 79 years from now. What is the present value of your windfall if the appropriate discount rate is 10 percent?

$805.51 To find the PV of a lump sum, we use:PV = FV / (1 + r)tPV = $1,500,000 / (1.10)79 PV = $805.51

You purchase a bond with an invoice price of $1,090. The bond has a coupon rate of 8.8 percent, and there are 5 months to the next semiannual coupon date. What is the clean price of the bond? Assume a par value of $1,000.

$1,082.67 Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six months is one-half of the annual coupon payment. There are 5 months until the next coupon payment, so one has months have passed since the last coupon payment. The accrued interest for thebond is: Accrued interest = $88/2 × 1/6 Accrued interest = $7.33 And we calculate the clean price as: Clean price = Dirty price − Accrued interest Clean price = $1,090 − 7.33 Clean price = $1,082.67

You purchase a bond with a coupon rate of 7.6 percent and a clean price of $1,090. If the next semiannual coupon payment is due in five months, what is the invoice price? Assume a par value of $1,000.

$1,096.33 Accrued interest is the coupon payment for the period times the fraction of the period that has passed since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six months is one-half of the annual coupon payment. There are five months until the next coupon payment, so one has passed since the last coupon payment. The accrued interest for the bond is: Accrued interest = $76/2 × 1/6 Accrued interest = $6.33 And we calculate the dirty price as:Dirty price = Clean price + Accrued interest Dirty price = $1,090 + 6.33Dirty price = $1,096.33

What is the future value of $300 in 16 years assuming an interest rate of 11 percent compounded semiannually?

$1,664.18 For this problem, we need to find the FV of a lump sum using the equation: FV = PV(1 + r)^t It is important to note that compounding occurs semiannually. To account for this, we will divide the interest rate by two (the number of compounding periods in a year), and multiply the number of periods by two. Doing so, we get: FV = $300[1 + (0.11/2)]^32 FV = $1,664.18

Mannix Corporation stock currently sells for $45 per share. The market requires a return of 11 percent on the firm's stock. If the company maintains a constant 7 percent growth rate in dividends, what was the most recent dividend per share paid on the stock?

$1.68 We are given the stock price, the dividend growth rate, and the required return, and are asked to find the dividend. Using the constant dividend growth model, we get: P0 = $45 = D0(1 + g)/(R − g) Solving this equation for the dividend gives us: D0 = $45(0.11 − 0.07)/(1.07)D0 = $1.68

An investment will pay you $20,000 in 7 years. The appropriate discount rate is 9 percent compounded daily. What is the present value?

$10,652.66 For this problem, we need to find the PV of a lump sum using the equation:PV = FV/(1 + r)^t It is important to note that compounding occurs daily. To account for this, we will divide the interest rate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periods by 365. Doing so, we get: PV = $20,000/[(1 + 0.09/365)^(7(365))] PV = $10,652.66

Synovec Company is growing quickly. Dividends are expected to grow at a rate of 22 percent for the next 3 years, with the growth rate falling off to a constant 5 percent thereafter. If the required return is 10 percent and the company just paid a $3.30 dividend. what is the current share price?

$106.77 With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 3, one year before the constant dividend growth begins as: P3 = D3(1 + g2)/(R − g2) P3 = D0(1 + g1)^3(1 + g2)/(R − g2) P3 = $3.30(1.22)^3(1.05)/(.10 − .05) P3 = $125.84 The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. The price of the stock today will be: P0 = $3.30(1.22)/1.10 + $3.30(1.22)2/1.102 + $3.30(1.22)3/1.103 + $125.84/1.103P0 = $106.77

You want to buy a new sports car from Muscle Motors for $47,000. The contract is in the form of an annuity due for 72 months at an APR of 9.00 percent. What will your monthly payment be?

$840.89 We need to use the PVA due equation, that is:PVAdue = (1 + r) PVAUsing this equation:PVAdue = $47,000 = [1 + (.0900/12)] × C {1 − [1/(1 + (.0900/12))^72]}/(.0900/12)$46,650.12 = $C {1 − [1/(1 + (.0900/12))72]}/(.0900/12)C = $840.89Notice, when we find the payment for the PVA due, we simply discount the PV of the annuity due back one period. We then use this value as the PV of an ordinary annuity

Bilbo Baggins wants to save money to meet three objectives. First, he would like to be able to retire 30 years from now with retirement income of $23,000 per month for 25 years, with the first payment received 30 years and 1 month from now. Second, he would like to purchase a cabin in Rivendell in 20 years at an estimated cost of $919,000. Third, after he passes on at the end of the 25 years of withdrawals, he would like to leave an inheritance of $750,000 to his nephew Frodo. He can afford to save $1,600 per month for the next 20 years. If he can earn a 10 percent EAR before he retires and a 7 percent EAR after he retires, how much will he have to save each month in Years 21 through 30?

$14,306.69 The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR based on monthly compounding, and then divide by 12. So, the pre-retirement APR is: EAR = .10 = [1 + (APR/12)]^12 − 1 APR = 12[(1.10)^(1/12) − 1] APR = 0.0957, or 9.57% And the post-retirement APR is: EAR = .07 = [1 + (APR/12)]^12 − 1APR = 12[(1.07)^(1/12) − 1]APR = 0.0678, or 6.78% First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is: PVA = $23,000{1 − [1/(1 + 0.0678/12)^(12(25))]}/(0.0678/12) PVA = $3,318,320.92 PV = $750,000/[1 + (0.0678/12)]^300 PV = $138,186.88 So, at retirement, he needs:$3,318,320.92 + 138,186.88 = $3,456,507.80 He will be saving $1,600 per month for the next 20 years until he purchases the cabin. The value of his savings after 20 years will be:FVA = $1,600[{[1 + (0.0957/12)]12(20) − 1}/(0.0957/12)] FVA = $1,149,214.76 After he purchases the cabin, the amount he will have left is:$230,214.76 − 919,000 = $230,214.76 He still has 10 years until retirement. When he is ready to retire, this amount will have grown to:FV = $230,214.76[1 + (0.0957/12)]12(10)FV = $597,118 So, when he is ready to retire, based on his current savings, he will be short:$3,456,507.80 − 597,118 = $2,859,389.99 This amount is the FV of the monthly savings he must make between years 20 and 30. So, finding the annuity payment using the FVA equation, we find his monthly savings will need to be:FVA = $2,859,389.99 = C[{[ 1 + (0.0957/12)]12(10) − 1}/(0.0957/12)]C = $2,897,628.35

A7X Corporation just paid a dividend of $1.50 per share. The dividends are expected to grow at 35 percent for the next 8 years and then level off to a growth rate of 7 percent indefinitely. If the required return is 13 percent, what is the price of the stock today?

$140.00 We can use the two-stage dividend growth model for this problem, which is: P0 = [D0(1 + g1)/(R − g1)]{1 − [(1 + g1)/(1 + R)]^T}+ [(1 + g1)/(1 + R)]^T[D0(1 + g2)/(R − g2)] P0 = [$1.50(1.35)/(0.13 − 0.35)][1 − (1.35/1.13)^8] + [(1.35)/(1.13)]^8[$1.50(1.07)/(0.13 − 0.07)] P0 = $140.00

You are planning to save for retirement over the next 25 years. To do this, you will invest $1,100 a month in a stock account and $800 a month in a bond account. The return of the stock account is expected to be 9 percent, and the bond account will pay 7 percent. When you retire, you will combine your money into an account with a return of 8 percent. How much can you withdraw each month from your account assuming a 20-year withdrawal period?

$15,735.88 We need to find the annuity payment in retirement. Our retirement savings ends and the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings. So, we find the FV of the stock account and the FV of the bond account and add the two FVs. Stock account: FVA = $1,100[{[1 + (.09/12)]^300 − 1} / (.09/12)] Stock account: FVA = $1,233,234.13 Bond account: FVA = $800[{[1 + (.07/12)]^300 − 1} / (.07/12)] Bond account: FVA = $648,057.35 So, the total amount saved at retirement is: $1,233,234.13 + 648,057.35 = $1,881,291.49 Solving for the withdrawal amount in retirement using the PVA equation gives us: PVA = $1,881,291.49 = $C[1 − {1/[1 + (.08/12)]^240}/(.08/12)] C = $1,881,291.49/119.5543 C = $15,735.88 withdrawal per month

Given an interest rate of 5.5 percent per year, what is the value at date t = 7 of a perpetual stream of $1,300 payments with the first payment at date t = 13?

$18,084.99 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 = $1,300/0.055 PV = $23,636.36 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 = $23,636.36/1.0555 PV = $18,084.99

A 13-year annuity pays $1,700 per month, and payments are made at the end of each month. The interest rate is 6 percent compounded monthly for the first six years and 4 percent compounded monthly thereafter. What is the present value of the annuity?

$189,425.59 This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last 7 years first. The PV of these cash flows is: PVA2 = $1,700 [{1 − 1/[1 + (.04/12)]^84}/(.04/12)] PVA2 = $124,370.77 Note that this is the PV of this annuity exactly 6 years from today. Now we can discount this lump sum to today. The value of this cash flow today is: PV = $124,370.77/[1 + (.06/12)]^72 PV = $86,848.41 Now we need to find the PV of the annuity for the first 6 years. The value of these cash flows today is: PVA1 = $1,700[{1 − 1 / [1 + (.06/12)]^72}/(.06/12)] PVA1 = $102,577.17 The value of the cash flows today is the sum of these two cash flows, so: PV = $86,848.41 + 102,577.17 PV = $189,425.59

The appropriate discount rate for the following cash flows is 7 percent compounded quarterly. Year Cash Flow 1$ 700 2 800 3 0 4 1,200

$2,258.54 The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR to make the interest rate comparable with the timing of the cash flows. Using the equation for the EAR, we get: EAR = [1 + (APR/m)]^m − 1 EAR = [1 + (.07/4)]^4 − 1 EAR = .0719, or 7.19% And now we use the EAR to find the PV of each cash flow as a lump sum and add them together: PV = $700/1.0719 + $800/1.0719^2 + $1,200/1.0719^4 PV = $2,258.54

When Marilyn Monroe died, ex-husband Joe DiMaggio vowed to place fresh flowers on her grave every Sunday as long as he lived. The week after she died in 1962, a bunch of fresh flowers that the former baseball player thought appropriate for the star cost about $4. Based on actuarial tables, "Joltin' Joe" could expect to live for 34 years after the actress died. Assume that the EAR is 11.4 percent. Also, assume that the price of the flowers will increase at 3.2 percent per year, when expressed as an EAR. Assuming that each year has exactly 52 weeks, what is the present value of this commitment? Joe began purchasing flowers the week after Marilyn died.

$2,515.41 To find the present value, we need to find the real weekly interest rate. To find the real return, we need to use the effective annual rates in the Fisher equation. So, we find the real EAR is: (1 + R) = (1 + r)(1 + h)1 + .114 = (1 + r)(1 + .032)r = .0795, or 7.95% Now, to find the weekly interest rate, we need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR/m)]^m − 1 We can solve for the APR. Doing so, we get: APR = m[(1 + EAR)^(1/m) − 1]APR = 52[(1 + .0795)^(1/52) − 1]APR = .0766, or 7.66% So, the weekly interest rate is: Weekly rate = APR/52Weekly rate = .0766/52Weekly rate = .0015, or .15% Now we can find the present value of the cost of the roses. The real cash flows are an ordinary annuity, discounted at the real interest rate. So, the present value of the cost of the roses is: PVA = C({1 − [1/(1 + r)]^t }/r)PVA = $4({1 − [1/(1 + .0015)]^(34(52))}/.0015)PVA = $2,515.41

The present value of the following cash flow stream is $7,000 when discounted at 9 percent annually. Year Cash Flow 1. $ 1,200 2 ? 3 1,500 4 2,100 What is the value of the missing cash flow?

$3,865.03 We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are: PV of Year 1 CF: $1,200/1.09 = $1,100.92 PV of Year 3 CF: $1,500/1.09^3 = $1,158.28 PV of Year 4 CF: $2,100/1.09^4 = $1,487.69 So, the PV of the missing CF is: $7,000 − 1,100.92 − 1,158.28 − 1,487.69 = $3,253.11 The question asks for the value of the cash flow in Year 2, so we must find the future value of this amount. The value of the missing CF is: $3,253.11(1.09)2 = $3,865.03

Suppose you know a company's stock currently sells for $80 per share and the required return on the stock is 9 percent. You also know that the total return on the stock is evenly divided between a capital gains yield and a dividend yield. If it's the company's policy to always maintain a constant growth rate in its dividends, what is the current dividend per share?

$3.44 We know the stock has a required return of 9 percent, and the dividend and capital gains yield are equal, so: Dividend yield = 1/2(.09) Dividend yield = .045 = Capital gains yield Now we know both the dividend yield and capital gains yield. The dividend is the stock price times the dividend yield, so: D1 = .045($80) D1 = $3.60 This is the dividend next year. The question asks for the dividend this year. Using the relationship between the dividend this year and the dividend next year: D1 = D0(1 + g) We can solve for the dividend that was just paid:$3.60 = D0(1 + .045)D0 = $3.60/1.045D0 = $3.44

You are planning to make monthly deposits of $110 into a retirement account that pays 7 percent interest compounded monthly. If your first deposit will be made one month from now, how large will your retirement account be in 14 years?

$31,244.03 This problem requires us to find the FVA. The equation to find the FVA is: FVA = C{[(1 + r)^t − 1]/r} FVA = $110[{[1 + (.07/12) ]^168 − 1}/(.07/12)] FVA = $31,244.03

Metallica Bearings, Incorporated, is a young start-up company. No dividends will be paid on the stock over the next 8 years because the firm needs to plow back its earnings to fuel growth. The company will pay a dividend of $8 per share 9 years from today and will increase the dividend by 7 percent per year thereafter. If the required return on this stock is 15 percent, what is the current share price?

$32.69 Here we have a stock that pays no dividends for 9 years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. It is important to remember the general constant dividend growth formula is: Pt = [Dt × (1 + g)]/(R − g) This means that since we will use the dividend in Year 9, we will be finding the stock price in Year 8. The dividend growth model is similar to the PVA and the PV of a perpetuity: The equation gives you the PV one period before the first payment. So, the price of the stock in Year 8 will be: P8 = D9/(R − g) P8 = $8/(.15 − .07) P8 = $100.00 The price of the stock today is simply the PV of the stock price in the future. We discount the future stock price at the required return. The price of the stock today will be: P0 = $100.00/1.158P0 = $32.69

E-Eyes.com just issued some new preferred stock. The issue will pay an annual dividend of $7 in perpetuity, beginning 10 years from now. If the market requires a 9 percent return on this investment, how much does a share of preferred stock cost today?

$35.81 The price of a share of preferred stock is the dividend payment divided by the required return. We know the dividend payment in Year 10, so we can find the price of the stock in Year 9, one year before the first dividend payment. Doing so, we get: P9 = $7/0.09 P9 = $77.78 The price of the stock today is the PV of the stock price in the future, so the price today will be: P0 = $77.78/(1.09)^9 P0 = $35.81

You are scheduled to receive $29,000 in two years. When you receive it, you will invest it for 8 more years at 5.5 percent per year. How much will you have in 10 years?

$44,505.91 We need to find the FV of a lump sum. However, the money will only be invested for 8 years, so the number of periods is 8. FV = PV(1 + r)^t FV = $29,000(1.055)^8 FV = $44,505.91

Antiques R Us is a mature manufacturing firm. The company just paid a dividend of $7, but management expects to reduce the payout by 5 percent per year indefinitely. If you require a return of 10 percent on this stock, what will you pay for a share today?

$44.33 The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. So, the price of the stock today will be: P0 = D0(1 + g)/(R − g)P0 = $7(1 − .05)/[(.10 − (− .05)]P0 = $44.33

Lohn Corporation is expected to pay the following dividends over the next four years: $9, $6, $5, and $4. Afterward, the company pledges to maintain a constant 6 percent growth rate in dividends forever. If the required return on the stock is 15 percent, what is the current share price?

$44.87 With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4, at the beginning of the constant dividend growth, as: P4 = D4 (1 + g)/(R − g) P4 = $4(1.06)/(.15 − .06)P4 = $47.11 The price of the stock today is the PV of the first four dividends, plus the PV of the Year 4 stock price. So, the price of the stock today will be: P0 = $9/1.15 + $6/1.152 + $5/1.153 + $4/1.154 + $47.11/1.154 P0 = $44.87

You want to have $42,000 in your savings account 6 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 7.9 percent interest, what amount must you deposit each year?

$5,739.70 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}$42,000 = $C[(1.079^6 − 1)/.079] We can now solve this equation for the annuity payment. Doing so, we get: C = $42,000/7.31745 C = $5,739.70

You want to have $2.5 million in real dollars in an account when you retire in 40 years. The nominal return on your investment is 12 percent and the inflation rate is 2.5 percent. What real amount must you deposit each year to achieve your goal?

$6,884.77 We first need to find the real interest rate on the savings. Using the Fisher equation, the real interest rate is: (1 + R) = (1 + r)(1 + h) 1 + .12 = (1 + r)(1 + .025) r = .0927, or 9.27% Now we can use the future value of an annuity equation to find the annual deposit. Doing so, we find: FVA = C{[(1 + r)^t − 1]/r} $2,500,000 = $C[(1.0927^40 − 1)/.0927 C = $6,884.77

Weismann Company issued 19-year bonds a year ago at a coupon rate of 9 percent. The bonds make semiannual payments and have a par value of $1,000. If the YTM on these bonds is 12 percent, what is the current bond price?

$780.69 To find the price of this bond, we need to realize that the maturity of the bond is 18 years. The bond was issued one year ago, with 19 years to maturity, so there are 18 years left on the bond. Also, the coupons are semiannual, so we need to use the semiannual interest rate and the number of semiannual periods. The price of the bond is: P = $45(PVIFA6.0%,36) + $1,000(PVIF6.0%,36) P = $780.69

You need a 20-year, fixed-rate mortgage to buy a new home for $240,000. Your mortgage bank will lend you the money at a 8.6 percent APR for this 240-month loan. However, you can afford monthly payments of only $850, 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 $850?

$792,394.67 The amount of principal paid on the loan is the PV of the monthly payments you make. So, the present value of the $850 monthly payments is: PVA = $850[(1 − {1/[1 + (.0860/12)]}^240)/(.0860/12)] PVA = $97,235.90 The monthly payments of $850 will amount to a principal payment of $97,235.90. The amount of principal you will still owe is: $240,000 − 97,235.90 = $142,764.10 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 20 years, which is the FV of the remaining principal will be: Balloon payment = $142,764.10 [1 + (.0860/12)]^240 Balloon payment = $792,394.67

What is the value today of $1,800 per year, at a discount rate of 10 percent, if the first payment is received 6 years from now and the last payment is received 28 years from today?

$9,928.40 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 23 payments, so the PV of the annuity is: PVA = $1,800{[1 − (1/1.10)^23] / 0.10}PVA = $15,989.79 Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is the PV at t = 5. To find the value today, we find the PV of this lump sum. The value today is: PV = $15,989.79 / 1.10^5PV = $9,928.40

Although appealing to more refined tastes, art as a collectible has not always performed so profitably. During 2016, a sculpture was sold at auction for a price of $10,318,500. Unfortunately for the previous owner, he had purchased it in 2012 at a price of $12,371,500. What was his annual rate of return on this sculpture?

-4.44% To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV/PV)1/t − 1r = ($10,318,500 / $12,371,500)1/4 − 1 r = -.0444, or -4.44% Notice that the interest rate is negative. This occurs when the FV is less than the PV.

An investment project has annual cash inflows of $5,800, $6,900, $7,700 for the next four years, respectively, and $9,000, and a discount rate of 14 percent. What is the discounted payback period for these cash flows if the initial cost is $9,000?

1.74 years When we use discounted payback, we need to find the value of all cash flows today. The value today of the project cash flows for the first four years is: Value today of Year 1 cash flow = $5,800/1.14 Value today of Year 1 cash flow = $5,087.72 Value today of Year 2 cash flow = $6,900/1.14^2 Value today of Year 2 cash flow = $5,309.33 Value today of Year 3 cash flow = $7,700/1.14^3 Value today of Year 3 cash flow = $5,197.28 Value today of Year 4 cash flow = $9,000/1.14^4 Value today of Year 4 cash flow = $5,328.72 To find the discounted payback, we use these values to find the payback period. The discounted first year cash flow is 5,087.72, so the discounted payback for an $9,000 initial cost is: Discounted payback = 1 + ($9,000 − 5,087.72)/$5,309.33 Discounted payback = 1.74 years

Hudson Corporation will pay a dividend of $3.40 per share next year. The company pledges to increase its dividend by 6.30 percent per year indefinitely. If you require a return of 9.60 percent on your investment, how much will you pay for the company's stock today?

103.03 Using the constant growth model, we find the price of the stock today is:P0 = D1/(R− g) P0 = $3.40/(.096 − .063) P0 = $103.03

Elliott Credit Corporation wants to earn an effective annual return on its consumer loans of 13 percent per year. The bank uses daily compounding on its loans. What interest rate is the bank required by law to report to potential borrowers?

12.22% The reported rate is the APR, so we need to convert the EAR to an APR as follows: EAR = [1 + (APR/m)]^m − 1 APR = m[(1 + EAR)^(1/m) − 1] APR = 365[(1.13)^(1/365) − 1] APR = .1222, or 12.22%

McConnell Corporation has bonds on the market with 14 years to maturity, a YTM of 11.0 percent, a par value of $1,000, and a current price of $1,216.50. The bonds make semiannual payments. What must the coupon rate be on these bonds?

14.07% Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: P = $1,216.50 = C(PVIFA5.5%,28) + $1,000(PVIF5.5%,28) Solving for the coupon payment, we get: C = $70.33 Since this is the semiannual payment, the annual coupon payment is: 2 × $70.33 = $140.66 And the coupon rate is the annual coupon payment divided by par value, so: Coupon rate = $140.66/$1,000Coupon rate = 0.1407, or 14.07%

The next dividend payment by Savitz, Incorporated, will be $4.65 per share. The dividends are anticipated to maintain a growth rate of 5 percent forever. If the stock currently sells for $51 per share, what is the required return?

14.12% We need to find the required return of the stock. Using the constant growth model, we can solve the equation for R. Doing so, we find:R = (D1/P0) + g R = ($4.65/$51) + .05 R = .1412, or 14.12%

You are looking at a one-year loan of $15,000. The interest rate is quoted as 12 percent plus 3 points. A point on a loan is simply 1 percent (one percentage point) of the loan amount. Quotes similar to this one are common with home mortgages. The interest rate quotation in this example requires the borrower to pay 3 points to the lender up front and repay the loan later with 12 percent interest. What rate would you actually be paying here?

15.46% Again, to find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be: Loan repayment amount = $15,000(1.12) Loan repayment amount = $16,800 The amount you will receive today is the principal amount of the loan times one minus the points .Amount received = $15,000(1 − .03) Amount received = $14,550 Now, we find the interest rate for this PV and FV.$16,800 = $14,550(1 + r) r = ($16,800/$14,550) − 1 r = .1546, or 15.46%

Burnett Corporation pays a constant $29 dividend on its stock. The company will maintain this dividend for the next 13 years and will then cease paying dividends forever. If the required return on this stock is 15 percent, what is the current share price?

161.91 The price of any financial instrument is the PV of the future cash flows. The future dividends of this stock are an annuity for 13 years, so the price of the stock is the PVA, which will be:P0 = $29(PVIFA 15%,13) P0 = $161.91

What is the payback period for the following set of cash flows? Year Cash Flow 0 −$ 6,300 1 2,300 2 1,900 3 2,700 4 3,000

2.78 years To calculate the payback period, we need to find the time that the project has recovered its initial investment. After two years, the project has created: $2300 + $1900 = $4200 in cash flows. The project still needs to create another: $6,300 − 4,200 = $2,100 in cash flows. During the third year, the cash flows from the project will be $2,700. So, the payback period will be 2 years, plus what we still need to make divided by what we will make during the third year. The payback period is: Payback = 2 + ($2,100/$2,700) Payback = 2.78 years

According to the Census Bureau, in October 2016, the average house price in the United States was $27,658. 8 years earlier, the average price was $21,408. What was the annual increase in the price of the average house sold?

3.25% To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV/PV)1/t − 1r = ($27,658/$21,408)1/8 − 1 r = .0325, or 3.25%

You're trying to save to buy a new $150,000 Ferrari. You have $35,000 today that can be invested at your bank. The bank pays 4.0 percent annual interest on its accounts. How long will it be before you have enough to buy the car?

37.11 years To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)t Solving for t, we get:t = ln(FV / PV) / ln(1 + r)t = ln ($150,000 / $35,000) / ln 1.040 t = 37.11 years

You expect to receive $6,000 at graduation in two years. You plan on investing it at 8 percent until you have $107,000. How long will you wait from now?

39.44 years To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)^t Solving for t, we get:t = ln(FV / PV) / ln(1 + r)t = ln($107,000 / $6,000) / ln(1.08) t = 37.44 So, the money must be invested for 37.44 years. However, you will not receive the money for another two years. From now, you'll wait:2 years + 37.44 years = 39.44 years

You're prepared to make monthly payments of $300, beginning at the end of this month, into an account that pays 11 percent interest compounded monthly. How many payments will you have made when your account balance reaches $18,921?

50 Here we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the number of payments. Using the FVA equation: FVA = $18,921 = $300[{[1 + (0.11/12)]^t − 1 } / (0.11/12)] Solving for t, we get: 1.009^t = 1 + [($18,921)/($300)](0.11/12) t = ln 1.578142 / ln 1.009 t = 50 payments

You have just won the lottery and will receive $570,000 in one year. You will receive payments for 24 years, and the payments will increase 4 percent per year. If the appropriate discount rate is 11 percent, what is the present value of your winnings?

6,437,460 We can use the present value of a growing annuity equation to find the value of your deposits today. Doing so, we find: PV = C {[1/(r − g)] − [1/(r − g)] × [(1 + g)/(1 + r)]^t} PV = $570,000{[1/(.11 − .04)] − [1/(.11 − .04)] × [(1 + .04)/(1 + .11)]^24} PV = $6,437,460

Suppose the real rate is 4.5 percent and the inflation rate is 2 percent. What rate would you expect to see on a Treasury bill?

6.59% The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation is: (1 + R) = (1 + r)(1 + h) R = (1 + .02)(1 + .045) − 1 R = .0659, or 6.59%

Your job pays you only once a year for all the work you did over the previous 12 months. Today, December 31, you just received your salary of $46,000 and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 2 percent of your annual salary in an account that will earn 11 percent per year. Your salary will increase at 5 percent per year throughout your career. How much money will you have on the date of your retirement 37 years from today?

667,291.34 Since your salary grows at 5 percent per year, your salary next year will be: Next year's salary = $46,000 (1 + .05) Next year's salary = $48,300 This means your deposit next year will be: Next year's deposit = $48,300(.02)Next year's deposit = $966 Since your salary grows at 5 percent, you deposit will also grow at 5 percent. We can use the present value of a growing perpetuity equation to find the value of your deposits today. Doing so, we find: PV = C {[1/(r − g)] − [1/(r − g)] × [(1 + g)/(1 + r)]^t} PV = $966{[1/(.11 − .05)] − [1/(.11 − .05)] × [(1 + .05)/(1 + .11)]^37} PV = $14,039.94 Now, we can find the future value of this lump sum in 37 years. We find: FV = PV(1 + r)tFV = $14,039.94(1 + .11)37FV = $667,291.34 This is the value of your savings in 37 years.

Suppose an investment offers to triple your money in 48 months (don't believe it). What rate of return per quarter are you being offered?

7.11% 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. FV = $3 = $1(1 + r)^(48/3) r = 0.0711, or 7.11%

An investment offers a total return of 12.5 percent over the coming year. Janice Yellen thinks the total real return on this investment will be only 5.0 percent. What does Janice believe the inflation rate will be over the next year?

7.14% The Fisher equation, which shows the exact relationship between nominal interest rates, real interest rates, and inflation is: (1 + R) = (1 + r)(1 + h) h = [(1 + 0.125)/(1 + 0.05)] − 1 h = 0.0714, or 7.14%

Chamberlain Company wants to issue new 15-year bonds for some much-needed expansion projects. The company currently has 8.4 percent coupon bonds on the market that sell for $1,070.88, make semiannual payments, and mature in 15 years. What coupon rate should the company set on its new bonds if it wants them to sell at par? Assume a par value of $1,000.

7.60% The company should set the coupon rate on its new bonds equal to the required return. The required return can be observed in the market by finding the YTM on outstanding bonds of the company. So, the YTM on the bonds currently sold in the market is: P = $1,070.88 = $42(PVIFAR%,30) + $1,000(PVIFR%,30) Using a spreadsheet, financial calculator, or trial and error we find:R = 3.8% This is the semiannual interest rate, so the YTM is:YTM = 2 × 3.8% YTM = 7.60%

Gabriele Enterprises has bonds on the market making annual payments, with 18 years to maturity, a par value of $1,000, and selling for $870. At this price, the bonds yield 10 percent. What must the coupon rate be on the bonds?

8.41% Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing equation and solve for the coupon payment as follows: P = $870 = C(PVIFA10.0%,18) + $1,000(PVIF10.0%,18) Solving for the coupon payment, we get:C = $84.15 The coupon payment is the coupon rate times par value. Using this relationship, we get:Coupon rate = $84.15/$1,000Coupon rate = 0.0841, or 8.41%

West Corporation issued 13-year bonds 2 years ago at a coupon rate of 9.4 percent. The bonds make semiannual payments. If these bonds currently sell for 98 percent of par value, what is the YTM?

9.70% Here we are finding the YTM of a semiannual coupon bond. The bond price equation is: P = 980 = $47(PVIFAR%,22) + $1,000(PVIFR%,22) Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error, we find: R = 4.8499% Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the bond, so: YTM = 2 × 4.8499% YTM = 9.70%

Bond J has a coupon rate of 4 percent. Bond K has a coupon rate of 8 percent. Both bonds have 9 years to maturity, make semiannual payments, and have a YTM of 7 percent.

If interest rates suddenly rise by 4 percent, what is the percentage price change of Bond J? If interest rates suddenly rise by 4 percent, what is the percentage price change of Bond K? If interest rates suddenly fall by 4 percent, what is the percentage price change of Bond J? If interest rates suddenly fall by 4 percent, what is the percentage price change of Bond K? Initially, at a YTM of 7 percent, the prices of the two bonds are: PJ = $20(PVIFA3.5%,18) + $1,000(PVIF3.5%,18) PJ = $802.15 PK = $40(PVIFA3.5%,18) + $1,000(PVIF3.5%,18) PK = $1,065.95 If the YTM rises from 7 percent to 11 percent: PJ = $20(PVIFA5.5%,18) + $1,000(PVIF5.5%,18) PJ = $606.39 PK = $40(PVIFA5.5%,18) + $1,000(PVIF5.5%,18) PK = $831.31 The percentage change in price is calculated as: Percentage change in price = (New price − Original price)/Original price ΔPJ% = ($606.39 − 802.15)/$802.15 ΔPJ% = -.2441, or -24.41% ΔPK% = ($831.31 − 1,065.95)/$1,065.95 ΔPK% = -.2201, or -22.01% If the YTM declines from 7 percent to 3 percent: PJ = $20(PVIFA1.5%,18) + $1,000(PVIF1.5%,18)PJ = $1,078.36 PK = $40(PVIFA1.5%,18) + $1,000(PVIF1.5%,18) PK = $1,391.81 ΔPJ% = ($1,078.36 − 802.15)/$802.15 ΔPJ% = .3443, or 34.43% ΔPK% = ($1,391.81 − 1,065.95)/$1,065.95ΔPK% = .3057, or 30.57% All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in interest rates.

A firm evaluates all of its projects by using the NPV decision rule. YearCash Flow 0 −$ 29,000 1 23,000 2 14,000 3 6,000

a. At a required return of 16 percent, what is the NPV for this project? b. At a required return of 35 percent, what is the NPV for this project? a. The NPV of a project is the PV of the outflows minus the PV of the inflows. The equation for the NPV of this project at a reuquired return of 16 percent is: NPV = − $29,000 + $23,000/1.16 + $14,000/1.16^2 + $6,000/1.16^3 NPV = $5,075.81 At a required return of 16 percent, the NPV is positive, so we would accept the project. b. The equation for the NPV of the project at a required return of 35 percent is: NPV = − $29,000 + $23,000/1.35 + $14,000/1.35^2 + $6,000/1.35^3 NPV = $-1,842.55 At a required return of 35 percent, the NPV is negative, so we would reject the project.

You deposit $1,900 at the end of each year into an account paying 10.1 percent interest.

a. How much money will you have in the account in 22 years? b. How much will you have if you make deposits for 44 years? a. Here we need to find the FVA. The equation to find the FVA is: FVA = C{[(1 + r)^t − 1]/r} FVA for 22 years = $1,900[(1.1010^22 − 1)/.1010] FVA for 22 years = $137,414.09 b. FVA = C{[(1 + r)t − 1]/r}FVA for 44 years = $1,900[(1.101044 − 1)/.1010] FVA for 44 years = $1,278,589.22 Notice that because of exponential growth, doubling the number of periods does not merely double the FVA.

The Jackson-Timberlake Wardrobe Company just paid a dividend of $1.28 per share on its stock. The dividends are expected to grow at a constant rate of 4 percent per year indefinitely.

a. If investors require a return of 9 percent on the company's stock, what is the current price? b. What will the price be in 6 years? a. The constant dividend growth model is:Pt = Dt × (1 + g)/(R − g) So the price of the stock today is:P0 = D0 (1 + g)/(R − g) P0 = $1.28 (1.04)/(.09 − .04) P0 = $26.62 b. We can do the same thing to find the dividend in Year 7, which gives us the price in Year 6, so:P6 = D6 (1 + g)/(R − g) P6 = D0 (1 + g)7/(R − g)P6 = $1.28 (1.04)7/(.09 − .04) P6 = $33.69

A project that provides annual cash flows of $15,000 for 5 years costs $45,000 today.

a. If the required return is 9 percent, what is the NPV for this project? b. Determine the IRR for this project. a. The NPV of a project is the PV of the outflows minus by the PV of the inflows. Since the cash inflows are an annuity, the equation for the NPV of this project at an 9 percent required return is: NPV = −$45,000 + $15,000(PVIFA9%, 5)NPV = $13,344.77 b. We would be indifferent to the project if the required return was equal to the IRR of the project, since at that required return the NPV is zero. The IRR of the project is: 0 = −$45,000 + $15,000(PVIFAIRR, 5)IRR = 19.86%

An investment project costs $17,200 and has annual cash flows of $3,500 for six years.

a. What is the discounted payback period if the discount rate is zero percent? b. What is the discounted payback period if the discount rate is 4 percent? c. What is the discounted payback period if the discount rate is 18 percent? a. R = 0%:($17,200/$3,500) = 4.91 years discounted payback = regular payback = 4.91 years b. R = 4%:3500/1.04 + 3500/1.042 + 3500/1.043 + 3500/1.044 + 3500/1.045 = $15,581.38 $3,500/1.046 = $2,766.1 discounted payback = 5 + ($17,200 − 15,581.38)/$2,766.1 = 5.59 years c. R = 18%:$3,500(PVIFA18%, 6) = $12,241.61 The project never pays back.

An investment project provides cash inflows of $650 per year for 11 years.

a. What is the project payback period if the initial cost is $2,600? b. What is the project payback period if the initial cost is $3,900? c. What is the project payback period if the initial cost is $7,800? a. To calculate the payback period, we need to find the time that the project requires ro recover its initial investment. The cash flows in this problem are an annuity, so the calculation is simpler. If the initial cost is $2,600, the payback period is: Payback = 4 + ($0/$650) Payback = 4 years There is a shortcut to calculate the future cash flows or an annuity. Just divide the initial cost by the annual cash flow. For the $2,600 cost, the payback period is: Payback = $2,600/$650 Payback = 4 years b. For an initial cost of $3,900, the payback period is: Payback = $3,900/$650 Payback = 6 years c. The payback period for an initial cost of $7,800 is a little trickier. Notice that the total cash inflows after 11 years will be: Total cash inflows = 11($650) Total cash inflows = $7,150 If the initial cost is $7,800, the project never pays back. Notice that if you use the shortcut for annuity cash flows, you get: Payback = $7,800/$650 Payback = 12 years This answer does not make sense since the cash flows stop after 11 years, so again, we must conclude the payback period is never.

Find the EAR in each of the following cases:

a.8% compounded quarterly b.5% compounded monthly c.8% compounded daily d.15% with continuous compounding For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR/m)]^m − 1 a. EAR = [1 + (.08/4)]^4 − 1 EAR = .0824, or 8.24% b. EAR = [1 + (.05/12)]1%2 − 1 EAR = .0512, or 5.12% c. EAR = [1 + (.08/365)]^365 − 1 EAR = .0833, or 8.33% d. To find the EAR with continuous compounding, we use the equation: EAR = e^q − 1 EAR = e^.15 − 1 EAR = .1618, or 16.18%

Find the APR, or stated rate, in each of the following cases:

a.An effective interest of 5% compounded semiannually b.An effective interest of 14% compounded monthly c.An effective interest of 16% compounded weekly d.An effective interest of 19% with continuous compounding Here we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR/m)]^m − 1 We can now solve for the APR. Doing so, we get:APR = m[(1 + EAR)^(1/m) − 1] a. APR = 2[(1.05)^(1/2) − 1] APR = .0494, or 4.94% b. APR = 12[(1.14)^(1/12) − 1] APR = .1317, or 13.17% c. APR = 52[(1.16)1/52 − 1] APR = .1486, or 14.86% d. We can solve the continuous compounding EAR equation. Doing so we get: EAR = eq − 1 APR = ln(1 + EAR) APR = ln(1 + .19) APR = .1740, or 17.40%

Fuente, Incorporated, has identified an investment project with the following cash flows. Year Cash Flow 1 $ 600 2 950 3 1,100 4 1,325

a.If the discount rate is 11 percent, what is the future value of these cash flows in year 4? b.What is the future value at a discount rate of 17 percent? c.What is the future value at discount rate of 26 percent? a.To solve this problem, we must find the FV of each cash flow and add them. To find the FV of a lump sum, we use: FV = PV(1 + r)^tFV@11% = $600(1.11)3 + $950(1.11)2 + $1,100(1.11) + $1,325 FV@11% = $4,537.07 b.FV = PV(1 + r)tFV@17% = $600(1.17)3 + $950(1.17)2 + $1,100(1.17) + $1,325 FV@17% = $4,873.42 c.FV = PV(1 + r)tFV@26% = $600(1.26)3 + $950(1.26)2 + $1,100(1.26) + $1,325 FV@26% = $5,419.45

McCann Company has identified an investment project with the following cash flows. Year. Cash Flow 1$ 800 2 1,050 3 1,300 4 1,130

a.If the discount rate is 8 percent, what is the present value of these cash flows? b.What is the present value at 19 percent? c.What is the present value at 29 percent? a. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV/(1 + r)^t PV@8% = $800/1.08 + $1,050/1.08^2 + $1,300/1.08^3 + $1,130/1.08^4 PV@8% = $3,503.51 b. PV = FV/(1 + r)tPV@19% = $800/1.19 + $1,050/1.19^2 + $1,300/1.19^3 + $1,130/1.19^4 PV@19% = $2,748.68 c. PV = FV/(1 + r)t PV@29% = $800/1.29 + $1,050/1.292 + $1,300/1.293 + $1,130/1.294 PV@29% = $2,264.77

A 10-year annuity of twenty $9,000 semiannual payments will begin 10 years from now, with the first payment coming 10.5 years from now.

a.If the discount rate is 9 percent compounded monthly, what is the value of this annuity 7 years from now? b.What is the current value of the annuity? a. The cash flows in this problem are semiannual, so we need the effective semiannual rate. The interest rate given is the APR, so the monthly interest rate is: Monthly rate = .09/12 Monthly rate = .007500 To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the exponent, we will use 6 months. The effective semiannual rate is: Semiannual rate = (1.007500)^6 − 1 Semiannual rate = .045852, or 4.59% We can now use this rate to find the PV of the annuity. The PV of the annuity is: PVA @ Year 10 = $9,000{[1 − (1/1.05^20 )]/.045852} PVA @ Year 10 = $116,211.66 Note, this is the value one period (six months) before the first payment, so it is the value at year 10. So, the value at the various times the questions asked for uses this value 10 years from now. PV @ Year 7 = $116,211.66/1.045852^6 PV @ Year 7 = $88,803.02 b. PV @ t = 0 = $116,211.657799/1.045852^20 PV @ t = 0 = $47,407.07

You have just purchased a new warehouse. To finance the purchase, you've arranged for a 30-year mortgage loan for 80 percent of the $2,700,000 purchase price. The monthly payment on this loan will be $16,000.

a.What is the APR on this loan? b.What is the EAR? a. Here we are finding the interest rate for an annuity cash flow. We are given the PVA, number of periods, and the amount of the annuity. We should also note that the PV of the annuity is not the amount borrowed since we are making a down payment on the warehouse. The amount borrowed is: Amount borrowed = .80($2,700,000) Amount borrowed = $2,160,000 Using the PVA equation: PVA = $2,160,000 = $16,000[{1 − [1 / (1 + r)]360}/ r] Unfortunately this equation cannot be solved to find the interest rate using algebra. To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. If you use trial and error, remember that increasing the interest rate lowers the PVA, and decreasing the interest rate increases the PVA. Using a spreadsheet, we find: r = .675% The APR is the monthly interest rate times the number of months in the year, so: APR = 12(.675%) APR = 8.10% b. The EAR is: EAR = (1 + .00675)12 − 1 EAR = .0841, or 8.41%

You are looking at an investment that has an effective annual rate of 12 percent.

a.What is the effective semiannual return? b.What is the effective quarterly return? c.What is the effective monthly return? Here we need to convert an EAR into interest rates for different compounding periods. Using the equation for the EAR, we get: EAR = [1 + (APR/m)]^m− 1 a. EAR = .12 = (1 + r)^2 − 1; r = (1.12)^(1/2) − 1 EAR = .0583, or 5.83% per six months b. EAR = .12 = (1 + r)^4 − 1; r = (1.12)^(1/4) − 1 EAR = .0287, or 2.87% per quarter c. EAR = .12 = (1 + r)^12 − 1; r = (1.12)^(1/12) − 1 EAR = .0095, or .95% per month Notice that the effective six month rate is not twice the effective quarterly rate because of the effect of compounding.

Live Forever Life Insurance Company is selling a perpetuity contract that pays $1,400 monthly. The contract currently sells for $67,000.

a.What is the monthly return on this investment vehicle? b.What is the APR? c.What is the effective annual rate? a. 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$67,000 = $1,400/r We can now solve for the interest rate as follows: r = $1,400/$67,000 r = .0209, or 2.09% per month b. The interest rate is 2.09% per month. To find the APR, we multiply this rate by the number of months in a year, so: APR = 12(2.09%) APR = 25.07% c. And using the equation to find an EAR: EAR = [1 + (APR/m)]^m − 1EAR = [1 + 0.0209]^12 − 1 EAR = .2817, or 28.17%

Suppose you are going to receive $14,000 per year for 9 years. The appropriate interest rate is 11 percent.

a.What is the present value of the payments if they are in the form of an ordinary annuity? b.What is the present value if the payments are an annuity due? c.Suppose you plan to invest the payments for 9 years, what is the future value if the payments are an ordinary annuity? d.Suppose you plan to invest the payments for 9 years, what is the future value if the payments are an annuity due? a. If the payments are in the form of an ordinary annuity, the present value will be: PVA = C({1 − [1/(1 + r)^t]}/r)) PVA = $14,000[{1 − [1/(1 + .11)]9}/.11] PVA = $77,518.67 b. If the payments are an annuity due, the present value will be: PVAdue = (1 + r)PVA PVAdue = (1 + .11)$77,518.67 PVAdue = $86,045.72 c. We can find the future value of the ordinary annuity as: FVA = C{[(1 + r)^t − 1]/r} FVA = $14,000.00{[(1 + .11)9 − 1]/.11} FVA = $198,295.61 d. If the payments are an annuity due, the future value will be: FVAdue = (1 + r)FVA FVAdue = (1 + .11)$198,295.61 FVAdue = $220,108.13

Big Dom's Pawn Shop charges an interest rate of 25 percent per month on loans to its customers. Like all lenders, Big Dom must report an APR to consumers.

a.What rate should the shop report? b.What is the effective annual rate? a. The APR is the interest rate per period times the number of periods in a year. In this case, the interest rate is 25 percent per month, and there are 12 months in a year, so we get: APR = 12(25%) APR = 300.00% b.To find the EAR, we use the EAR formula: EAR = [1 + (APR/m)]^m − 1 EAR = (1 + .25)^12 − 1 EAR = 13.5519, or 1,355.19% Notice that we didn't need to divide the APR by the number of compounding periods per year. We do this division to get the interest rate per period, but, in this problem, we are already given the interest rate per period.

A local finance company quotes a 15 percent interest rate on one-year loans. So, if you borrow $20,000, the interest for the year will be $3,000. Because you must repay a total of $23,000 in one year, the finance company requires you to pay $23,000/12, or $1,916.67, per month over the next 12 months.

a.What rate would legally have to be quoted? b.What is the effective annual rate? 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 = $1,917{(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 = 2.219% per month a.So the APR is:APR = 12(2.219%) APR = 26.63% b.And the EAR is:EAR = (1.02219)^12 − 1 EAR = .3013, or 30.13%

In 1904, the first Putting Green Championship was held. The winner's prize money was $110. In 2016, the winner's check was $1,163,000. (Do not round intermediate calculations.)

a.What was the percentage increase per year in the winner's check over this period? b.If the winner's prize increases at the same rate, what will it be in 2049? Explanation To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)t a. Solving for r, we get: r = (FV/PV)1/t − 1 r = ($1,163,000/$110)1/112 − 1 r = .0863, or 8.63% b. To find the FV of the first prize, we use: FV = PV(1 + r)t FV = $1,163,000(1.0863)33 FV = $17,835,187.48

You have just made your first $2,500 contribution to your retirement account. Assume you earn a return of7 percent and make no additional contributions.

a.What will your account be worth when you retire in 45 years? b.What will your account be worth if you wait 9 years before contributing? Explanation To find the FV of a lump sum, we use:FV = PV(1 + r)t a. FV = $2,500(1.07)^45 FV = $52,506.13 b. FV = $2,500(1.07)^36 FV = $28,559.86

Even though most corporate bonds in the United States make coupon payments semiannually, bonds issued elsewhere often have annual coupon payments. Suppose a German company issues a bond with a par value of €1,000, 11 years to maturity, and a coupon rate of 6 percent paid annuallly. If the yield to maturity is 10 percent, what is the current price of the bond?

€740.20 The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this problem assumes an annual coupon. The price of the bond will be: P = €$60({1 − [1/(1 + 0.10)]^11 }/0.10) + €1,000[1/(1 + 0.10)^11] P = €740.20