[1] (30 points) EFFECTIVE RATES  Solution
{a} You have the opportunity to invest $5,000 for 4 years in one of two annuities. Annuity Q will let you withdraw $425 per quarter (starting in one quarter) and Annuity M will let you withdraw $140 per month (starting in one month). At the end of the 4-year period, both investments will have nothing left over. For both investments, calculate the total amount withdrawn, the APR and the EAR. Which should you choose and why?
{b} Now suppose you learn about another 4-year, $5,000 investment. Annuity QX allows you to withdraw $225 per quarter for the first 8 quarters and then $650 per quarter for the last 8 quarters. For QX, calculate the EAR.
{c} While riding in your car listening to your cheap AM radio, a news reporter say “A new government report says that ‘last month, prices fell by #@$!* per cent. This is an annual rate of deflation of 24 per cent.’” The interference made it impossible for you to hear the missing number. So you pull safely off the road, whip out your financial calculator, and compute the missing number as ????
 

[2] (20 points) EFFECTIVE RATES  Solution
Today (February 23, 2010) you retire and take your $1,000,000 pension with you. You invest it today with an insurance company that gives you two choices: Plan A lets you take out $60,000 every six months starting today and continuing for 10 years (last withdrawal would be August 23, 2019). The withdrawals are the beginning and middle of each year. Plan B would be similar except that the first 10 semi-annual withdrawals would be $90,000 and the last 10 would be only $25,000 each.
{a} Calculate the APR and EAR of each plan.
{b} Ignoring the time value of money for a minute, calculate the total amount withdrawn from both plans.
{c} Which plan do you choose and why?
 

[3] (35 points) TIME VALUE OF MONEY  Solution
Today (September 24, 2009) you make the first of 6 annual deposits of $20 (last one on September 24, 2014) into a bank account that will pay 10% through September 24, 2012 and then 15% thereafter. You also make 5 annual deposits of $X a year starting on September 24, 2014 and ending September 24, 2018 (the last $20 and the first $X are both made on the same day).
{a} How big must X be in order to be able to withdraw $90 a year in perpetuity (first withdrawal is September 24, 2021)?
{b} Assume the same interest rate pattern and the same $90 infinite withdrawals. Instead of the $20 and $X annuities of {a}, what uniform annual deposit D from September 24, 2009 through September 24, 2018 (with a double one on September 24, 2014) would be necessary?
{c} Even though you planned to take out $90 each time, suppose a crooked bank teller skims off another $5 each time you withdrew $90. How many full withdrawals can you take out before running out of funds?
 

[4] (30 points) TIME VALUE OF MONEY  Solution
In order to save for your retirement, today (February 23, 2010) you make the first of what you expect to be 18 equal annual deposits (last one is February 23, 2027). The deposits must be of sufficient size to enable you to withdraw $100,000 per year for 20 years starting on February 23, 2027) – the same day as the last deposit). You expect that your investment will earn 8% a year until February 23, 2027 and then you expect that you will have to move your funds to another account that pays only 6% a year.
You start making the deposits as planned but on February 23, 2016 and 2021 you have to skip those two deposits but you make all of the rest. Everything else goes according to plan. You make the first four planned withdrawals. Just after you make the fourth withdrawal (February 23, 2030, it hits you that you are not going to have enough of a nest egg to make it because of the two deposits you skipped.
You make a quick call to your rich nephew and ask for a lump sum that day to make up for the short fall. How much must the nephew send you on February 23, 2030 so that you can make all of the remaining planned withdrawals?
 

[5] (35 points) CAPITAL BUDGETING  Solution
XC Sports is considering replacing a piece of equipment with a newer, more efficient model. The existing equipment was purchased 3 years ago for $480 and is being depreciated using straight line to a salvage value of $40 over an originally expected life of 10 years. The proposed new equipment has a purchase price of $840 and would be depreciated using straight line to a salvage value of $70 over an expected life of 7 years. The new machine would require additional net working capital of $49.2. The firm has a marginal tax rate of 40% and a required rate of return of 16%.
XC estimates that the new equipment would increase annual sales by $350 and increase annual operating expenses by $124. XC is unsure of how much it can sell the existing equipment for but knows that this is a critical factor in its decision to replace the equipment. Calculate minimum amount that the firm would need to receive in order for the replacement to clear the 16% hurdle.
 

[6] (30 points) CAPITAL BUDGETING  Solution
Emmo’s is contemplating replacing an existing piece of equipment with an newer model. The existing machinery was purchased 6 years ago for $780. It is being depreciated over an original life of 15 years to a $30 salvage value. It can be sold today for $300.
Emmo’s expects that the new equipment will decrease annual operating expenses by $150 and have no effect on sales. The replacement would require additional net working capital of $72. Emmo’s uses the straight-line method, has a marginal tax rate of 40% and has a required hurdle rate of 15%.
The firm is unsure of how much it will have pay for the new equipment. Their policy is to depreciate this type of equipment over a 9 year life to a salvage value of 10% of the original price.
Calculate the maximum amount Emmo’s would be willing to pay for the new equipment.
 

[6.5] (20 points) CAPITAL BUDGETING
{a} Briefly explain why the assumed reinvestment of the NPV or IRR method is better than the assumed reinvestment rate of the other one and why this leads to a preference for one over the other.
{b} Assume you calculate the NPV of a project using straight-line depreciation. If you were to recalculate it again using an accelerated method (such as MACRS or double declining balance) what would happen to the NPV and why? (Hint: what is happening to the timing of the cash flows?)
 

[7] (40 points) CAPITAL BUDGETING UNDER RISK  Solution
Mags, Inc. is a firm operating in a single line of business. It is planning to purchase a new piece of machinery. The cost of the project is $1,400 and the equipment would be depreciated using the straight-line method to a $200 salvage value over a life of 6 years. The firm's marginal tax rate is 35% and additional net-working-capital of $150 would be required.
Below are the forecasts of the expected change in sales revenues less expenses for each year:

The risk-free rate of interest is 3%. The project has a beta coefficient of 2.5. The market's risk premium is 6%. The firm also found the table below from an old copy of a Fin 125 exam that may or may not prove useful: If and when you use the table, apply your a to all cash flows, both operating and non-operating.

{a} The missing (a)’s are .5, .3, .9, .7, and .8. Put them in the right order in the chart before you use the chart.
Read parts {b} and {c} before proceeding.
{b} Compute the NPV assuming the actual life is 4 years and the equipment can be scrapped at that time for $300. Repeat assuming an actual life of 15 years and no scrap value (keep your same a) and assume any remaining book value is taken as depreciation in the 7th year.
{c} Now repeat the two NPV calculations assuming that Mags is very diversified across several different businesses.
 
 

[8] (49 points) CAPITAL BUDGETING UNDER RISK  Solution
Asics Corp. is considering purchasing a new piece of equipment for $50. It would be depreciated over a 4-year life to a salvage value of $10 using the straight-line method. Additional net-working capital of $5 would be needed. The firm has a marginal tax rate of 50%. In addition, the risk-free rate of interest is 4%, the market risk-premium is 5% and the equipment beta is 1.4. The firm has made the following forecasts of the annual change in sales minus operating costs if the equipment is purchased:

The following table may or may not be useful:

Fill in the gray cells with these values (in the right order, of course): .8, .3, .5, .9.
Use an a of 1.0 for any non-operating cash flows.
{a} Assume that the firm is diversified across a wide-range of industries. Calculate the NPV of the purchase assuming an expected life of 7 years (assume any remaining book value would be taken as a single depreciation charge in year 5). Then calculate the NPV assuming an expected life of only 2 years and that the firm can sell the equipment at the end of the second year for $18.
{b} Repeat the two NPV calculations assuming that the firm operates in a single line of business.
{c} In which scenario {a} or {b} is unsystematic risk a factor? What happens to it in the other scenario? Give two examples of unsystematic risk.
 

[9] (25 points) RISK PREMIUMS AND BOND VALUATION  Solution
Bond S is a 20-year straight, nonconvertible bond with a $1,000 par value and an annual coupon rate of 16%, payable semi-annually. It’s currently selling for $892.43..
Bond C is a 20-year convertible bond of the same company with a $1,000 par value and an annual coupon rate of 12%, compounded semi-annually. It is convertible into the company’s common stock at a conversion price of $20 per share.
{a} Calculate Bond S’s yield to maturity.
{b} Suppose the (annual) yield premium between bonds C and S is currently 2.5%. Calculate the current price of bond C.
{c} Calculate your IRR (expressed as an EAR) if you buy Bond C at its current price and convert it 6 years later when the price of the stock is $26 per share.
{d} What would likely happen to the current yield premium if the long-term outlook for some of the company’s main products became brighter. Why?
 

[10] (25 points) RISK PREMIUMS - YIELD CURVE
{a} Draw a downward sloping term structure of interest rates and label both axes
{b} What does the shape of the graph suggest about market expectations of future interest rate movements? Explain what a risk premium is and why one exists between the long-term bond and short-term bonds. Clearly explain your answer.
{c} From the point of view of an issuer, what are the advantages and disadvantages of issuing long-term and issuing short-term securities, given this term structure?
 

[11] (15 points) RISK PREMIUMS AND BOND VALUATION Solution
Explain the concept of a risk premium as it applies to a firm’s long-term subordinated debentures vs. its long-term mortgage bonds. Why does the premium exist in the example that follows below?
Calculate the risk premium between Bond S and Bond M; both bonds have a par value of $1,000. Bond S is a subordinated debenture that matures in 30 years. Its annual coupon rate is 8%, payable semi-annually, and it is currently selling for $1059.34. Bond M is a mortgage bond that also matures in 30 years. Its annual coupon rate is 9.5%, payable semi-annually, and it is currently selling for $1,484.32.
Explain what is likely to happen to the premium as the outlook for the economy begins to improve and why
 

DUE TO NUMBERING SYSTEM THERE IS NO PROBLEM 12
 

[13] (12 points) RISK PREMIUMS AND BOND VALUATION Solution
Ten years ago you bought a bond ($1,000 par, 15 year maturity and 6% annual coupon rate) for $987.65.
{a} If the coupon is paid semi-annually, find the yield to maturity (EAR).
{b} You sell it today when the yield on comparable bonds is 5% per year compounded semi-annually. Find the IRR (EAR) you earned over the holding period.
 

[14] (35 points)BOND VALUATION  Solution
Today Nylon, Inc. issues new 25-year subordinated debentures. The $1,000 par value bonds have an annual coupon rate of 12%, payable semi-annually and are callable at a call price of $1075 after 5 years. They also convertible into Nylon’s common stock at a price of $25 a share. The bonds are issued for $1,015.98.
PARTS {b} – {e} ARE INDEPENDENT OF EACH OTHER
{a} What is a subordinated debenture?
{b} You buy a bond today. What’s your yield to maturity?
{c} You hold the bond for 6.5 years and sell it in the open market. If the yield to maturity at that time is 14% a year, compounded semi-annually, what IRR did you earn?
{d} If the bond is called at the end of 6.5 years, what IRR did you earn?
{e} If you convert the bond at the end of 6.5 years, what IRR did you earn? Assume a common stock price of $29 a share.
{f} For part {d}, assume that Nylon can finance the call with a new issue of 18.5 year bonds issued at par. They would carry an annual coupon rate of 9% payable semi-annually. Legal fees would amount to an additional $10 per bond. Calculate the NPV per bond of the decision to call.
 

[15] (30 points) SUPERGROWTH - EQUITY VALUATION Solution
Today you buy a $1,000 par value bond of the Kane Company. The annual coupon rate is 8%, payable semiannually and the bond matures in 25 years. The yield to maturity is 8.5% a year, compounded semiannually. The bond is convertible into Kane’s common stock at a conversion price of $100 a share.
Yesterday Kane paid a dividend per share of $2.50. The earnings and dividends are expected to grow at 25% for the next 10 years before declining to a 6% annual rate for the indefinite future.
{a} If common stock investors require an 18% return and you expect the forecasts to hold, calculate the IRR of your investment if you expect to convert the bond in 7 years.
{b} If the forecasts still hold and you keep the bond to maturity and convert on the last day, what is your IRR? Why is this answer bigger or smaller than part {a}?
 

[16] (30 points) SUPERGROWTH - EQUITY VALUATION Solution
Today you buy a $1,000 par value bond of Bunky’s Burgers for $950. The bond pays an annual coupon rate of 8%, compounded semi-annually and matures in 20 years. The bond is convertible into Bunky’s common stock at a conversion price of $10 a share.
Yesterday Bunky’s paid a dividend per share of $7.60. The firm forecasts that its earnings and dividends will decline at 20% a year for the next 5 years and then decline at 10% a year for another 4 years before stabilizing at a positive growth rate of 5% a year for the indefinite future.
{a} If common stock investors require an 18% return and you expect the forecasts to hold, calculate the IRR (expressed ss an EAR) of your investment if you expect to convert the bond in 6 years.
{b} If the forecasts still hold and you keep the bond to maturity and convert on the last day, what is your IRR (express as an EAR?
 

[17] (70 points) COST OF CAPITAL  Solution
Huckleberry SmartPhones is planning its 2010 capital budget and needs your advice. The firm believes that the capital structure relations shown below are optimal and will be maintained.

The firm has a marginal tax rate of 40% and has $15,000,000 of retained earnings available for investment this year. Huckleberry’s stock currently has a dividend yield of 18% and its earnings and dividends are expected to grow indefinitely at 8%.
The firm can raise funds under these conditions:
BONDS: Up to $13 million in new bonds can be issued to yield the investor 8% per year, compounded semi-annually. The annual coupon rate is 9% a year, payable semiannually. The bonds have a par value of $1,000 and mature in 20 years. Flotation costs would be $48.96 per bond. An additional $26 million of bonds would cost the firm a before-tax APR of 2.52% higher than the first issue of bonds. Still another additional $26 million would have a before-tax cost (APR) of 4.52% higher than the first issue of bonds. Any additional bonds would have a before-tax cost (APR) of 7.52% above the first issue of bonds.
PREFERRED STOCK: Any size issue can sold to net the firm $125 per share. Par value is $100 and the dividend rate is 25% paid annually. The yield to the investor is 17.24%.
COMMON STOCK: Any size issue can be sold with flotation costs and underpricing equal to 20% of the current stock price.
The firm is considering six potential projects with the following forecasted cash flows:

{a} Compute Huckleberry’s marginal cost of capital for each segment of the marginal cost schedule and display your results on a CLEARLY LABELED graph.
{b} For project F, assume a life of 6 years and uniform annual cash flows of $5.43 million. Calculate its IRR.
{c} On the same graph, plot the firm’s IRR schedule and indicate which projects are acceptable. Compute the average cost of capital for the capital budget you are advocating.
{d} For the preferred stock, calculate the flotation costs charged by the underwriters.
{e} Calculate the NPV of project F. Clearly indicate your discount rate.
(f) Why did Huckleberry accept projects whose IRR is less than the required rate of return on their common stock?
{g} Draw and label the graph from the last lecture depicting what happens to ke, ki and k0 as the level of debt increases. If the firm is currently operating close to the right side of the optimal range, where are they likely to get their next increase in capital from and why?
 

[18] (70 points) COST OF CAPITAL  Solution
Psyleron is planning its forthcoming capital budget and needs your advice. The firm believes that the capital structure relations shown below are optimal and will be maintained.

The firm has a marginal tax rate of 40% and has $20,000,000 of retained earnings available for investment this year. Three years ago Psyleron paid a common stock dividend of $5.513 per share and yesterday it paid one of $6.944. Assume this rate of growth continues for the indefinite future. The common stock is currently trading for $75 a share.
The firm can raise funds under these conditions:
BONDS: Up to $20 million in new 25-year $1,000 par bonds can be issued with an annual yield to maturity of 10.32%, compounded semiannually. The annual coupon rate is 12% a year, payable semiannually. Flotation costs would be $64.98 per bond. Any additional bonds would have an annual after-tax cost of 1.8% higher than the first $20 million.
PREFERRED STOCK: New $50 par value preferred stock can be issued with an annual dividend of 16% to produce an annual yield of 12%. The first $25 million requires flotation costs of $9.53 and anything beyond that requires flotation costs of $19.61 per share.
COMMON STOCK: Up to $30 million can be sold with flotation costs of $25 a share. Any additional shares would require flotation costs of $30.82.
The firm is considering five potential projects with the following forecasted cash flows (assume a 5-year life for all projects):

{a} Compute Psyleron’s marginal cost of capital for each segment of the marginal cost schedule and display your results on a CLEARLY LABELED graph.
{b} For project C, assume a life uniform annual cash flows of $7.11 million. Calculate its IRR.
{c} On the same graph, plot the firm’s IRR schedule and indicate which projects are acceptable. Compute the average cost of capital for the capital budget you are advocating.
{d} Calculate the NPV of project E. Clearly indicate your discount rate.
(e) Why did Psyleron reject a project whose IRR is greater than the average cost of capital found in {c}?

[1] i_QQ = 3.87%, EAR_Q = 16.40%, APR_Q = 15.48%, i_M = 1.28% per month, EAR_M = 16.46%, APR_M = 15.33%; (b) IRR=3.37% per quarter, EAR_QX = 14.19%; (c) x=-2.261%

[2] i_A = 2.008%, EAR_A = 4.06%, APR_A = 4.016; IRR_B = 2.2057 per period, EAR_B = 4.46%, APR_B = 4.41%, choose B

[3] (a) $600 needed 2020 x = $24.29; (b) D=21.36; (c) 21 full payments, #22 is less than full

[4] D=$32,464.72; missing two D's at 2030 = $151,513.24

[5] DCF = 162, outlay = 750 - .6MV, MV = $112.88

[6] DCF=.1X-50, BVold=480, DCF = 70+.04X, outlay = X-300, proceeds sale old = 372, X=827.39

{7} outlay = 1550, Cost = 1400, DCF = 746, s = 238.29, a = .8, n=4 years NPV_{not diversified} = 1062.85 NPV_diversified = 743.05; n=15 yrs NPV_nd = 5331.95 and NPV_div = 2171.24

{8} outlay = 55, D CF=40, s = 21.91, a = .8, Diversified: k'=11%, n=7yrs NPV=130.81; n=2 yrs, NPV=37.04; NOT-DIVERSIFIED n= 2 yrs NPV=32.17, n=7yrs, NPV=134.66

{9} i=18%, i_c = 15.5%, P_c = 785.60, r=10.59%, gap widens

{11} P_S = 1,059.34, P_M = 1,484.32, i_S = 7.5% per year, i_M = 6% per year, Premium = 1.5%

{13} (a) i = 3.06%, EAR = 6.22%, (b) P = $1,043.76, r = 3.24% per period, EAR = 6.59%

{14}(b) i=5.9% per period or 11.80% per year; (c) n=37 periods, P=868.83, r=10.19% per yr csa; (d) IRR=12.42%/year csa; (e) r=13.25% per yr; (f) outlay=85, savings = 15 per period, NPV=182.93

[15] Price of bond = $948.52, Price of stock (year 7) = $165.35 so conversion value is $1653.51, IRR over 7 yrs = 7.42% per period or 14.83% per year, Price of stock year 25 = $492.89 and IRR = 5.77% per period or 11.55% per year

{16}(a) P₆ = 12.04 per share or conv value=1,204; r=5.81%/period or EAR=11.96%; (b) P₂₀ = 22.57, IRR = 5.28%, or EAR=10.83%

[17] Break points: debt 20, 60 and 100; equity 60; component costs debt 8.48% per year or 5.1% after tax, 6.6%, 7.8%, 9.6% (all after-tax); preferred 20% ({d} flotation costs on preferred =$20 per share), retained earnings 26% and common stock 30.5%, marginal costs: (0-20) 11.8%, (20-60) 12.8%, (60-100) 14.7% and (100+) 15.9%; accept B, D, F and E and raise $90 million at an ACC of 13.2%; IRR of F = 16.0%; NPV of F = $1.26 million using a discount rate of 13.75%

{18} Break points $50 and $125; Cost debt 6.6% and 8.4%, pf'd stock 14.0 and 17.0; equity 18.0, 23.0 and 25.0; Accept A-E-B and raise $110 million, NPV_E = 3.68 using DCF_E = 15.63 and k_E = 13.98%