Business Math

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Chapter 5
Interest Rates

5-1.Your bank is offering you an account that will pay 20% interest in total for a two-year deposit. Determine the equivalent discount rate for a period length of

a.Six months.

b.One year.

c.One month.

a.Since 6 months is [pic] of 2 years, using our rule [pic]

So the equivalent 6 month rate is 4.66%.

b.Since one year is half of 2 years [pic]

So the equivalent 1 year rate is 9.54%.

c.Since one month is [pic] of 2 years, using our rule [pic]

So the equivalent 1 month rate is 0.763%.

5-2.Which do you prefer: a bank account that pays 5% per year (EAR) for three years or

a.An account that pays 2[pic] every six months for three years?

b.An account that pays 7[pic] every 18 months for three years?

c.An account that pays [pic] per month for three years?

If you deposit $1 into a bank account that pays 5% per year for 3 years you will have [pic] after 3 years.

a.If the account pays [pic] per 6 months then you will have [pic] after 3 years, so you prefer [pic] every 6 months.

b.If the account pays [pic] per 18 months then you will have [pic] after 3 years, so you prefer 5% per year.

c.If the account pays [pic] per month then you will have [pic] after 3 years, so you prefer [pic] every month.

5-3.Many academic institutions offer a sabbatical policy. Every seven years a professor is given a year free of teaching and other administrative responsibilities at full pay. For a professor earning $70,000 per year who works for a total of 42 years, what is the present value of the amount she will earn while on sabbatical if the interest rate is 6% (EAR)?

Timeline:
|0 |7 |14 | | | |42 | | | | | | | | | |0 |1 |2 | | | |8 | | | | | | | | | |0 |1 |2 | | | |304 | | | | | | | |48 |0.75 % |20,092.39 |–500 |0 |

Thus, your remaining balance is $20,092.39.

If you prepay an extra $100 today, your will lower your remaining balance to $20,092.39 – 100 = $19,992.39. Though your balance is reduced, your required monthly payment does not change. Instead, you will pay off the loan faster; that is, it will reduce the payments you need to make at the very end of the loan. How much smaller will the final payment be? With the extra payment, the timeline changes:

That is, we will pay off by paying $500 per month for 47 months, and some smaller amount, $500 – X, in the last month. To solve for X, recall that the PV of the remaining cash flows equals the outstanding balance when the loan interest rate is used as the discount rate:

[pic]

Solving for X gives

[pic]

So the final payment will be lower by $143.14.

You can also use the annuity spreadsheet to determine this solution. If you prepay $100 today, and make payments of $500 for 48 months, then your final balance at the end will be a credit of $143.14:

|N |I |PV |PMT |FV | |48 |0.75 % |19,992.39 |-500 |143.14 |

(2) The extra payment effectively lets us exchange $100 today for $143.14 in four years. We claimed that the return on this investment should be the loan interest rate. Let’s see if this is the case:

[pic], so it is....
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