Question

Loan 1: $6,180.26 with interest rate of 6.550%

Loan 2: $11,346.96 with interest rate of 5.410%

Loan 3: $14,044.21 with interest rate of 6.840%

A) You have decided you would like to clear out all of your student loans in 10 years by making monthly payments on your loans. How much are you paying monthly to clear your debt in 10 years?

B) You found a company that will consolidate each individual loan at 5.200% What is your new monthly payment?

C) Using the 5.200% you decide to make payments of $500 to clear your loans faster, how long will it take to pay them off?

Answer 1

A)EMI for loan 1:

EMI = P × r × (1 + r)ⁿ /{(1 + r)ⁿ - 1 }

Where,

P = Principal= $ 6,180.26

r = rate of interest = 6.55 % or 0.0655 p.a. or 0.0655/12 = 0.005458 p.m.

n = 10 years x 12 months = 120 periods

EMI = $ 6,180.26 x 0.005458 x (1 + 0.005458)¹²⁰ / {(1 + 0.005458) ¹²⁰ -1}

       = $ 6,180.26 x 0.005458 x (1.005458) ¹²⁰ / {(1.005458) ¹²⁰ -1}

       = $ 6,180.26 x 0.005458 x 1.921717/ (1.921717-1)

       = $ 6,180.26 x 0.005458 x 1.921717/ 0.921717

      = $ 64.82703/ 0.921717

      = $ 70.33293 or $ 70.33

EMI for loan 2:

P = $ 11,346.96

r = 5.41 % or 0.0541 p.a. or 0.0541/12 = 0.004508 p.m.

n = 120 periods

EMI = $ 11,346.96 x 0.004508 x (1 + 0.004508)¹²⁰ / {(1 + 0.004508) ¹²⁰ -1}

       = $ 11,346.96 x 0.004508 x (1.004508) ¹²⁰ / {(1.004508) ¹²⁰ -1}

       = $ 11,346.96 x 0.004508 x 1.715636/ (1.715636-1)

       = $ 11,346.96 x 0.004508 x 1.715636/ 0.715636   

     = $ 87.76489/ 0.715636           

     = $ 122.6389 or $ 122.64

EMI for loan 3:

P = $ 14,044.21

r = 6.84 % or 0.0684 p.a. or 0.0684/12 = 0.0057 p.m.

EMI = $ 14,044.21 x 0.0057 x (1 + 0.0057)¹²⁰ / {(1 + 0.0057) ¹²⁰ -1}

       = $ 14,044.21 x 0.0057 x (1.0057) ¹²⁰ / {(1.0057) ¹²⁰ -1}

       = $ 14,044.21 x 0.0057 x 1.9779/ (1.9779-1)

       = $ 14,044.21 x 0.0057 x 1.9779/ 0.9779

      = $ 158.3384/ 0.9779

      = $ 161.9094 or $ 161.91

Total monthly payment = $ 31.0931 + $ 122.6389 + $ 161.9094 = $ 315.6415 or $ 315.64

B)EMI for loan 1 @ 5.2 %: $ 66.157

EMI for loan 2 @ 5.2 %: $ 121.46

EMI for loan 3 @ 5.2 %: $ 150.34

C)

EMI = P × r × (1 + r)ⁿ /{(1 + r)ⁿ - 1 }

Where,

P = Principal= $ 6,180.26 + $ 11,346.96 + $ 14,044.21 = $ 31,571.43

r = rate of interest = 5.2 % or 0.052 p.a. or 0.052/12 = 0.004333 p.m.

n = No of periods

$ 500 = $ 31571.43 x 0.004333 x (1 + 0.004333)ⁿ / {(1 + 0.004333) ⁿ -1}

      $ 500/ $ 31,571.43 x 0.004333 = (1.004333) ⁿ / {(1.004333) ⁿ -1}

      $ 500/ $ 136.80953 = (1.004333) ⁿ / {(1.004333) ⁿ -1}

      3.6547162 = (1.004333) ⁿ / {(1.004333) ⁿ -1}

      3.6547162 x {(1.004333) ⁿ -1} = (1.004333) ⁿ

      3.6547162 x (1.004333) ⁿ - 3.6547162 = (1.004333) ⁿ

     Suppose (1.004333) ⁿ = t

3.6547162 t - 3.6547162 = t

3.6547162 t – t = 3.6547162

2.6547162 t = 3.6547162

t = 3.6547162/2.6547162 = 1.376688

(1.004333) ⁿ = 1.376688

Taking log of both sides we get,

Log (1.004333) ⁿ = Log 1.376688

n x 0.0018777328135 = 0.13883552687

n = 0.13883552687/0.0018777328135 = 73.93784988 or 74 months