contestada

A 10-year loan of 2000 is to be repaid with payments at the end of each year. It can be repaid under the following two options. Equal annual payments at an annual effective rate of 8.07%. Installments of 200 each year plus interest on the unpaid balance at an annual effective rate of i The sum of the payments under option (i) equals the sum of payments under option (ii). Calculate i.

Respuesta :

ogbe2k3

Answer:

The answer to the question is i= 9%

Step-by-step explanation:

From the example given, let recall the following formula,

10 year loan of =2000

The equal annual payments at an annual effective rate = 8.07%

The installment for each year =200

Them

2000 = x/1.0807 + x/1.0807^2 +..... x/1.0807^10

2000= x(6.6889)

x= $299 per month

The payment under under each option is the same:

299×10= 200 × 10 + I

I (Total Interest paid) = $990

I= (2000 × i) + (1800 × i) + (1600 × i) +

$990= i(2000 + 1800 +........ +200)

$990= i × 11000

Therefore i = 9%

The interest paid on a loan is dependent on the type of annuity, which can be annuities-due, ordinary  annuity, or perpetuities.

  • The interest under option (ii) is i = 11%

Reasons:

The annual effective rate for the first option, r = 8.07%

The payment made each year under option (i) = P

The amount loaned, PV = 2,000

Payment made each year under option (ii) = 200 + Interest on the unpaid balance

Number of years of the loan = 10 years

The sum of payment under option (i) = The sum of payment under option (ii)

Required:

The interest rate under option (ii)

Solution:

Using the ordinary annuities payment, we have;

The total payment is given by the formula;

[tex]\displaystyle Payment, P = \mathbf{\frac{r\times PV}{1 - (1 + r)^{-n}}}[/tex]

Therefore;

[tex]\displaystyle Annual \ payment, P = \frac{0.0807 \times 2000}{1 - (1 + 0.0807)^{-10}} \approx 299[/tex]

The equal annual payment, P ≈ $299

The sum of payment by the first option ≈ $299 × 10 = $2,990

The payment made each year under option (ii) = $200

The interest paid in first year = i × 1800

The interest payed in second year = i × 1600...

Therefore, we have;

I = i×(1800 + 1600 + 1400 + 1200 + 1000 + 800 + 600 + 400+ 200)

I = i×9000

Sum of payment made in option (ii) = 200 × 10 + i × 9000 = 2000 + 9000·i

The sum of payment under option (i) = The sum of payment under option (ii)

Which gives;

2,000 + 9,000·i = 2,990

9,000·i = 2,990 - 2000 = 990

[tex]\displaystyle i = \mathbf{\frac{990}{9,000}} = 0.11[/tex]

The interest under option (ii) is i = 0.11 = 11%

Learn more about loan repayment here:

https://brainly.com/question/3260195

Otras preguntas