Question

Because unintended lane changes by distracted drivers are responsible for 43% of all highway fatalities, Ford Motor Co. and Volvo launched a program to jointly develop a technology to prevent accidents by sleepy or distracted drivers. A device costing $260 tracks lane markings and sounds an alert during lane change. If these devices are included in 100,000 new cars per year beginning 3 years from now, determine the present worth of the cost over a 10-year period at an interest rate of 10% per year.

Answer 1

Answer:

present cost of the device: $   114,634,777.81

Explanation:

we are going to add a device worth 260 dollars on 100,000 cars three years from now, every year.

The cash outflow will be for: 260 x 100,000 = 26,000,000 every year

Timeline:

    1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th)      

<----/----/----/(/----/----/----/----/----/----/----/)---->

The first step is to calcualte the value of the annuity that begings in 3 years:

Notice during a seven years period it has 8 payment so it will be an annuity due:

$C \times \frac{1-(1+r)^{-time} }{rate}(1+r) = PV$

C 26,000,000

time 8 years

rate 0.1

$26000000 \times \frac{1-(1+0.1)^{-8} }{0.1} = PV$

PV $152,578,889.2600

Now, we calcualte the present value of this annuity which begins 3 years from now:

  1st 2nd Annuity-due

<----(/----/----/)--------->

We discount as a lump sum:

$\frac{Nominal}{(1 + rate)^{time} } = PV$  

Nominal: 152,578,889.2600

time   3 years

rate  0.1

$\frac{152,578,889.2600}{(1 + 0.1)^{3} } = PV$  

PV   114,634,777.81