) The force of interest is given by: d(t)= 0.08−0.001t 0≤t <3 0.025t −0.04 3≤t <5...

) The force of interest is given by: d(t)= 0.08−0.001t 0≤t <3 0.025t −0.04 3≤t <5 0.03 5≤t ⎧ ⎨ ⎪ ⎩ ⎪ i) Calculate the present value at time 2 of a payment of Ksh 1,000,000 at time 10. ii) Calculate the annual effective rate of interest from time 2 to time 10 equivalent to the force of interest function in (i) above.

Solutions

Expert Solution

First lets see what force of interest is.

The “force of interest” is that interest rate that compounds continuously, rather than compounding after a fixed time period (such as compounding every six months).

Now, we know that the Future value of a PV at an interest rate compounding continuously (force of interest) i is given by

FV=PVeⁿⁱ,

where PV is the present value, e is exponential, n is number of years and i is the force of interest.

It is given to us that force of interest is

0.08 − 0.001t, when 0 ≤ t < 3

0.025t − 0.04, when 3 ≤ t < 5

0.03, when t ≥5

Using this and the above formula for present value, we get

Present Value of a payment at time 2 of 1000000 at time 10 is

1000000*exp( -[0.2355 - 0.158])*exp ( -[0.1125 + 0.0075])*exp ( -[0.3 - 0.15])

= 1000000*exp(-0.3475) = 706,452.

B. Lets say that the annual effective rate of interest is i , then

706452(1 + i)⁸ = 1000000

⇒ i = 4.44%

Rahul Sunny answered 1 year ago

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