The owner states that she can repay a loan at $1300 per month for the next 3 years and then $2600 per month after 2 years. If the bank charge 8.25%APR how much would it be willing to lend the business owner?

The bank would be willing to lend $ 86,141.35

Explanation:

The concept of calculation of time value of money is the basis of this question.Loan amount would be the present value of monthly repayment of loan.      

Steps are as follows:

1: We have to Calculate present value of monthly repayment for first 3 years     by following formula  

Present value=Monthly Payment * Present value of annuity of 1     =$1,300*31.794659      

=$41,333.06        

Working is as follows:          

Present value of annuity of 1=(1-(1+i)^-n)/i      

                                              =(1-(1+0.006875)^-36)/0.00687                    .                                              =                                             =31.794659  

Where  i8.25%/12=0.006875

              n3*12=36  

Step-2:Calculation of present value of monthly repayment for next 2 years      

Present value=Monthly Payment * Present value of annuity of 1*Present value of 1  

=$         2,600*22.0549002*0.781412    

=$ 44,808.29        

Working:          

Present value of annuity of 1=(1-(1+i)^-n)/i              =(1-(1+0.006875)^-24)/0.006875i8.25%/12=0.006875         =22.0549002                   n2*12=24  

Present value of 1=(1+0.006875)^-36        

=0.78141172        

Step-3:Calculation of loan amount        

Loan amount=Present value of monthly repayments      

=$ 41,333.06+$ 44,808.29      

=$ 86,141.35  

The bank would be willing to lend the business owner a sum of $86,141.35

Explanation: For this question, we shall be calculating the time value of money. The amount of the loam that the bank would lend to the business owner would be the present value of monthly repayment of loan.      

We calculate thus:

Present value = Monthly Payment X Present value of annuity of 1     =$1,300 X 31.794659      

=$41,333.06        

We calculate for annuity thus:          

Present value of annuity of 1 =(1-(1+i)^-n)/i

i = 8.25%/12 = 0.0825/12 = 0.006875

n = 3 X 12 = 36

Therefore:

= (1-(1+0.006875)^-36)/0.006875 =31.794659  

Now, we calculate the present value of monthly repayment for next 2 years:  

Present value = Monthly Payment X Present value of annuity of 1 X Present value of 1  

= $2,600 X 22.0549002 X 0.781412 = $ 44,808.29        

We calculate thus:          

Present value of annuity of 1 = (1-(1+i)^-n)/i

i = 8.25%/12 = 0.0825/12 = 0.006875

n = 2*12 = 24

= (1-(1+0.006875)^-24)/0.006875    

= 22.0549002                  

Present value of 1 = (1+0.006875)^-36 = 0.78141172        

Now, we calculate the loan amount: Loan amount = Present value of monthly repayments      

= $ 41,333.06 + $ 44,808.29      

= $ 86,141.35