The force of interest 8(t) at any time t, measured in years, is given by: 8(t) = { 0.04e0.02t 0.03 0.001t for 0 < t≤ 10 for 10 < t≤ 15 0.045 for t > 15 (i) Derive, and simplify as far as possible, expressions in terms of t for V(t), where V(t) is the present value of a unit sum of cash flow made at time t. You should derive separate expressions for the three sub-intervals. (ii) Hence, making use of the result in part (i), calculate the value at time t = 3 of a payment of £2,500 made at time t = 15. (iii) Calculate, to the nearest 0.01%, the constant nominal annual rate of interest convertible half-yearly implied by the transaction in part (ii). (iv) Making use of the result in part (i), calculate the present value of a payment stream p(t) paid continuously from time t = 15 to t = 20 at a rate of payment at time t given by: p(t) = 300e 0.02t
The force of interest 8(t) at any time t, measured in years, is given by: 8(t) = { 0.04e0.02t 0.03 0.001t for 0 < t≤ 10 for 10 < t≤ 15 0.045 for t > 15 (i) Derive, and simplify as far as possible, expressions in terms of t for V(t), where V(t) is the present value of a unit sum of cash flow made at time t. You should derive separate expressions for the three sub-intervals. (ii) Hence, making use of the result in part (i), calculate the value at time t = 3 of a payment of £2,500 made at time t = 15. (iii) Calculate, to the nearest 0.01%, the constant nominal annual rate of interest convertible half-yearly implied by the transaction in part (ii). (iv) Making use of the result in part (i), calculate the present value of a payment stream p(t) paid continuously from time t = 15 to t = 20 at a rate of payment at time t given by: p(t) = 300e 0.02t
Essentials of Business Analytics (MindTap Course List)
2nd Edition
ISBN:9781305627734
Author:Jeffrey D. Camm, James J. Cochran, Michael J. Fry, Jeffrey W. Ohlmann, David R. Anderson
Publisher:Jeffrey D. Camm, James J. Cochran, Michael J. Fry, Jeffrey W. Ohlmann, David R. Anderson
Chapter5: Probability: An Introduction To Modeling Uncertainty
Section: Chapter Questions
Problem 4P: Suppose that we have two events, A and B, with P(A) = 0.50, P(B) = 0.60, and P(A B) = 0.40. a. Find...
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PLEASE, WRITE THE SOLUTIONS ON PAPER, EXPLAINING THE ENTIRE PROCESS, THE ONLY AND CORRECT ANSWERS ARE FOR (i) V(t) = exp ( -2e^0.02t + 2 ) for 0 < t <10, V(t) = exp( -0.0928 - 0.03t - 0.0005t^2) for 10 < t <15, V(t) = exp ( 0.0197 - 0.045t ) for 15 < t.
AND (ii) C3 = £2500 x 0.5877 = £1,469.16
AND (iii) i^2 = 4.48% pa
AND (iv) PV = £988.38
Transcribed Image Text:The force of interest 8(t) at any time t, measured in years, is given by:
8(t) =
{
0.04e0.02t
0.03 0.001t
for
0 < t≤ 10
for
10 < t≤ 15
0.045
for
t > 15
(i)
Derive, and simplify as far as possible, expressions in terms of t for V(t), where V(t) is
the present value of a unit sum of cash flow made at time t. You should derive separate
expressions for the three sub-intervals.
(ii)
Hence, making use of the result in part (i), calculate the value at time t = 3 of a payment of
£2,500 made at time t = 15.
(iii) Calculate, to the nearest 0.01%, the constant nominal annual rate of interest convertible
half-yearly implied by the transaction in part (ii).
(iv) Making use of the result in part (i), calculate the present value of a payment stream p(t)
paid continuously from time t = 15 to t = 20 at a rate of payment at time t given by:
p(t) = 300e 0.02t
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