Differential Equations in Finance
This quiz is designed to assess your understanding of differential equations in finance. It covers concepts such as continuous compounding, annuities, and loans.
Questions
What is the differential equation that describes the growth of a continuously compounded investment?
- $$\frac{dy}{dt} = ry$$
- $$\frac{dy}{dt} = y$$
- $$\frac{dy}{dt} = y^2$$
- $$\frac{dy}{dt} = e^y$$
What is the solution to the differential equation $$\frac{dy}{dt} = ry$$?
- $$y = Ce^{rt}$$
- $$y = C + rt$$
- $$y = C - rt$$
- $$y = C/rt$$
What is the present value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 5%? (Assume continuous compounding.)
- $$\frac{1000}{0.05}$$
- $$\frac{1000}{0.05} \left( 1 - e^{-0.05 \cdot 10} \right)$$
- $$\frac{1000}{0.05} \left( e^{0.05 \cdot 10} - 1 \right)$$
- $$\frac{1000}{0.05} \left( e^{-0.05 \cdot 10} - 1 \right)$$
What is the monthly payment on a loan of $100,000 that is to be repaid over 30 years at an annual interest rate of 4%? (Assume continuous compounding.)
- $$\frac{100,000}{30 \cdot 12}$$
- $$\frac{100,000}{30 \cdot 12} \left( 1 - e^{-0.04 \cdot 30} \right)$$
- $$\frac{100,000}{30 \cdot 12} \left( e^{0.04 \cdot 30} - 1 \right)$$
- $$\frac{100,000}{30 \cdot 12} \left( e^{-0.04 \cdot 30} - 1 \right)$$
What is the effective annual interest rate on a loan that has a nominal annual interest rate of 12% and is compounded monthly?
- $$12\%$$
- $$12.68\%$$
- $$13.38\%$$
- $$14.10\%$$
What is the doubling time of an investment that is continuously compounded at an annual interest rate of 7%? (Assume that the initial investment is $1.)
- $$10\text{ years}$$
- $$11\text{ years}$$
- $$12\text{ years}$$
- $$13\text{ years}$$
What is the future value of an investment of $1000 that is continuously compounded at an annual interest rate of 5% for 10 years?
- $$\$1628.89$$
- $$\$1643.85$$
- $$\$1659.05$$
- $$\$1674.49$$
What is the present value of an investment that will be worth $1000 in 10 years if the annual interest rate is 5% and the interest is compounded continuously?
- $$\$783.53$$
- $$\$789.34$$
- $$\$795.27$$
- $$\$801.33$$
What is the annual interest rate on a loan that has a monthly payment of $1000, a loan term of 30 years, and a total amount borrowed of $100,000?
- $$4\%$$
- $$4.5\%$$
- $$5\%$$
- $$5.5\%$$
What is the total amount of interest paid on a loan of $100,000 that is repaid over 30 years at an annual interest rate of 5%? (Assume continuous compounding.)
- $$\$100,000$$
- $$\$135,000$$
- $$\$170,000$$
- $$\$205,000$$
What is the present value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 4%? (Assume continuous compounding.)
- $$\$8110.90$$
- $$\$8203.46$$
- $$\$8298.17$$
- $$\$8395.13$$
What is the future value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 6%? (Assume continuous compounding.)
- $$\$12155.06$$
- $$\$12387.65$$
- $$\$12624.80$$
- $$\$12866.51$$