Introduction to Finance - Busfin 1030 - Problem Set 3

Introduction to Finance

BUSFIN 1030

Professor Schlingemann

Problem Set 3

SOLUTIONS

Problem 1:

You are deciding among three cars to use as a company car. The garage offers you a lease deal and two different options for purchasing the car. You are completely indifferent among these cars except for their costs.  Once you have decided which car to take, you will always take the same car again at the end of its useful life. The pre-tax residual value (salvage value) for the car at the end of its useful life is equal to (100 / n) % of the purchase price, where n is equal to the lifetime of the car. All cars are fully depreciated according to its lifetime on a straight-line basis with no half-year convention. The discount rate is 7% and the tax rate is 20%. Assume that you pay the price of the car up front, and the annual costs at the end of the year. The costs of leasing occur at the end of each period. (For example, if you purchase Car B, you pay $18,000 in year 0 and $1,000 in year 1, year 2, etc. If you lease car A, you pay $4,650 in year1, year 2, etc.) Note: consider all relevant cash flows for this problem.

                        Lease A        Purchase B                Purchase C

Purchase price                   ---                $18,000                $45,000

Total annual costs*        $4,650                $900                            ---

Lifetime of the car        3 years                5 years                        18 years

* Includes all after-tax costs like fuel, wear and tear, maintenance, etc.

Using the Equivalent Annual Cost (EAC) method, which of the cars should you decide to drive always?

ANSWER:

First find the relevant cash flows associated with each purchase option:

Purchase B:

Depreciation tax shield = 20% × $3,600 = $720

After-tax salvage value = $3,600 – 20% × $3,600 = $2,880

PVB = 18,000 + [pic 1]$16,684.64

EACB ⇒ [pic 2]16,684.64 ⇒ EACB = $4,069.23

Purchase C:

Depreciation tax shield = 20% × $2,500 = $500

After-tax salvage value = $2,500 – 20% × $2,500 = $2,000

PVC = 45,000 + [pic 3]$39,378.73

EACC ⇒ [pic 4]39,378.73 ⇒ EACC = $3,914.74

Purchase C has the lowest cost per year (note that the lease is already in $ per year).

Problem 2:

You are thinking about investing your money in the stock market. You have the following two stocks in mind: stock A and stock B.  You know that the economy can either go in recession or it will boom. Being an optimistic investor, you believe the likelihood of observing an economic boom is two times as high as observing an economic depression. You also know the following about your two stocks:

1. Calculate the expected return for stock A and stock B
2. Calculate the total risk (variance and standard deviation) for stock A and for stock B
3. Calculate the expected return on a portfolio consisting of equal proportions in both stocks.
4. Calculate the expected return on a portfolio consisting of 10% invested in stock A and the remainder in stock B.
5. Calculate the covariance between stock A and stock B.
6. Calculate the correlation coefficient between stock A and stock B.
7. Calculate the variance of the portfolio with equal proportions in both stocks using the covariance from answer e.
8. Calculate the variance of the portfolio with equal proportions in both stocks using the portfolio returns and expected portfolio returns from answer c.

ANSWER

1. p(boom) = 2/3 and p(recession)=1/3 (Note that probabilities always add up to 1)

        E(RA) = 2/3 × 0.10 + 1/3 × 0.06 = 0.0867 (8.67%)

        E(RB) = 2/3 × -0.02 + 1/3 × 0.40 = 0.12 (12%)

b)         SD(RA) = [2/3 × (0.10-0.0867)2 + 1/3 × (0.06-0.0867)2]0.5= 0.018856 (1.886%)

        SD(RB) = [2/3 × (-0.02-0.12)2 + 1/3 × (0.40-0.12)2]0.5 = 0.19799 (19.799%)

1. Portfolio weights: WA=0.5 and WB=0.5:

        E(RP) = 0.5 × 0.0867 + 0.5 × 0.12 = 0.10335 (10.335%)

1. Portfolio weights: WA=0.1 and WB=0.9:

        E(RP) = 0.1 × 0.0867 + 0.9 × 0.12 = 0.11667 (11.667%)

1. COV (RA,RB) =

      2/3 × (0.10-0.0867) × (-0.02-0.12)  + 1/3 × (0.06-0.0867) × (0.40-0.12) = –0.0037333

f)        CORR(RA,RB) = –0.0037333 / (0.018856 × 0.19799) = –1  (Rounding! Remember the correlation coefficient cannot be less than –1)

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