Question

**Do not round in this problem. Use at least 5 significant figures.

A large entertainment company, let’s call them Wisney! Ok, so Wisney is considering buying a new piece of land in the Everglades and planning to put a new campground…but make it fun! The land they are buying has a stipulation that nothing can be built on the property until it is paid in full. Wisney is wealthy and has cash available but due to COVID-19 is being a bit more conservative about spending large amounts of money on new projects, therefore they will take out a loan with a nominal rate of 4.78%.

1. What is the effective interest rate if the interest is compounded semi-annually and payment are made semi-annually? (Hint: The effective rate should be in the same period as the payment schedule)
   1. If the land costs $2,800,000. What is the semi-annual payment needed to pay it off in 5 years?
2. What is the effective interest rate if the interest is compounded monthly and payments are made semi-annually?
   1. (5 points) If the land costs $2,800,000. What is the semi-annual payment needed to pay it off in 5 years?

Answer 1

(a) Effective Interest rate = ( 1 + i/n)^n - 1,

where i = nominal interest rate and n = Number of periods. in this case, our interest is being compounded semin annually, then the number of period will be 2 and nominal interest rate is 4.78%.

therefore, Effective Interest rate = ( 1 + 0.0478/2)^2 - 1 = 4.837%

(a.1) For calculation of Amount to be paid semi annually, we will use this formula--

B = i x A / {1 - (1 + i)^-N},

where B = amount to be paid semi annually,

i = interest rate

N = no. of period in the loan,here we will take it 5 x 2 = 10, as it has to be paid semi annually and in 5 years.

Thus, 0.04837 x 2,800,000 { 1- (1 + 0.04837)^-10 } = 133,840 / 0.37647 = 355,513.055489 ~ 355,513.

Therefore, a payment of 355,513 semi annually for 5 years will repay our loan of 2,800,000.

(b) Interest is compounded monthly, hence again the same formula will apply, but no. of periods now will be 12 --

Effective Interest rate = ( 1 + i/n)^n - 1

= (1 + 0.0478/12)^12 - 1 = 4.886%

(b.1) We will use the same formula(a.1) -->

B = i x A / {1 - (1 + i)^-N},

where B = amount to be paid semi annually,

i = interest rate

N = no. of period in the loan,here we will take it 5 x 2 = 10, as it has to be paid semi annually and in 5 years.

Thus, 0.04886 x 2,800,000 { 1- (1 + 0.04886)^-10 } = 136,808 / 0.37938 = 360,609.41536 ~ 360,609.

Let me know in the comments if you have any feedback or suggestions!

Hope it helps. Thanks :)