Test 1 solution sketches

Note for multiple-choice questions: Choose the closest

answer

PV of Perpetuity

Alma receives $1,000 annually forever. The stated annual discount rate is 7%, compounded every 6 months. Determine the present value given the following assumptions.

PV of Perpetuity

Alma receives the first payment three months from today. Subsequent payments are made yearly.

EAR = (1.035)2 - 1 = 7.1225% PV = 1000/.071225 * (1.071225)3/4

= $14,783.53

PV of Perpetuity

Alma receives $500 every six months, starting today.

PV = 500 + 500/.035 = $14,785.71

Internal Rate of Return

Joey invests $50,000 today. In return he receives $800 one year from now, followed by a payment 10% higher the next year. These yearly payments continue forever, with each payment 10% higher than the previous payment. What is the internal rate of return of this investment?

Internal Rate of Return

50,000 = 800 / (IRR - .1) 50000*IRR – 5000 = 800 50000*IRR = 5800 IRR = 0.116 or 11.6%

Choosing Investments Ryan’s Carnivals has exactly $100,000.

Ryan must invest one or more of the following three options. The effective annual discount rate is 15%. Investment I: He invests $60,000 today and

receives $92,000 two years from today Investment II: He invests $51,000 today and

receives $80,000 one year from today Investment III: He invests $X today and

receives $(1.25*X) one year from today

Choosing Investments X can be any non-negative amount that

Ryan wants. Which investments should Ryan invest positive amounts of money in? Must invest in III since exactly $100K must

be invested today and I and II total to more than $100K

Choosing Investments

III only: NPV = 100000(1.25)/1.15 – 100000 = 8,700

I and III: NPV = 92000/(1.152) – 60000 + 40000(1.25)/1.15 – 40000 = 13,043

II and III: NPV = 80000/1.15 – 51000 + 49000(1.25)/1.15 – 49000 = 22,826 Choose II and III

Yield to Maturity

Roberto buys a zero-coupon bond with a face value of $5,000. The face value will be paid in 7 years. The current price of the bond is $2,500. What is the yield to maturity (expressed as an effective annual rate)? 2500 = 5000/(1+r)7

(1+r)7 = 2 r = 10.41%

Compounding interest

Jerry makes a $1,000,000 investment that will earn 12% per year. However, he does not know if his interest is being compounded yearly or monthly. How much more will he earn in interest over the next two years if interest is compounded monthly (versus yearly)?

Compounding interest

Interest if monthly:1,000,000*(1.0124 – 1) = 269,735

Interest if annually:1,000,000*(1.122 – 1) = 254,400

Difference = $15,335

Firm Value Shannon has made a major scientific

breakthrough that will lead to increased profits for her company as follows: $500,000 one year from today $200,000 two years from today Each subsequent year: 5% lower than the

previous year These extra profits will be received by

the firm forever.

Firm Value

How much does this breakthrough increase the firm’s current value if the effective annual discount rate is 8%? (In $1000s):

500/1.08+[200/(.08-(-.05))]*1/1.08=1,887

Increases the firm’s value by $1.887 million

Nominal Payments Byron is set to receive a real

payment of $50,000 (in today’s dollars) five years from now. Inflation will be 6% for the first three years, and 8% per year thereafter. The nominal payment five years from now will be ____. 50,000 * (1.06)3 (1.08)2 = $69,460

Yield to Maturity A bond with a face value of $1,000 is

currently selling for $800. Annual coupon payments of $50 are to be paid later today and one year from today. The bond matures in one year. What is the yield to maturity (expressed as an effective annual rate)? 800 = 50 + 50/(1+r) + 1000/(1+r) 750 = 1050/(1+r) r = 1050/750 – 1 = 0.4 or 40%

Payback periods

Ali bought a new Green Pimus Hybrid vehicle today for $30,000. He will receive $4,500 in benefits each year, forever, starting one year from today. Assuming an effective annual discount rate of 10%, in how many years will the net present value of the investment turn from negative to positive.

Payback periods 30000 = 4500/.10 * (1 – 1/(1.1)T) 2/3 = 1 – 1/(1.1)T

(1.1)T = 3 T = ln(3)/ln(1.1) = 11.53 ≈ 12 years

Or we could just guess and check some numbers to see what the best answer is (I will start with 10 since it is in the middle)…

Payback periods

Let T=10: 4500/.1 * [1 – 1/1.110] = 27,650.56 Must be more than 10

Let T=12: 4500/.1 * [1 – 1/1.112] = 30,661.61

Retirement Planning

Booker is planning his retirement. He will make a single deposit 5 years from now so that he is able to withdraw $300,000 per year in the future, starting 20 years from today and ending 40 years from today. How much must the deposit be so that the account balance will be $0 after the final withdrawal? Assume a SAIR of 9%, compounded monthly.

Retirement Planning EAIR = (1 + .09/12)12 = 1.0938069 PV of withdrawals (in $1000s):

300/.09381*[1 – 1/(1.09381)21]* 1/1.0938119 = 493.56

PV of deposit (in $1000s): 493.56 Amount of deposit in 5 years (in

$1000s):493.56(1.0938069)5 = 772.75

Equivalent Annual Cost

Gareth buys the Hamburger Broiler 3000 machine for $5,000 today. The machine lasts for 14 years, and has maintenance costs of $600 each three times: in 5 years, 9 years, and 12 years. The effective annual discount rate is 9%. What is the equivalent annual cost of this machine?

Equivalent Annual Cost

PV of costs:5000 + 600/(1.09)5 + 600/(1.09)9 + 600/(1.09)12 = $5,879.54

EAC:5879.54 = EAC/.09 * [1 – 1/(1.09)14]5879.54 = 7.78615 * EAC

EAC = $755.13

Internal Rate of Return Diane must choose between two

mutually exclusive investments, A or B. If she invests in A, she pays $600 today and

receives $720 one year from now. If she invests in B, she pays $2,000 today

and receives $2,200 one year from now. Diane’s effective annual discount rate is 5% Assume that any remaining money is

invested in something that has a NPV of 0

Parts (a) and (b) What is the

internal rate of return for Investment A? 600 * (1 + IRR) =

720 1 + IRR = 1.2 IRR = .2 = 20%

What is the internal rate of return for Investment B? 2000 * (1 + IRR) =

2200 1 + IRR = 1.1 IRR = .1 = 10%

Part (c)

Which investment should Diane choose? NPVA = 720/1.05 – 600 = 85.71 NPVB = 2200/1.05 – 2000 = 95.24

Choose B due to higher NPV