STAT2902 Financial Mathematics Assignment 2
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2902 Financial Mathematics
Assignment 2
Due Date: March 6, 2023 by 17:00 (Late submission will not be accepted)
1. An annuity-immediate is paid monthly at a rate of $200,000 per annum for 12 years. The rate of interest is 6% per annum compounded semiannually in the first 4 years and 4% per annum convertible quarterly for the remaining 8 years. Calculate the accumulation of the annuity at the end of 12 years.
2. An annuity provides for 25 annual payments. The first payment of 1,000 is made immediately and the remaining payments increase by 5% per year. Interest is calculated at 6% per year. Calculate the present value of the annuity.
3. Plato donated 40,000 toward future scholarships for students in his Academy. The scholarships will pay 1,000 at the end of year 1, 1,500 at the end of year 2, 2,000 at the end of year 3 and so on, with the amount increasing 500 each year until the scholarship reaches 5,000. The annual scholarship will remain at 5,000 until the fund is depleted. If the account balance is less than 5,000 at the time of payment, that amount will be rewarded as a smaller scholarship and the account will be closed. The scholarship fund earns interest at an effective annual rate of 8%. Determine how many full 5,000 scholarships will be awarded.
4. Find the present value of a perpetuity under which a payment of 1 is made at the end of the first year, 2 at the end of the second year, increasing until a payment of n is made a the end of the nth year, and thereafter payments are level at n per year forever. Express the answer in terms of nl and i.
5. Follow the steps below to obtain an approximation formula for a) :
a) Prove that
i(m) s i +
by considering the Taylor Series expansion of (1 + i) and keeping only second order terms.
b) Prove that
1 1 m - 1
i(m) i 2m
by using the results of (a) and the approximation s 1 - x assuming x is sufficiently small.
c) Show that
a) s anl + (1 - vn )
with the results of part (b).
6. Show that ()2nl = 2()nl + .
7. A perpetuity has payments at the end of each 4-year period. The first payment at the end of 4 years is 1. Each subsequent payment is 5 more than the previous payment. Given that v4 = 0.75, calculate the present value of this perpetuity.
8. The present value of a perpetuity paying 1 at the end of every three years is 125/91. Find i.
9. If 3a) = 2a = 45s) , find i.
10. If nl = 4 and nl = 12, find δ .
11. A 5-year annuity has the following schedule of payments: On each January 1 500 On each March 1 1000 On each May 1 1500 On each July 1 2000 On each September 1 2500 On each November 1 3000
Express the present value of this annuity on January 1 just before the first payment in terms of actuarial symbols.
12. Evaluate (I¯)2l given a constant force of interest δ = 0.05.
13. The force of interest δt is given by
,.0.05, 0 < t < 10,
δt = .0.005t, 10 < t < 20
.(0.003t + 0.0001t2 , t > 20
a) Calculate the present value of a unit sum of money due at t = 25.
b) Calculate the effective rate of interest per unit time from t = 19 to t = 20.
c) A continuous payment stream is paid at the rate of e-0 .03t per unit time between t = 0 and time t = 5. Calculate the present value of that payment stream.
14. Demonstrate the following
a) Show that
d
b) Find di anl evaluated at i = 0.
2023-03-03