Important formulas

Accumulation function

Amount of interest earned over n periods

Examples

1. Find the accumulated value of $900 invested for 10 years if the force of interest is 5%.

Ans. 1483.85

2. Fund A accumulates at a simple interest rate of 10%. Fund B accumulates at a simple discount rate of 5%. Find the point in time at which the forces of interest of the two funds are equal.

Ans. t = 5

3. Find the accumulated value of $1 at the end of n years if .

Ans. 1+n

Nov. 2001

#1 Ernie makes deposits of 100 at time 0, and X at time 3. The fund grows at a force of interest The amount of interest earned from time 3 to time 6 is X. Calculate X.

Ans. 784.59

May 2000

#37 A customer is offered an investment where interest is calculated according to the following force of interest:

The customer invests 1000 at time t = 0. What nominal interest, compounded quarterly, is earned over the first four-year period?

Ans.3.39%

Nov. 2001

#28 Payments are made to an account at a continuous rate of (8k+tk) where .

Interest is credited at a force of interest . After 10 years, the account is worth 20,000. Calculate k.

Ans.111

May 2001

#45 At time t =0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest . Fund Y accumulates at a nominal rate of discount of 8% per annum convertible semiannually. At time t = 5, the accumulated value of Fund X equals the accumulated value of Fund Y. Determine k.

Ans.102.07

Nov. 2000

#53 At time 0, K is deposited into Fund X, which accumulates at a force of interest . At time m, 2K is deposited into Fund Y, which accumulates at an annual effective interest rate of 10%. At time n, where n>m, the accumulated value of each fund is 4K. Determine m.

Ans. 1.58

May 2001

#49 Tawny makes a deposit into a bank account which credits interest at a nominal interest rate of 10% per annum, convertible semiannually. At the same time, Fabio 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 Fabio’s account has accumulated to Z. Determine Z.

Ans.1953

May 2003

#1 Bruce deposits 100 into a bank account. His account is credited interest at a nominal rate of interest i convertible semiannually. At the same time, Peter deposits 100 into a separate account. Peter’s account is credited interest at a force of interest of . After 7.25 years, the value of each account is 200. Calculate (i -).

Ans. 0.23%