# Finance

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You have just purchased a new warehouse. To finance the purchase, you have arranged for a 30-year mortgage loan for 80 percent of the \$2,800,000 purchase price. The monthly payment on the loan will be \$22,000.

a. What is the effective annual rate (EAR) on this loan?

• Finance -

80% of2800000 = 2240000

2240000 = 22000( 1 - (1+i)^-360)/i
101.81818.. = ( 1 - (1+i)^-360)/i

101.818182i = 1 - (1+i)^-360
(1+i)^-360 = 1 - 101.818182 i

Now that is tough equation to solve.
In the "olden days" we used interpolation.
Even Wolfram, at least in its simple version, cannot handle it
http://www.wolframalpha.com/input/?i=solve+1%2F%281%2Bx%29%5E360+%3D+1+-+101.81x

let's try some values:
i = .04
PV = 22000(1 - (1.04)^-360)/.04 = 550,000 way off! , expecting appr 224,000
i = .01
PV = 22000(1 - 1.01^-360)/.01 = 2,138,803.28
i = .0005
PV = 22000(1 - 1.005^-360)/.005 = 3,669.415 , rate is too low

.01 was very close

let i = .0099
PV = 22000(1 - 1.0099^-360)/.0099 = 2,158.164 , even better

let i = .0095
PV = 22000(1 - 1.0095^-360)/.0095 = 2,238,801, a bit too low
let i = .0094
PV = 22000(1 - 1.0094^-360)/.0094 = 2,259794 , a bit too high

do you get the idea?
we could get as close as we want with a good calculator
but the monthly rate has to be between .0095 and .0094
I will guess at .00945
so the effective annual rate is 12(.00945) = .1134
or 11.34%