An investment is made for one year in a fund whose accumulat

An investment is made for one year in a fund whose accumulation function is a second degree polynomial. The nominal rate of interest earned during the first half of the year is 5% convertible semi-annually and the effective rate of interest earned for the entire year is 7%. Find delta_1.5 and delta_.

Solution

Answer :

ASince accumulation function is a second degree polynomial.

So Let the accumulation function a(t) =  a + bt + ct². (where a must be 1 since a(0) = 1).

The nominal rate of interest earned during the first half of the year is 5% convertible semiannually.

so that  a(.5) = 1 + .5b + .25c = 1+ 0.05/2

=> 1 + .5b + .25c = 1.025 ------------------- ( 1 )

and the effective rate of interest earned for the entire year is 7%

so that a(1) = 1 + b + c = 1+ 7% = 1 + 0.07 = 1.07

=> 1 + b + c = 1.07---------------------------- ( 2 )

Now solve for ( 1 ) and ( 2 ) we get b = 0.3 and c = 0.4

We have delata(t) = a\' (t) / a(t) = ( b + 2ct ) /( a + bt + ct² )

= > delata(t) = ( 0.3 + 0.8t)  /( 1 + 0.3t + 0.4t² )

When t = 1.5 , delta(1.5) = 0.6383

and when t = 5 , delta(5) = 0.344