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?
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
not bad
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%
To calculate the Effective Annual Rate (EAR) on a loan, you need to consider two factors: the nominal interest rate and the compounding period. The nominal interest rate is the yearly interest rate, and the compounding period is the frequency at which interest is calculated and added to the loan balance.
In this case, we are given the monthly payment on the loan, but we need to find the nominal interest rate and the compounding period. Since the loan term is 30 years, we can assume that the compounding period is monthly.
To find the nominal interest rate, we need to use the loan payment formula:
Loan Payment = Principal * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
- Loan Payment is $22,000
- Principal is 80% of $2,800,000, which is $2,240,000
- r is the monthly interest rate
- n is the number of compounding periods, which is 30 years * 12 months
Rearranging the formula to solve for r, we get:
r = (Loan Payment * ((1 + r)^n - 1)) / (Principal * (1 + r)^n)
To find the nominal interest rate, we can use an iterative approach, such as the Newton-Raphson method or trial and error:
1. Start with an initial guess for r (e.g., 0.01, or 1%)
2. Plug in the values into the formula and calculate the left-hand side of the equation.
3. Compare the calculated value with the right-hand side of the equation (Loan Payment * ((1 + r)^n - 1)) / (Principal * (1 + r)^n).
4. If the calculated value is equal to the right-hand side of the equation, the initial guess is the correct nominal interest rate.
5. If the calculated value is higher, increase the guess for r, and if it is lower, decrease the guess for r. Repeat steps 2-5 until you find the correct nominal interest rate.
Once you have found the nominal interest rate, you can convert it into the Effective Annual Rate (EAR), which takes into account the compounding:
EAR = (1 + r)^c - 1
Where:
- r is the nominal interest rate
- c is the number of compounding periods (in this case, 12 since it's a monthly compounding period)
Substitute the values into the formula to calculate the Effective Annual Rate (EAR).
Keep in mind that this calculation assumes that the loan payment remains constant over the entire loan term and does not consider any additional fees or charges associated with the loan. It's always advisable to consult with a financial professional or use specialized loan calculators to get more accurate results.