You are shopping for a car and read the following advertisement in the newspaper

Question

You are shopping for a car and read the following advertisement in the newspaper: "Own a new Tesla! No money down. Five annual payments of just $20,000." You have shopped around and know that you can buy a Tesla for cash for $85,000. Assume that you must make the annual payments at the end of each year. Draw the timeline for the advertised offer. What interest rate is the dealer advertising (i.e. what is the IRR of the loan in the advertisement)? Assume that the interest rate is between 0% and 20%. Start with an initial guess of 10% and report your next 4 guesses using the bisection method. What is the present value of a $20,000 per year, 5 year annuity using the interest rate found in Part (b)?

Explanation / Answer

(a) The timeline is as follows.

(b)

Initial cash flow = 85,000

Annual cash flows = - 20,000

IRR is that discount rate which will equate the sum of discounted cash flows of $20,000 per year equal to $85,000.

With approximation, we shall use the high-low method with 10% & 20% discount rates. Using this method,

IRR = RL + [NPVL / (NPVL - NPVH)] x (RH - RL)

Where

RL: Lower discount rate (10%)

RH: Higher discount rate (20%)

NPVL: NPV corresponding to lower discount rate

NPVH: NPV corresponding to higher discount rate

The NPV calculations as follow.

So,

IRR = 0.10 + [9,184 / (9,184 - 38,096)] x (0.20 - 0.10)

= 0.10 - (0.3177 x 0.10)

= 0.0682 = 6.82%

(c)

Present value interest factor of annuity (r%, N years) = [1 - (1 + r)-N] / r

Present value interest factor of annuity, PVIFA (6.82%, 5 years)

= [1 - (1.0682)-5] / 0.0682 = 4.12

So, present value of the annuity = $20,000 x 4.12 = $82,400

Year Cash Flow Discount Factor @10% Discounted CF @10% Discount Factor @20% Discounted CF @20% 0 85,000 1.0000 85,000 1.0000 85,000 1 -20,000 0.9091 -18,182 0.8333 -15,152 2 -20,000 0.8264 -16,529 0.6944 -11,478 3 -20,000 0.7513 -15,026 0.5787 -8,696 4 -20,000 0.6830 -13,660 0.4823 -6,588 5 -20,000 0.6209 -12,418 0.4019 -4,991 NPV 9,184 NPV 38,096

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