Question

In October 2012, the average house price in the USA was $223,500. In October 2003, the average price was $285,000. What was the nominal (APR) annual change in the average selling price compounded monthly? What was the per month rate of change in the average selling price? What was the EAR annual rate of change?

Answer 1

As we see that the average selling price of house has decreased over a period of 9 years hence the annual percentage rate would be negative for such scenario.
Given,

Average house price in October 2012,FV = $223500
Average house price in October 2003,PV = $285000

FV = PV*(1+r/n)^n*t
Where r = Annual percentage rate
n = no. of times compounded in a year = 12
t = time period for which the future value is calculated = 9

223500 = 285000(1+r/12)^12*9
(1+r/12)^12*9= 223500/285000
(1+r/12)¹⁰⁸ = 0.78421
1+r/12 = 0.78421^1/108
r = (0.9978-1)*12
r = -2.70%
Nominal Annual Percentage rate compounded monthly is -2.70%
b) Per month rate of change = Annual Percentage rate/12
      = -2.70/12
      = -0.225%
Per month rate of change is -0.23%
c) Effective Annual rate is the actual interest rate earned or paid over a given period due to the effect of compounding.

Effective Annual Rate of change = (1+r/n)ⁿ-1
where r = Annual Percentage rate = -2.70% or -0.027
n= compounding period = 12

EAR = (1+(-0.027/12))¹²-1
      = (1+(-0.00225))¹²-1
= (0.99775)¹²-1
      = -0.02667
   = -2.67%

Effective Annual rate = -2.67%