Question

Q1/ You deposit $1 into a bank account which credits interest at a nominal interest rate of 10% per annum, convertible semiannually. At the same time, your classmate deposits $1000 into a different bank account, which is credited with simple interest.

At the end of 5 years, the forces of interest on the two accounts are equal, and your classmate’s account has accumulated to Z . Determine Z.

(A) 1800 (B) 1854 (C) 1900 (D) 1953 (E) 2000

Q2/ You deposit 100 into a savings account at time 0, which pays interest at a nominal rate of i, compounded semiannually. Your friend deposits 200 into a different savings account at time 0, which pays simple interest at an annual rate of i.

You and your friend earn the same amount of interest during the last 6 months of the 10th year. Calculate i.

(A) 8% (B) 7.4% (C) 6.8% (D) 6% (E) 5.2%

Q3/ Money is deposited in a bank. For the first 4 years interest accumulates at annual nominal rate of 6% convertible monthly. For the next 6 years it accumulates at a force of interest of 5%. For the 10 year period what is the equivalent nominal discount rate convertible quarterly?

(A) 4.9% (B) 5.2% (C) 5.4% (D) 5.7% (E) 5.9%

Answer 1

Here the correct option would be option D i.e. 1953.

Let me explain how, your account has 10% of nominal interest compounded semi-annually i.e. 5% in every 6 months. Therefore the actual annual effective rate would be (1.05^2 -1) = 10.25% or 0.1025.

So here effective rate of interest would be "In 1.1025", as per the first problem after 5 years forces of interest on two account would be the same. For our problem the simple interest is depended on a force which is time or "t".

So, simple interest rate = d/dt [ 1 + it ] = i / (1 + it)

where in "i" s the interest rate and "t" is the time.

As per the problem at the end of 5 years the forces of interest is the same which means when "t" or time is 5

Therefore the same can be equated as "In 1.1025 = i / (1 +i5) = 0.09758 ( compound interest for 5 years ) = i /(1+ i5) { 0.09758 was computed by the formula Delta t = In (1 +i ) where is i = 0.1025 and t is 5 }

by solving this get i as 0.190549

therefore Z or your classmates accumulated amount after 5 years would be 1000 ( 1 + 5i) =1000 ( 1 + 5*0.190549)

= 1000 + 952.745 = 1953 (Round off ) .