Question

A businessman wants to buy a truck. The dealer offers to sell the truck for either $145,000 now, or 7 yearly payments of $24,000. What interest rate would make these two options financially equivalent?

Answer 1

Cash price of truck or Present value of truck= 145000

Annual payments for truck on credit (P) = 24000

number of yearly payments (n) =7

To be two options financlly equivalent rate should be such at which present value of second option is equal to present value of first option that is $145000

PV of second option or PV of annuity formula = P*(1-(1/(1+i)^n))/i

24000*(1-(1/(1+i)^7))/i

i is that interest rate at which PV is equal to 145000

145000 =24000*(1-(1/(1+i)^7))/i

We will calculate it by trial and error method and interpolaiton formula

assume i is 3.5%

PV =24000*(1-(1/(1+3.5%)^7))/3.5%

=146749.0555

Assume i is 4%

PV =24000*(1-(1/(1+4%)^7))/4%

=144049.3121

Interpolation formula for rate calculation = Lower rate + ((upper rate -lower rate)/(upper value - lower value)*(upper value - actual loan value))

3.5% + ((4%-3.5%)/(146749.0555-144049.3121)*(146749.0555-145000))

=0.03823929952 or 3.82%

So interest rate must be 3.82% to be both options financlly equivalent.

Excel function = rate(number of periods, annuity or payment per period, -present value)

=rate(7, 24000, -145000)

=3.82%

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