ODEs: In a period of falling interest rates, the (continuously compounded) short interest rate as a function of t is r(t) = 1 20 + t 2 per annum. (a) What is the value at t = 0 of a contract that makes a payment at time t = 10 of USD 1,000? An investor deposits USD 300 in a bank account at time 0, reinvests all interest payments and also additionally continuously invests USD 300 per annum, until the total value of the deposits reaches USD 3312. At that point the investor stops making additional deposits, but still lets the interest payments accumulate in the account. From our discussion in class, the ODE for the value of deposits, V , over time is then dV dt = r(t)V (t) + I(t), where I(t) = 300 until V (t) reaches V = 3, 312, at which point I(t) instantaneously switches to I(t) = 0. (b) Derive an expression for the value of the asset as a function of time, V (t), t ≥ 0. 2. ODES: In a period of falling interest rates, the (continuously compounded) short interest rateas a function of t is1r(t) :20 +per annum.(a) What is the value at t = 0 of a contract that makes a payment at time t = 10 of USD1,000?An investor deposits USD 300 in a bank account at time 0, reinvests all interest paymentsand also additionally continuously invests USD 300 per annum, until the total value of thedeposits reaches USD 3312. At that point the investor stops making additional deposits, butstill lets the interest payments accumulate in the account. From our discussion in class, theODE for the value of deposits, V, over time is thendVr(t)V(t)+I(t),dtwhere I(t) = 300 until V(t) reaches V = 3, 312, at which point I(t) instantaneously switchesto I(t) = 0.(b) Derive an expression for the value of the asset as a function of time, V(t), t > 0.

Answered: 2. ODES: In a period of falling…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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ODEs: In a period of falling interest rates, the (continuously compounded) short interest rate as a function of t is r(t) = 1 20 + t 2 per annum. (a) What is the value at t = 0 of a contract that makes a payment at time t = 10 of USD 1,000? An investor deposits USD 300 in a bank account at time 0, reinvests all interest payments and also additionally continuously invests USD 300 per annum, until the total value of the deposits reaches USD 3312. At that point the investor stops making additional deposits, but still lets the interest payments accumulate in the account. From our discussion in class, the ODE for the value of deposits, V , over time is then dV dt = r(t)V (t) + I(t), where I(t) = 300 until V (t) reaches V = 3, 312, at which point I(t) instantaneously switches to I(t) = 0. (b) Derive an expression for the value of the asset as a function of time, V (t), t ≥ 0.