Mathematics for Management -- Supplementary Electronic Materials

Future Value & Present Value of an Income Flow

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1. Money is transferred continuously into an account at the constant rate of \(1200\) GEL per year. The account earns interest at the annual rate of \(8\%\) compounded continuously. How much will be in the account approximately at the end of \(2\) years (round to the nearest integer)?

\( 2400 \) GEL \( 2571 \) GEL \( 2603 \) GEL \( 3102 \) GEL

2. Find the integral below which gives the future value of a continuous income stream of \(50\) GEL per year over the next \(15\) years, assuming an annual interest rate of \(3\%\) compounded continuously.

\( \int^{15}_0 \, 50 {\rm{e}}^{0.45 t} \, \textrm{d} t \) \( \int^{15}_0 \, 50 {\rm{e}}^{-0.03 t} \, \textrm{d} t \) \( \int^{15}_0 \, 50 {\rm{e}}^{15 (0.03 - t)} \, \textrm{d} t \) \( \int^{15}_0 \, 50 {\rm{e}}^{0.03 (15 - t)} \, \textrm{d} t \)

3. Find the future value after \(10\) years of a continuous income stream of \(1200\) GEL per year deposited in an account paying \(6\%\) annual interest, compounded continuously.

\( 20000 \) GEL \( 20000 ( {\rm{e}}^{0.6} - 1 ) \) GEL \( 1200 {\rm{e}}^{0.6} \) GEL \( 20000 {\rm{e}}^{0.6} \) GEL

4. Money is transferred continuously into an account at the constant rate of \(200\) GEL per year. Assume the account earns interest at the annual rate of \(6\%\) compounded continuously. Compute the future value of the income stream over a \(18\) year period rounded to the nearest integer.

\( 5953 \) GEL \( 6160 \) GEL \( 6482 \) GEL \( 7153 \) GEL

5. Money is transferred continuously into an account at the constant rate of \(150\) GEL per month. Assume the account earns interest at the annual rate of \(6.5\%\) compounded continuously. Compute the future value of the income stream over a \(20\) year period rounded to the nearest integer.

\( 5953 \) \( 6160 \) \( 6482 \) \( 7153 \)

6. Assuming an interest rate of \(5\%\), compute the approximate present value of a constant income stream of \(100000\) GEL per year for \(10\) years rounded to the nearest integer.

\( 601745 \) \( 692291 \) \( 734871 \) \( 786939 \)

7. Find the present value of a continuous income stream of \(500\) GEL per month deposited for 20 years into an account paying \(4\%\) annual interest, compounded continuously.

\( 120000 \) GEL \( 120000 {\rm{e}}^{-0.8} \) GEL \( 12500 (1 - {\rm{e}}^{-0.8}) \) GEL \( 150000 (1 - {\rm{e}}^{-0.8}) \) GEL

8. Assume an investment costs \(4 000\) GEL and is estimated to generate income at the constant rate of \(f(t) = 4000\) GEL per year. If the prevailing annual interest rate remains fixed at \(5\%\) compounded continuously over the next \(5\) years, what is the net income generated (i.e. the present value minus the costs) approximately rounded to the nearest integer?

\( 13151 \) \( 13274 \) \( 13696 \) \( 13878 \)

9. Assume an investment costs \(1 000\) GEL and is estimated to generate income at the rate of \(f(t) = 3000 {\rm{e}}^{0.03 t}\) GEL per year. If the prevailing annual interest rate remains fixed at \(5\%\) compounded continuously over the next \(5\) years, what is the net income generated (i.e. the present value minus the costs) approximately rounded to the nearest integer?

\( 13151 \) \( 13274 \) \( 13696 \) \( 13878 \)

10. Assume that a company manufactures a certain type of machine and it estimates that each machine will generate a continuous income stream whose rate in the \(t\)th year of operation will be \(15 - 2t\) million GEL per year. Assuming that the lifetime of a machine is about \(7\) years and that money can be invested at the annual rate of \(8\%\), compounded continuously, which of the integrals below represent the fair market price for each machine?

\( \int^7_0 \left( 15 - 2t \right) {\rm{e}}^{0.08 t} \textrm{d} t \) \( \int^7_0 \left( 15 - 2t \right) {\rm{e}}^{0.08 (7 - t)} \textrm{d} t \) \( \int^7_0 \left( 15 - 2t \right) {\rm{e}}^{-0.08 t} \textrm{d} t \) \( \int^7_0 \left( 15 - 2t \right) {\rm{e}}^{0.08 (t - 7)} \textrm{d} t \)

Your grade is: __ of 10

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Solution: 1c; 2d; 3b; 4c; 5b; 6d; 7c; 8b; 9c; 10c

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.