1+EAR= (1+r/k)k

So, for 10% APR compounded monthly, the EAR is

1+EAR= (1+0.1/12)12 = 1.10471

=> EAR= 10.47%

For 10% compounded annually, the EAR is

1+EAR= (1+0.1)=1.1

* EAR= 10% (remains the same).

For 9% compounded daily

1+EAR= (1+0.09/365)365 = 1.09416

* EAR= 9.4%

5-8. You can earn $50 in interest on a $1000 deposit for eight months. If the EAR is the same regardless of the length of the investment, how much interest will you earn on a $1000 deposit for a. 6 months.

b. 1 year.

c. 1 1/2 years.

Sol:

Since we can earn $50 interest on a $1000 deposit,

Rate of interest is 5%

Therefore, EAR = (1.05)12/8 -1 =7.593%

a) 1000(1.075936/12 – 1) = 37.27

b) 1000(1.07593−1) = 75.93

c) 1000(1.075933/2 −1) = 116.03

5-12. Capital One is advertising a 60-month, 5.99% APR motorcycle loan. If you need to borrow $8000 to purchase your dream Harley Davidson, what will your monthly payment be? Sol:

Discount rate for 12 months is,

5.99/12 = 0.499167%

C= 8000/[1/0.004991(1-1/(1+0.004991)60)] = $154.63

5-16. You have just purchased a home and taken out a $500,000 mortgage. The mortgage has a 30-year term with monthly payments and an APR of 6%. a. How much will you pay in interest, and how much will you pay in principal, during the first year? b. How much will you pay in interest, and how much will you pay in principal, during the 20th year (i.e., between 19 and 20 years from now)? Sol:

a. APR of 6%/12 = 0.5% per month.

Payment = 500,000/[(1/.005)(1- 1/1.005360)]= $2997.75

Total annual payments = 2997.75 × 12 = $35,973.

Loan Balance after 1 year is 2997.75[1/0.005(1- 1/1.005348)] = $493,860. Therefore,

500,000 – 493,860 = $6140 is principal repaid in first year. Interest paid in 1st year is 35,973 – 6140 = $29833.

b. Loan balance in 19 years (or 360 – 19×12 = 132 remaining pmts) is 2997.75[1/0.005(1- 1/1.005192)]= $289,162

Loan Balance in 20 years = 2997.75[1/0.005(1- 1/1.005120)] = $270,018 Therefore,

Principal repaid = 289,162 – 270,018 = $19,144, and Interest repaid =$35,973 – 19,144 = $16,829.

5-20. Oppenheimer Bank is offering a 30-year mortgage with an APR of 5.25%. With this mortgage your monthly payments would be $2000 per month. In addition, Oppenheimer Bank offers you the following deal: Instead of making the monthly payment of $2000 every month, you can make half the payment every two weeks (so that you will make 52 ⁄ 2 = 26 payments per year). With this plan, how long will it take to pay off the mortgage of $150,000 if the EAR of the loan is unchanged? Sol:

For every 2 weeks payment = 2000/2 = 1000.

1 year = 26 weeks.

Therefore,

(1.0525)1/26 = 1.001970.

So, discount rate = 0.1970%.

Here, PV of loan payments is the outstanding balance.

150, 000= (1000/0.001970)[1- 1/(1.001970)N]

If we solve for N,

We get N= 177.98.

So, it takes 178 months to pay off the mortgage. If we decide to pay for 2 weeks, then 178*2= 356 weeks.

5-24. You have credit card debt of $25,000 that has an APR (monthly compounding) of 15%. Each month you pay the minimum monthly payment only. You are required to pay only the outstanding interest. You have received an offer in the mail for an otherwise identical credit card with an APR of 12%. After considering all your alternatives, you decide to switch cards, roll over the outstanding balance on the old card into the new card, and borrow additional money as well. How much can you borrow today on the new card without changing the minimum monthly payment you will be required to pay?

Sol:

Here the discount rate = 15/12 = 1.25%.

Assuming that monthly payment is the interest we get,

25,000*0.15/12= $312.50.

This is perpetuity. So the amount can be borrowed at the new interest rate is...