BFC2140 question

[Sam_95]

Right well, this question seems very easy, but I'm having a pretty terrible mental blockage (probably from having crammed all day)

Joanne has signed up for a 2 year car loan which requires payments of $200
per month for the first year and payments of $400 per month during the
second year. The annual interest rate is 12% and payments begin in one
month. How much has Joanne borrowed today?
A. $6,246.34
B. $6,389.78
C. $6,428.57
D. $6,753.05

[Reckoner]

Draw a timeline. Discount the $200 for two years as you normally would with the ordinary annuity formula. This brings the first 2 years of payments to t=0. Now do exactly the same annuity formula for the $400. What time period does this bring the $400 payments too? We need them to be in the same period to add them properly. Also be careful of what interest rate you use (monthly rather than per annum). It has to be consistent throughout the calculation as well.

If you still need a bit of help I'll post the numbers.

[Sam_95]

Still don't get it

[Reckoner]

PV_{0, Total}=PV_{0,$200 payments} + PV_{0, $400 payments}

As the payments are monthly, but the interest rate is pa, the monthly interest rate is 0.12/12=0.01 per month.

PV_{0,$200 payments}= (200/0.01)*(1-1/(1.01^12)) =2,251.0155     (12 monthly payments of $200 - ordinary annuity formula)

Now, for the $400 payments, the structure is still exactly the same, so we can still use the annuity formula

PV_{1,$400 payments}= (400/0.01)*(1-1/(1.01^12)) =4502.0310.

So can we just add these values together to get the total NPV? Not quite, because the $200 payments are an ordinary annuity starting today (so are discounted to today), but the $400 ordinary annuity starts in one years time so the PV we calculated is the PV in one years time. So to get them into the same time period, we must discount the $400 annuity a further year to get it in today's dollars.

Here again we have to use the monthly interest rate to do this (so its consistent), so we discount the $4502 back 12 months.

PV_{0,$400 payments}= 4502.0310/(1.01^12) =3995.3239

So now out PVs are in the same time period, so we can add them to get the loan amount

PV_{0, Total}= 2,251.0155  + 3995.3239  =  $6,246.34. Hence, A.

I hope this helps

[Sam_95]

You're a champions, thank you very much, explained it perfectly.