Question

Today is 1 January 2015. Anna just used $200,000 ($100,000 is from Anna’s own investment and $100,000 is raised by taking a 5-year loan) to purchase a cafe franchise. To operate this business, Anna needs to pay rent, maintenance costs, labour costs and loan repayments. She took a 5-year $100,000 loan from the bank.

- Anna will make 60 monthly repayments at the end of each month.

- Anna can have a one year interest-only-period at the beginning of the loan. Anna's repayments will be interest-only1 for the first year (i.e., first 12 payments will be interest-only payments), followed by payments of principal plus interest for the following 4 years.

- During the interest-only period, the interest rate will be j12 = 3% which means Anna needs to repay 3%/12 ∗ 100000 by the end of each month.

- Anna needs to pay $4,000 by the end of each month after the interest-only period.

- This package has an annual fee of $200. The package fee is paid by the end of January of each year during the following five year period (from 1 January 2015 to 31 December 2019).

What is the equation to find the implied annual nominal rate of interest payable monthly (i.e., j12) charged by the bank for this loan package?

Answer 1

What is the equation to find the implied annual nominal rate of interest payable monthly (i.e., j12) charged by the bank for this loan package?

Nominal Rate of Interest Payable Monthly=R

Monthly Effective Interest =i=R/12

Cash Flow at the end of Month 1=((3%*100000)/12)+$200=$$450

Cash Flow at the end of each month from month 1 to month 12=$450

Uniform Series Present Worth Factor =USPWF=(P/A,i,N)=(((1+i)^N)-1)/(i*((1+i)^N))

N=12

USPWF=(P/A,i,12)=(((1+i)^12)-1)/(i*((1+i)^12))
A=Present Value of Interest Only Plus annual fees = 450*(((1+i)^12)-1)/(i*((1+i)^12))

Monthly Payments during balance 4 years(48 months)=$4000+$200=$4200

N=48

USPWF=(P/A,i,48)=(((1+i)^48)-1)/(i*((1+i)^48))

Present Value of Interest+Principal payments at end of Year 1=4200*(((1+i)^48)-1)/(i*((1+i)^48))

B=Present Value of Interest+Principal payments TODAY =(4200/((1+i)^12))*(((1+i)^48)-1)/(i*((1+i)^48))

$100000=A+B=450*(((1+i)^12)-1)/(i*((1+i)^12))+(4200/((1+i)^12))*(((1+i)^48)-1)/(i*((1+i)^48))

Nominal Rate of Interest Payable Monthly=R=i*12

450*(((1+i)^12)-1)/(i*((1+i)^12))+(4200/((1+i)^12))*(((1+i)^48)-1)/(i*((1+i)^48))=$100,000

By using Excel IRR function , we can get the value of i=0.015751=1.5751%

Nominal Annual Interest Rate=18.90%