Chapter 4

Josh purchased a condominium 5 yr ago for $$\$ 180,000$$. He made a down payment of \(20 \%\) and financed the balance with a 30 -yr conventional mortgage to be amortized through monthly payments with an interest rate of \(7 \% /\) year compounded monthly on the unpaid balance. The condominium is now appraised at $$\$ 250,000$$. Josh plans to start his own business and wishes to tap into the equity that he has in the condominium. If Josh can secure a new mortgage to refinance his condominium based on a loan of \(80 \%\) of the appraised value, how much cash can Josh muster for his business? (Disregard taxes.)

David owns $$\$ 20,000$$ worth of 10 -yr bonds of Ace Corporation. These bonds pay interest every 6 mo at the rate of \(7 \%\) /year (simple interest). How much income will David receive from this investment every \(6 \mathrm{mo}\) ? How much interest will David receive over the life of the bonds?

HomE Eight years ago, Kim secured a bank loan of $$\$ 180,000$$ to help finance the purchase of a house. The mortgage was for a term of \(30 \mathrm{yr}\), with an interest rate of \(9.5 \%\) /year compounded monthly on the unpaid balance to be amortized through monthly payments. What is the outstanding principal on Kim's house now?

Maya paid $$\$ 10,000$$ for a 7 -yr bond issued by a city. She received interest amounting to $$\$ 3500$$ over the life of the bonds. What rate of (simple) interest did the bond pay?

Olivia plans to secure a 5-yr balloon mortgage of $$\$ 200,000$$ toward the purchase of a condominium. Her monthly payment for the \(5 \mathrm{yr}\) is calculated on the basis of a 30 -yr conventional mortgage at the rate of \(6 \% /\) year compounded monthly. At the end of the 5 yr, Olivia is required to pay the balance owed (the "balloon" payment). What will be her monthly payment, and what will be her balloon payment?

Emilio is securing a 7 -yr Fannie Mae "balloon" mortgage for $$\$ 280,000$$ to finance the purchase of his first home. The monthly payments are based on a 30 -yr amortization. If the prevailing interest rate is \(7.5 \% /\) year compounded monthly, what will be Emilio's monthly payment? What will be his "balloon" payment at the end of 7 yr?

If the cost of a semiprivate room in a hospital was $$\$ 580 /$$ day 5 yr ago and hospital costs have risen at the rate of \(8 \% /\) year since that time, what rate would you expect to pay for a semiprivate room today?

Sarah secured a bank loan of $$\$ 200,000$$ for the purchase of a house. The mortgage is to be amortized through monthly payments for a term of \(15 \mathrm{yr}\), with an interest rate of \(6 \% /\) year compounded monthly on the unpaid balance. She plans to sell her house in 5 yr. How much will Sarah still owe on her house?

Today a typical family of four spends $$\$ 600 /$$ month for food. If inflation occurs at the rate of \(5 \%\) lyear over the next \(6 \mathrm{yr}\), how much should the typical family of four expect to spend for food 6 yr from now?

Four years ago, Emily secured a bank loan of $$\$ 200,000$$ to help finance the purchase of an apartment in Boston. The term of the mortgage is \(30 \mathrm{yr}\), and the interest rate is \(9.5 \% /\) year compounded monthly. Because the interest rate for a conventional 30 -yr home mortgage has now dropped to \(6.75 \%\) /year compounded monthly, Emily is thinking of refinancing her property. a. What is Emily's current monthly mortgage payment? b. What is Emily's current outstanding principal? c. If Emily decides to refinance her property by securing a 30 -yr home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of 6.75\%/year compounded monthly, what will be her monthly mortgage payment? d. How much less would Emily's monthly mortgage payment be if she refinances?

The Kwans are planning to buy a house 4 yr from now. Housing experts in their area have estimated that the cost of a home will increase at a rate of \(5 \%\) year during that period. If this economic prediction holds true, how much can the Kwans expect to pay for a house that currently costs $$\$ 210,000$$ ?

Five years ago, Diane secured a bank loan of $$\$ 300,000$$ to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was \(30 \mathrm{yr}\), and the interest rate was \(9 \%\) /year compounded monthly on the unpaid balance. Because the interest rate for a conventional 30 -yr home mortgage has now dropped to \(7 \% /\) year compounded monthly, Diane is thinking of refinancing her property. a. What is Diane's current monthly mortgage payment? b. What is Diane's current outstanding principal? c. If Diane decides to refinance her property by securing a 30 -yr home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of \(7 \% /\) year compounded monthly, what will be her monthly mortgage payment? d. How much less would Diane's monthly mortgage payment be if she refinances?

Three years ago, Samantha secured an adjustable-rate mortgage (ARM) loan to help finance the purchase of a house. The amount of the original loan was $$\$ 150,000$$ for a term of \(30 \mathrm{yr}\), with interest at the rate of \(7.5 \% /\) year compounded monthly. Currently the interest rate is \(7 \% /\) year compounded monthly, and Samantha's monthly payments are due to be recalculated. What will be her new monthly payment? Hint: Calculate her current outstanding principal. Then, to amortize the loan in the next \(27 \mathrm{yr}\), determine the monthly payment based on the current interest rate.

The managers of a pension fund have invested $$\$1.5$$ million in U.S. government certificates of deposit that pay interest at the rate of \(5.5 \%\) /year compounded semiannually over a period of 10 yr. At the end of this period, how much will the investment be worth?

Five and a half years ago, Chris invested $$\$ 10,000$$ in a retirement fund that grew at the rate of \(10.82 \% /\) year compounded quarterly. What is his account worth today?

After making a down payment of $$\$ 25,000$$, the Meyers need to secure a loan of $$\$ 280,000$$ to purchase a certain house. Their bank's current rate for 25 -yr home loans is \(11 \%\) /year compounded monthly. The owner has offered to finance the loan at \(9.8 \% /\) year compounded monthly. Assuming that both loans would be amortized over a 25 -yr period by 300 equal monthly installments, determine the difference in the amount of interest the Meyers would pay by choosing the seller's financing rather than their bank's.

Jodie invested $$\$ 15,000$$ in a mutual fund 4 yr ago. If the fund grew at the rate of \(9.8 \% /\) year compounded monthly, what would Jodie's account be worth today?

The Martinezes are planning to refinance their home. The outstanding balance on their original loan is $$\$ 150,000$$. Their finance company has offered them two options: Option A: A fixed-rate mortgage at an interest rate of 7.5\%/year compounded monthly, payable over a 30 -yr period in 360 equal monthly installments. Option B: A fixed-rate mortgage at an interest rate of \(7.25 \% /\) year compounded monthly, payable over a 15 -yr period in 180 equal monthly installments. a. Find the monthly payment required to amortize each of these loans over the life of the loan. b. How much interest would the Martinezes save if they chose the 15-yr mortgage instead of the 30 -yr mortgage?

A young man is the beneficiary of a trust fund established for him 21 yr ago at his birth. If the original amount placed in trust was $$\$ 10,000$$, how much will he receive if the money has earned interest at the rate of \(8 \% /\) year compounded annually? Compounded quarterly? Compounded monthly?

Find how much money should be deposited in a bank paying interest at the rate of \(8.5 \% /\) year compounded quarterly so that, at the end of \(5 \mathrm{yr}\), the accumulated amount will be $$\$ 40,000$$.

An individual purchased a 4-yr, $$\$ 10,000$$ promissory note with an interest rate of \(8.5 \%\) /year compounded semiannually. How much did the note cost?

The parents of a child have just come into a large inheritance and wish to establish a trust fund for her college education. If they estimate that they will need $$\$ 100,000$$ in \(13 \mathrm{yr}\), how much should they set aside in the trust now if they can invest the money at \(8 \frac{1}{2} \% /\) year compounded (a) annually, (b) semiannually, and (c) quarterly?

Anthony invested a sum of money 5 yr ago in a savings account that has since paid interest at the rate of \(8 \% /\) year compounded quarterly. His investment is now worth $$\$ 22,289.22$$. How much did he originally invest?

In the last 5 yr, Bendix Mutual Fund grew at the rate of \(10.4 \% /\) year compounded quarterly. Over the same period, Acme Mutual Fund grew at the rate of \(10.6 \% /\) year compounded semiannually. Which mutual fund has a better rate of return?

Fleet Street Savings Bank pays interest at the rate of \(4.25 \%\) /year compounded weekly in a savings account, whereas Washington Bank pays interest at the rate of \(4.125 \%\) /year compounded daily (assume a 365 day year). Which bank offers a better rate of interest?

The proprietors of The Coachmen Inn secured two loans from Union Bank: one for $$\$ 8000$$ due in 3 yr and one for $$\$ 15,000$$ due in \(6 \mathrm{yr}\), both at an interest rate of \(10 \%\) /year compounded semiannually. The bank has agreed to allow the two loans to be consolidated into one loan payable in 5 yr at the same interest rate. What amount will the proprietors of the inn be required to pay the bank at the end of 5 yr? Hint: Find the present value of the first two loans.

Find the effective rate of interest corresponding to a nominal rate of \(9 \% /\) year compounded annually, semiannually, quarterly, and monthly.

Juan is contemplating buying a zero coupon bond that matures in \(10 \mathrm{yr}\) and has a face value of $$\$ 10,000 .$$ If the bond yields a return of \(5.25 \% /\) year, how much should Juan pay for the bond?

Maxwell started a home theater business in 2005 . The revenue of his company for that year was $$\$ 240,000$$. The revenue grew by \(20 \%\) in 2006 and by \(30 \%\) in 2007 . Maxwell projected that the revenue growth for his company in the next 3 yr will be at least \(25 \% /\) year. How much does Maxwell expect his minimum revenue to be for \(2010 ?\)

Online retail sales stood at $$\$ 23.5$$ billion for the year 2000 . For the next 2 yr, they grew by \(33.2 \%\) and \(27.8 \%\) per year, respectively. For the next \(6 \mathrm{yr}\), online retail sales were projected to grow at \(30.5 \%, 19.9 \%\), \(24.3 \%, 14.0 \%, 17.6 \%\), and \(10.5 \%\) per year, respectively. What were the projected online sales for 2008 ?

The inflation rates in the U.S. economy for 2003 through 2006 are \(1.6 \%, 2.3 \%, 2.7 \%\), and \(3.4 \%\), respectively. What was the purchasing power of a dollar at the beginning of 2007 compared to that at the beginning of \(2003 ?\)

Investment A offers a \(10 \%\) return compounded semiannually, and investment \(\mathrm{B}\) offers a \(9.75 \%\) return compounded continuously. Which investment has a higher rate of return over a 4-yr period?

Leonard's current annual salary is $$\$ 45,000$$. Ten yr from now, how much will he need to earn in order to retain his present purchasing power if the rate of inflation over that period is \(3 \% /\) year compounded continuously?

Maria, who is now 50 yr old, is employed by a firm that guarantees her a pension of $$\$ 40,000 /$$ year at age \(65 .\) What is the present value of her first year's pension if the inflation rate over the next \(15 \mathrm{yr}\) is \(6 \% / \mathrm{year}\) compounded continuously? 8\%/year compounded continuously? \(12 \% /\) year compounded continuously?

The simple interest formula \(A=P(1+r t)\) [Formula (1b)] can be written in the form \(A=\) Prt \(+P\), which is the slope-intercept form of a straight line with slope \(P r\) and \(A\) -intercept \(P\). a. Describe the family of straight lines obtained by keeping the value of \(r\) fixed and allowing the value of \(P\) to vary. Interpret your results. b. Describe the family of straight lines obtained by keeping the value of \(P\) fixed and allowing the value of \(r\) to vary. Interpret your results.

Suppose an initial investment of $$\$ P$$ grows to an accumulated amount of $$\$ A$$ in \(t\) yr. Show that the effective rate (annual effective yield) is $$ r_{\text {eff }}=(A / P)^{1 / t}-1 $$ Use the formula given in Exercise 71 to solve Exercises \(72-76 .\)

Martha invested $$\$ 40,000$$ in a boutique 5 yr ago. Her investment is worth $$\$ 70,000$$ today. What is the effective rate (annual effective yield) of her investment?

Georgia purchased a house in January 2000 for $$\$ 200,000$$. In January 2006 she sold the house and made a net profit of $$\$ 56,000$$. Find the effective annual rate of return on her investment over the 6 -yr period.

Steven purchased 1000 shares of a certain stock for $$\$ 25,250$$ (including commissions). He sold the shares 2 yr later and received $$\$ 32,100$$ after deducting commissions. Find the effective annual rate of return on his investment over the 2 -yr period.

Nina purchased a zero coupon bond for $$\$ 6724.53 .$$ The bond matures in \(7 \mathrm{yr}\) and has a face value of $$\$ 10,000$$. Find the effective annual rate of interest for the bond. Hint: Assume that the purchase price of the bond is the initial investment and that the face value of the bond is the accumulated amount.

Carlos invested $$\$ 5000$$ in a money market mutual fund that pays interest on a daily basis. The balance in his account at the end of 8 mo ( 245 days) was \(\$ 5170.42\). Find the effective rate at which Carlos's account earned interest over this period (assume a 365-day year).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. When simple interest is used, the accumulated amount is a linear function of \(t\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If interest is compounded annually, then the effective rate is the same as the nominal rate.