Selected QuestionsTime Value of Money

Q9. Calculating the Number of Periods You’re trying to save to buy a new$160,000 Ferrari. You have $30,000 today that can be invested at your bank. Thebank pays 4.7 percent annual interest on its accounts. How long will it be beforeyou have enough to buy the car?

To answer this question, we can use either the FV or the PV formula. Both will give the sameanswer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)tSolving for t, we get:t = ln(FV / PV) / ln(1 + r)FV = $160,000 = $30,000(1.047)tt = ln($160,000 / $30,000) / ln 1.047t = 36.45 years

Q6.Calculating Interest Rates Assume the total cost of a college education willbe $290,000 when your child enters college in 18 years. You presently have$35,000 to invest. What annual rate of interest must you earn on your investmentto cover the cost of your child’s college education?

To answer this question, we can use either the FV or the PV formula. Both will give the sameanswer since they are the inverse of each other. We will use the FV formula, that is:FV = PV(1 + r)tSolving for r, we get:r = (FV / PV)1 / t – 1r = ($290,000 / $35,000)1/18 – 1r = 0.1247 or 12.47%

Q18.You have just made your first $5,000 contributionto your individual retirement account. Assuming you earn a 11 percent rate of returnand make no additional contributions, what will your account be worthwhen you retire in 45 years? What if you wait 10 years before contributing?

To find the FV of a lump sum, we use:FV = PV(1 + r)tFV = $5,000(1.11)45FV = $547,651.21If you wait 10 years, the value of your deposit at your retirement will be:FV = $5,000(1.11)35FV = $192,874.26

Cost of Money

Q4. Calculating Annuity Present Value An investment offers $7000 per year for15 years, with the first payment occurring one year from now. If the required returnis 9 percent, what is the value of the investment? What would the value beif the payments occurred for 40 years? For 75 years? Forever?

To find the PVA, we use the equation:PVA = C({1 – [1/(1 + r)]t } / r )PVA@15 yrs: PVA = $7,000{[1 – (1/1.09)15 ] / .09} = $56,424.82PVA@40 yrs: PVA = $7,000{[1 – (1/1.09)40 ] / .09} = $75,301.52PVA@75 yrs: PVA = $7,000{[1 – (1/1.09)75 ] / .09} = $77,656.48To find the PV of a perpetuity, we use the equation:PV = C / rPV = $7,000 / .09PV = $77,777.78Notice that as the length of the annuity payments increases, the present value of the annuityapproaches the present value of the perpetuity. The present value of the 75-year annuity and thepresent value of the perpetuity imply that the value today of all perpetuity payments beyond 75 yearsis only $121.30

Q9. Calculating Annuity Values Biktimirov Bank offers you a $50000, seven-yearterm loan at 8 percent annual interest. What will your annual loan payment be?

Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate theannuity payment. Using the PVA equation:PVA = C({1 – [1/(1 + r)]t } / r)$50,000 = C{[1 – (1/1.08)7 ] / .08}We can now solve this equation for the annuity payment. Doing so, we get:C = $50,000 / 5.20637C = $9,603.62

Q17. Calculating Future Values Corn Credit Bank is offering 4.5 percent compoundeddaily on its savings accounts. If you deposit $5,000 today, how muchwill you have in the account in 5 years? In 10 years? In 20 years?

For this problem, we simply need to find the FV of a lump sum using the equation:FV = PV(1 + r)tIt is important to note that compounding occurs daily. To account for this, we will divide the interestrate by 365 (the number of days in a year, ignoring leap year), and multiply the number of periodsby 365. Doing so, we get:FV in 5 years = $5,000[1 + (.045/365)]5(365) = $6,261.53FV in 10 years = $5,000[1 + (.045/365)]10(365) = $7,841.34FV in 20 years = $5,000[1 + (.045/365)]20(365) = $12,297.33

Bonds

Q5. Coupon Rates Mustaine Enterprises has bonds on the market making annual

payments, with 13 years to maturity, and selling for $1051. At this price, thebonds yield 6.8% percent. What must the coupon rate be on Mustaine’s bonds?

Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricingequation and solve for the coupon payment as follows:P = $1,051 = C(PVIFA6.8%,16) + $1,000(PVIF6.8%,16)Solving for the coupon payment, we get:C = $73.33The coupon payment is the coupon rate times par value. Using this relationship, we get:Coupon rate = $73.33 / $1,000Coupon rate = .0733 or 7.33%

18. Bond Yields Lifehouse Software has 8 percent coupon bonds on the marketWith 36 years to maturity. The bonds make semiannual payments and currentlysell for 93 percent of par. What is the current yield on Lifehouse’s bonds? TheYTM? The effective annual yield?

The current yield is:Current yield = Annual coupon payment / PriceCurrent yield = $80 / $930Current yield = 0.0860 or 8.60%The bond price equation for this bond is:P0 = $930 = $40(PVIFAR%,36) + $1,000(PVIFR%,36)Using a spreadsheet, financial calculator, or trial and error we find:R = 4.39%This is the semiannual interest rate, so the YTM is:YTM = 2 ´ 4.39%YTM = 8.78%The effective annual yield is the same as the EAR, so using the EAR equation from the previouschapter:Effective annual yield = (1 + 0.0439)2 – 1Effective annual yield = 8.97%

25. Finding the Maturity You’ve just found a 10 percent coupon bond on themarket that sells for par value. What is the maturity on this bond?

The maturity is indeterminate. A bond selling at par can have any length of maturity.

Stock Valuation

Q2. Stock Values The next dividend payment by SAF, Inc., will be $1.90 pershare. The dividends are anticipated to maintain a 5 percent growth rate, forever.If SAF stock currently sells for $47.00 per share, what is the required return?

We need to find the required return of the stock. Using the constant growth model, we can solve theequation for R. Doing so, we find:R = (D1 / P0) + gR = ($1.90 / $47.00) + .055R = .0954 or 9.54%

Q7. Stock Valuation Kiessling Corp. pays a constant $17 dividend on its stock. Thecompany will maintain this dividend for the next eight years and will then cease paying dividends forever. If the required return on this stock is 11 percent, whatis the current share price?

The price of any financial instrument is the present value of the future cash flows. The futuredividends of this stock are an annuity for eight years, so the price of the stock is the present value ofan annuity, which will be:P0 = $17.00(PVIFA11%,8)P0 = $87.48

Q9. Stock Valuation Smashed Pumpkin Farms (SPF) just paid a dividend of $3.00 on its stock. The growth rate in dividends is expected to be a constant 7.5 percentper year, indefinitely. Investors require an 18 percent return on the stock forthe first three years, a 12 percent return for the next three years, and then a 13percent return, thereafter. What is the current share price for SPF stock?

If the company uses straight voting, the board of directors is elected one at a time. You will need toown one-half of the shares, plus one share, in order to guarantee enough votes to win the election.So, the number of shares needed to guarantee election under straight voting will be:Shares needed = (300,000 shares / 2) + 1Shares needed = 150,001And the total cost to you will be the shares needed times the price per share, or:Total cost = 150,001 ´ $63Total cost = $9,450,063If the company uses cumulative voting, the board of directors are all elected at once. You will need1/(N + 1) percent of the stock (plus one share) to guarantee election, where N is the number of seatsup for election. So, the percentage of the company’s stock you need will be:Percent of stock needed = 1 / (N + 1)Percent of stock needed = 1 / (4 + 1)Percent of stock needed = .20 or 20%So, the number of shares you need to purchase is:Number of shares to purchase = (300,000 × .20) + 1

Number of shares to purchase = 60,001And the total cost to you will be the shares needed times the price per share, or:Total cost = 60,001 ´ $63Total cost = $3,780,063