Chapter 6: Varying payments

Question 1

PV of a cashflow

Answer 1

If there is a cashflow of c at time t then:
PV
=c * v(t)
=c * e^ - integral(δ(s)ds from 0 to t

if the force of interest is constant then this resolves to c* e^-δt

Question 2

A(t)

Answer 2

=c * e^ integral(δ(s)ds from 0 to t
=1/v(t)

Question 3

A(t1,t2)

Answer 3

For 0<=t1<=t2, accumulated value at time t2 of an investment of 1 unit at time t1.

A(0,t)=A(t)

A(t1,t2)=e^integral(δ(s)ds) from t1 to t2

=e^δ(t2-t1) for constant δ

Question 4

Present Value of a payment stream, paid continuously at a rate of ρ(t) pa from t=a to t =b

Answer 4

=Integral of ρ(t)v(t)dt
=Integral of ρ(t)e^ -integral δ(s)ds [0 to t] dt
both from a to b

If the force of interest is constant this simplifies to:
Integral of ρ(t)e^ -δ(t-a)dt

Question 5

Accumulated value at time n of a payment stream that is paid continuously at the rate of ρ(t) pa between t=a and b

Answer 5

integral from a to b of ρ(t)A(t,n)dt
=integral from a to b of ρ(t)e^integ t to n of δ(s)ds dt

if constant:
=integral from a to b of ρ(t)e^integ δ(n-t) dt