1. P5–4 **Future values** For each of the cases shown in the following table, calculate the future value of the single cash flow deposited today at the end of the deposit period if the interest is compounded annually at the rate specified.

Case |
Single cash flow |
Interest rate |
Deposit period (years) |

A | $ 200 | 5% | 20 |

B | 4,500 | 8 | 7 |

C | 10,000 | 9 | 10 |

D | 25,000 | 10 | 12 |

E | 37,000 | 11 | 5 |

F | 40,000 | 12 | 9 |

P5–11 **Present values** For each of the cases shown in the following table, calculate the present value of the cash flow, discounting at the rate given and assuming that the cash flow is received at the end of the period noted.

Case |
Single cash flow |
Discount rate |
End of period (years) |

A | $ 7,000 | 12% | 4 |

B | 28,000 | 8 | 20 |

C | 10,000 | 14 | 12 |

D | 150,000 | 11 | 6 |

E | 45,000 | 20 | 8 |

P5–20 **Present value of an annuity** Consider the following cases.

Case |
Amount of annuity |
Interest rate |
Period (years) |

A | $ 12,000 | 7% | 3 |

B | 55,000 | 12 | 15 |

C | 700 | 20 | 9 |

D | 140,000 | 5 | 7 |

E | 22,500 | 10 | 5 |

a. Calculate the present value of the annuity, assuming that it is

· (1) An ordinary annuity.

· (2) An annuity due.

b. Compare your findings in parts **a**(1) and **a**(2). All else being identical, which type of annuity—ordinary or annuity due—is preferable? Explain why.

P5–22 **Retirement planning** Hal Thomas, a 25-year-old college graduate, wishes to retire at age 65. To supplement other sources of retirement income, he can deposit $2,000 each year into a tax-deferred individual retirement arrangement (IRA). The IRA will earn a 10% return over the next 40 years.

a. If Hal makes annual end-of-year $2,000 deposits into the IRA, how much will he have accumulated by the end of his sixty-fifth year?

b. If Hal decides to wait until age 35 to begin making annual end-of-year $2,000 deposits into the IRA, how much will he have accumulated by the end of his sixty-fifth year?

c. Using your findings in parts **a** and **b,** discuss the impact of delaying making deposits into the IRA for 10 years (age 25 to age 35) on the amount accumulated by the end of Hal’s sixty-fifth year.

d. Rework parts **a, b,** and **c,** assuming that Hal makes all deposits at the beginning, rather than the end, of each year. Discuss the effect of beginning-of-year deposits on the future value accumulated by the end of Hal’s sixty-fifth year.

P5–37 **Compounding frequency, time value, and effective annual rates** For each of the cases in the following table:

a. Calculate the future value at the end of the specified deposit period.

b. Determine the *effective annual rate, EAR.*

c. Compare the nominal annual rate, *r*, to the effective annual rate, EAR. What relationship exists between compounding frequency and the nominal and effective annual rates?

Case |
Amount of initial deposit |
Nominal annual rate, r |
Compounding frequency, m(times/year) |
Deposit period (years) |

A | $ 2,500 | 6% | 2 | 5 |

B | 50,000 | 12 | 6 | 3 |

C | 1,000 | 5 | 1 | 10 |

D | 20,000 | 16 | 4 | 6 |

P5–53 **Rate of return and investment choice** Clare Jaccard has $5,000 to invest. Because she is only 25 years old, she is not concerned about the length of the investment’s life. What she is sensitive to is the rate of return she will earn on the investment. With the help of her financial advisor, Clare has isolated four equally risky investments, each providing a single amount at the end of its life, as shown in the following table. All the investments require an initial $5,000 payment.

Investment |
Single amount |
Investment life (years) |

A | $ 8,400 | 6 |

B | 15,900 | 15 |

C | 7,600 | 4 |

D | 13,000 | 10 |

a. Calculate, to the nearest 1%, the rate of return on each of the four investments available to Clare.

b. Which investment would you recommend to Clare, given her goal of maximizing the rate of return?