16. An investor borrows money from a bank in order to invest in a business venture. The initial loan is $500,000, with further loans of $250,000 made in 6 months' time and $250,000 made in 12 months' time. The business venture will provide the investor with an income of $2 million in exactly 10 years' time and $3 million in exactly 15 years' time. The bank offers a force of interest, 7 (t), as a function of time t (measured in years) which is given by: a 8(t)= 10.04 for 0 2 Derive expressions for v(t) which cover all values of t. 16. An investor borrows money from a bank in order to invest in a business venture. The initial loan is $500,000, with further loans of $250,000 made in 6 months' time and $250,000 made in 12 months' time. The business venture will provide the investor with an income of $2 million in exactly 10 years' time and $3 million in exactly 15 years' time. The bank offers a force of interest, 8(t), as a function of time t (measured in years) which is given by: g Pgup 3 10.04 for 0 2 Derive expressions for v(t) which cover all values of t. 17. The force of interest, 5 (t), is a function of time and at any time t, measured in years, is given by the formula: 8(t)= 0.09 -0.003t Osts 10 t>10 0.06 a. Calculate the corresponding constant effective annual rate of interest for the period from t = 0 to t = 10. b. Express the rate of interest in part (a) as a nominal rate of discount per annum convertible half-yearly. c. Calculate the accumulation at time t = 15 of 1,500 invested at time t - 5. 16. An investor borrows money from a bank in order to invest in a business venture. The initial loan is $500,000, with further loans of $250,000 made in 6 months' time and $250,000 made in 12 months' time. The business venture will provide the investor with an income of $2 million in exactly 10 years' time and $3 million in exactly 15 years' time. The bank offers a force of interest, 7 (t), as a function of time t (measured in years) which is given by: a 8(t)= 10.04 for 0 2 Derive expressions for v(t) which cover all values of t. 16. An investor borrows money from a bank in order to invest in a business venture. The initial loan is $500,000, with further loans of $250,000 made in 6 months' time and $250,000 made in 12 months' time. The business venture will provide the investor with an income of $2 million in exactly 10 years' time and $3 million in exactly 15 years' time. The bank offers a force of interest, 8(t), as a function of time t (measured in years) which is given by: g Pgup 3 10.04 for 0 2 Derive expressions for v(t) which cover all values of t. 17. The force of interest, 5 (t), is a function of time and at any time t, measured in years, is given by the formula: 8(t)= 0.09 -0.003t Osts 10 t>10 0.06 a. Calculate the corresponding constant effective annual rate of interest for the period from t = 0 to t = 10. b. Express the rate of interest in part (a) as a nominal rate of discount per annum convertible half-yearly. c. Calculate the accumulation at time t = 15 of 1,500 invested at time t - 5