Rates word problem

Question

If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by M = M0(1.07763)t

(a)  What is the effective annual rate for this account?

(b) What is the nominal rate?

Answer 1

If the balance, M, at time t in years, of a bank account that compounds its interest payments monthly is given by M = Mo(1.07763)t

(a) What is the effective annual rate for this account?

(b) What is the nominal rate?

First of all list all the factors pertaining to the math problem:

Balance at ime (t) = Mo (1.07763)^t

Effective Annual Rate = EAR =0.07763

Formula = (1 + i/n)^n -1

i = nominal interest rate

n= the number of compounding periods per year

The formula balance with monthly compounding is M = Mo (1 + i/12)^12^t

(Step 1)

Find the effective annual rate (EAR):

M = Mo (1.07763)^t

Formula for annual growth Mo(1 + EAR)^t

Solve for EAR: EAR = 1.07763 - 1 = 0.07763

(convert from decimal to percentage) 0.07763 x 100 = 7.763 %

(Step 2)

Find the nominal rate:

Use the formula EAR = (1+ i/n)^n - 1 and substitute the known values.

EAR = 0.07763 and n = 12 (monthly compounding)

EAR = (1+ i/n)^n - 1

0.07763 = (1 + i/12)^12 -1 (add 1 on both sides)

1.07763 = (1 + i/12)^12

Next take the 12^12th root of both sides:

(1.07763)^1/12 = 1 + i/12

1.0062999 = 1 + i/12 subtraction 1 from both sides

0.0062999 = 1/12

Multiply by 12 to find the nominal rate (i)

i = 0.0062999 x 12 = 0.0755988 roundoff as specified

i = 0.0756 (convert from decimal to percentage) 0.0756 x 100 = 7.56 %

Solution:

The Effective Annual Rate (EAR) = 7.763 % The nominal rate (i) = 7.56 %

I hope the mathematical calculations (step by step approach) was helpful. Please let me know if you require any more assistance. If anyone in my neighborhood is interested in setting up an in-person math tutoring session. I look forward to hearing from them. Have an amazing day. Doris H.