Question

Problem 1.10 You deposit $5,000 in an account that earns 5% interest compounded annually in years 1 and 2, and thereafter a continuous rate δ(t) = 2/(t + 1) (t ≥ 0). What is the value of the account after 5 years?

Problem 1.11 Suppose an initial investment of $100 grows according to the accumulated amount function A(t) = 100(1 + 0.05t) (t ≥ 0). (a) Find the effective rate of interest earned during the 5th year i5. (b) Find the force of interest δ(t). (c) Find the “average rate” (= equivalent annual effective rate) of interest earned during the first five years.

Answer 1

Here P=$5,5000

and n=2 and r=5%

for first two years the amount become

A=5512.5 is the amount in first two years

For next three years the rate is given by

= ln (A(5)/A(2))

so

A(5) = A(2)e^1.38629

= 5512.5*e^1.38629

= 22049.903

=22050

1.11

it+1 can be calculated as = [A(t+1) – A(t)]/A(t) At the end of year 5, i5= [A(5) – A(4)]/A(4)

A(5) = 100 (1 +0.05*5) = 125

A(4) =00 (1 +0.05*4) = 120

i5 = (125 - 120)/120 = = 0.0417 = 4.17%

b) Force of interest is given as = ln( 1+0.0417) = 0.0408

c) By the end of year 5, the fund has grown to 1.25. The equivalent annual rate,compounded annually, is 1.25^1/5= 1.0456. So the rate is 0.0456 = 4.56%