The fund earns a simple interest i, if the annual effective interest rate for year 5 is 1.2 times of the annual effective rate for year 10 determine i.

As the interest rate of 5 years is 1.2 times of the annual effective rate for 10 years, it can be written as

Taking 1/5 of  whole equation

Solving the equation gives

 (2) The present value of a payment 1004 at time t-month from now is the sum of the present values of the following payments: (a) 500 at the end of 4 months, (b) 400 at the end of 10 months, and (c) 100 at the end of 16 months.  Determine t if the annual effective interest is 5%.

Solution

Present Value Formula is below ;

                                                                     Effective interest rate is 5% 

(3) Brian has $1,000 to invest at time 0 and two ways to invest it. 

Investment option (A): it provides the force of interest at 1.08/(1+0.08) at time t.

Investment option (B): it provides a 5% annual effective interest rate.

Brian can invest any portion of his money in either (A) or (B), and can transfer any portion of his money between the two investments at any time. What is the maximum amount Brian can accumulate at t =15.  (Hint: option (A) is better in the early years but worse is the later years, so you need to find the cross over time first).

Solution

When investing money you want to apply the largest force of interest possible to your money. The force of interest is given for investment A. We need to convert to a force of interest for investment B. Since the effective rates does not vary for investment B   the force of interest is constant and equal to ln(1.05)which is approximately equal to 0.04879. Therefore as long as the force of interest for investment A is greater than 0.4879 then we invest in A.

We find that t<1

So, we invest in A for 8 years and then B for the remaining 12 years

By inspection we see that,

A (t) = 1+0.08t, so that accumulated value at time 15 is;

1000(1+0.08times3) + ({1.05}¹²

Approx, $1241.79 max amount of return

4). ) Fund X pays nominal interest rate of 6% convertible monthly, (ii) Fund Y pays interest at a force of interest of 1/(t+12).  Jill deposits same amount of money into each fund now, at the end of 12 years, the accumulated amount of fund X is 1,000.  Determine the accumulated amount of fund Y.

Solution

For Y

Interest rate = 6% monthly Convertible

Effective interest rates = (1+0.06/12))^12-1

=5.8%

i.e Effective annual interest rates = 6.16%

Now principle = P

Compound Sum = F = 1000

So, F= P* (1+0.0616)^12 = P*2.0489

                                                                                                        a(12) = 1551.08