Calculating Loan Payments by converting PV to PMTS

CFA Level I Quant
#1

Hello,

Say you want to buy a car for $4,329 and the bank offers you a 5-year loan at 5% yearly interest. You want to calculate what the PMTs will be so you apply the PVannuity formula, and isolate PMT and you get a PMT of $1,000.

Constructing this you get:

t = 0: (4329) t = 1: 1000 t = 2: 1000 t = 3: 1000 t = 4: 1000 t = 5: 1000

The future value of this annuity will turn out to be $5,525, so the interest you paid was (5525-4329) => 1196. And to verify this: 4329*(1+5%)^5 => 5525

HOWEVER, attempting to construct an amortization table shows different results for interest paid :

Year Amt Available Ending Amt (+5%) PMT Amount available after PMT

1 4329 4545 1000 3545 2 3545 3723 1000 2723 3 2723 2859 1000 1859 4 1859 1952 1000 952 5 952 1000 1000 0

So the interest you paid was (5000-4329) = 671. Effectively, this is (5000/4329)(1/5) - 1 = 2.92%/year

So why is there a difference here? How is the bank effectively charging 5% compounded interest per year? Or does it only charge 5% on the AMT Available per year? Thank you.

#2

Did I pay the $5,000 all at once or did I spread it out over 5 years? Remember, a dollar in one year is not worth a dollar today!

(5,525/4,329)0.2 - 1 = 5%: this follows from the general identity FV = PV * (1+i)n

#3

Hey Breadmaker,

I am not as confused on the percentages, just on the total payout amounts. I think I am failing to see something of great importance, maybe its the excess hours of studying.

No you paid $1,000 per year under both scenarios. But I cannot seem to reach 5525 under the amortized scenario.

Sorry I am confused.

#4

Banks charge you interest only on the amount currently outstanding.

You didn’t have $4,329 outstanding for the full term.

#5

Thank you!

#6

You’re welcome.