Question

Calculate the present value at time t = 0 of a 5-year annuity paid continuously at a rate of h(t)=(1+t).

The force of interest at time t is δ =(2(1+t))/((t^2) +2t+1)

Answer 1

Present Value of Annuity @ time t = 0 ?
Life of the Annuity = 5 Years
Continuous Annuity
Rate of Payment of Annuity h(t) = 1+t
Interest = δ		=(2(1+t))/((t^2) +2t+1)
	=	(2(1+t))		(2(1+t))
		((t2) +2t+1)		(1+t)2
		2
		(t+1)
Continuous annuities. If the payments are being made continuously at the rate f(t) at
exact moment t, then the present value of an n-period continuous varying annuity is

₀ʃⁿ f(t)e ^{− 0ʃtδrdr}dt,

where δr is the force of interest.

Now ₀ʃ^t δrdr =₀ʃ^t2/(t+1)dt = [2ln(t+1) + C]₀⁵

= 2log(5+1) - 2Log(0+1)+c = 2log6-2log1=2*0.778=1.556

Therefore present value of annuity ₀ʃⁿ f(t)e ^{− 0ʃtδrdr}dt,

₀ʃ⁵ (1+t)e ^{– 1.556}dt = ₀ʃ⁵ (1+t)4.7398dt

=4.7398[t+t²/2]₀⁵ =4.7398[5+25/2-0-0/2]

=4.7398[10+25]/2

=4.7398X35/2

=82.9465

Present Value of a 5 year annuity paid continuously is 82.9465