FINC3331 chapter 5 HW/quiz

5-9 PRESENT AND FUTURE VALUES FOR DIFFERENT PERIODS Find the following values using the equations and then a financial calculator. Compounding/discounting occurs annually. a. An initial $600 compounded for 1 year at 6% b. An initial $600 compounded for 2 years at 6% c. The present value of $600 due in 1 year at a discount rate of 6% d. The present value of $600 due in 2 years at a discount rate of 6%

*a. FVN= PV (1+I)^N =600(1+.06)^1 = 636 * b. FVN= PV (1+I)^N =600(1+.06)^2 =674.16 *c . PV = C/(1+R)^N =600/(1+.06)^1 = 566.03 *d . PV = C/(1+R)^N =600/(1+.06)^2 = 533.99 or 534

question 4 would you rather have a savings account that pays 5% interest compounded semiannually or one that pays 5% interest compounded daily?

5% compounded daily - All other things being equal, more frequent compounding results in a higher future value

5-16 PRESENT VALUE OF A PERPETUITY What is the present value of a $600 perpetuity if the interest rate is 5%? If interest rates doubled to 10%, what would its present value be?

600/5%=12000 600/10%=6000

question 8 what is a loan amortization schedule and what are some ways these schedules are used?

A loan amortization schedule is a document showing the following information for each mortgage payment: the payment number, the interest paid from the payment, the amount of the payment applied to the principal, and the balance of the loan after the payment. Aside from being used as a basis or reference for computing the loan balance per period, this may also be used for proofing purposes to reconcile any differences or errors in the recorded balances. The loan amortization schedule is a very practical tool to use in loan computations.

5-13 TIME FOR A LUMP SUM TO DOUBLE How long will it take $300 to double if it earns the following rates? Compounding occurs once a year. a. 6% b. 13% c. 21% d. 100%

Amount Invested * (1 + Interest Rate)^Period = Desired Sum a. $300(1+.06)^n=600 $300 * 1.06^n = $600 1.06^n=600/300 1.06^n=2 n=ln2/ln1.06 n=11.9 b. n=ln2/ln1.13 n=5.67 c. n=ln2/ln1.21 n=3.64 d. n=ln2/ln2 n=1

your client is 35 years old, and she want to begin saving for retirement, with the first payment to come one year from now. she can save 8,500 per year, and you advise her to invest it in securities which you expect to provide an average annual return of 8 percent. if she follows your advice, how much money would she have at age 65

Annual payment =8500 interest 8% No. of years (65-35=30) FVA=PMT * ((1+I)^n)-1)/I =8500 * ((1+.08)^30)-1)/.08 =8500 * (10.062656890-1)/.08 =8500 * (9.062656890/.08) =8500 * 11.2832111 =962,907.29

5-20 PV OF A CASH FLOW STREAM A rookie quarterback is negotiating his first NFL contract. His opportunity cost is 7%. He has been offered three possible 4-year contracts. Payments are guar-anteed, and they would be made at the end of each year. Terms of each contract are as follows: As his adviser, which contract would you recommend that he accept.

Contract 2; PV = $12,358,739.18. 2mill/(1.07^1)=1,869,158 3mill/(1.07^2)=2,620,316 4.5mill/(1.07^3)=3,673,340 5.5mill/(1.07^4)=4,195,923 1,869,158+2,620,316+3,673,340+4,195,923=12,358,739.19

5-27 EFFECTIVE VERSUS NOMINAL INTEREST RATES Bank A pays 2% interest compounded annually on deposits, while Bank B pays 1.75% compounded daily. a. Based on the EAR (or EFF%), which bank should you use?

EFF%=(1+r/n)^n)-1 (1+.02/1)^1)-1 =1.02-1 =2% (1+.0175/365)^365)-1 =1.0176536-1 =1.77%

If you deposit $50,000 in an account that pays 7 percent annually, how much would you have after 14 years?

FV = PV * (1 + r)^n =50,000(1+.07)^14 =128,926.71

What is the future value of an annuity due that pays $1,300 per year for 4 years? The appropriate interest rate is 10 percent.

FV of annuity due= annual payment((1+r)^n)-1) x (1+r)/r 1,300((1+.10)^4)-1) *(1+.10)/.10 =6,636,63

5-11 GROWTH RATES Sawyer Corporation's 2018 sales were $5 million. Its 2013 sales were $2.5 million. a. At what rate have sales been growing? b. Suppose someone made this statement: "Sales doubled in 5 years. This represents a growth of 100% in 5 years; so dividing 100% by 5, we find the growth rate to be 20% per year." Is the statement correct?

FV=5 mill PV= 2.5 mill T=5 n= annually r={[(FV/PV)^(1/(t)(n)]-1}*n ={[(5,000,000/2,500,000)^(1/(5)(1)]-1}*1 +{[(5,000,000/2,500,000)^(1/5)]-1}*1 =(1.1487-1)*1 =0.1487 * 1 =14.87 b. This is not correct statement because it missed the effect of compounding which means you need to assume same growth on the incremental sales.We already calculated above that sales doubled in 5 years at 14.87% rate not 20%.

5-23 FUTURE VALUE FOR VARIOUS COMPOUNDING PERIODS Find the amount to which $500 will grow under each of these conditions: a. 12% compounded annually for 5 years b. 12% compounded semiannually(=2) for 5 years c. 12% compounded quarterly(=4) for 5 years d. 12% compounded monthly(=12) for 5 years e. 12% compounded daily(=365) for 5 years f. Why does the observed pattern of FVs occur?

FV=PV(1+r)^n a. Future value=500*(1+12%)^5 Future value = $ 881.17 b. Future value = 500*(1+12%/2)^(5 *2) Future value = $ 895.42 c. Future value = 500*(1+12%/4)^(5 *4) Future value = $ 903.06 d. Future value = 500*(1+12%/12)^(5 *12) Future value = $ 908.35 e. Future value = 500*(1+12%/365)^(5 *365) Future value = $ 910.97 f. The observed pattern of FVs occur due to change in compounded period i.e interest is grow more if the compunded interest grow frequentlly than interest grow less frequently, The pattern occur the frequency of growth rate is faster than previous one.

5-5 TIME TO REACH YOUR FINANCIAL GOAL You have $33,556.25 in a brokerage account, and you plan to deposit an additional $5,000 at the end of every future year until your account totals $220,000. You expect to earn 12% annually on the account. How many years will it take to reach your goal?

FV=PV(1+r/100)^n =33,556.25(1.12)^n FV=PV[(1+r)^n)-1]/r 220,000=(33556.25*(1.12)^n)+($5000[(1.12)^n-1]/0.12)aka 5000/.12=41,666.67 220,000=33556.25*(1.12)^n+$41,666.67[(1.12)^n-1] 220,000=33556.25*(1.12)^n+$41,666.67*(1.12)^n-41,666.67 (220,000+41,666.67)=(1.12)^n[33556.25+41,666.67] (1.12)^n=(220,000 +41,666.67)/(33556.25+41,666.67) (1.12)^n=3.478549866 n=log 3.478549866/log 1.12 =11 years

You have $41,137.85 in a brokerage account, and you plan to deposit an additional $5,500 at the end of every year until your account totals $200,000. You expect to earn 10.4 percent annually on the account. How long will it take to reach your goal?

FV=[PV(1+r)^n]+[P((1+r)^n)-1)/r)] 200,000=(41,137.85(1.104)^n)+(5,500[(1.104)^n)-1]/.104 aka 5,500/.104=52,884.62 200,000=41,137.85(1.104)^n+52,884.62[(1.104)^n -1] 200,000=41137.85(1.104)^n+52,884.62(1.104)^n - 52,884.62 (200,000+52,884.62)=(1.104)^n[41,137.85+52,884.62] (1.104)^n=(200,000+52,884.62)/[41,137.85+52,884.62] (1.104)^n=252,884.62/94,022.47 (1.104)^n=2.6896189815 n=log2.6896189815/log1.104 n=10 years

5-14 FUTURE VALUE OF AN ANNUITY Find the future values of these ordinary annuities. Compounding occurs once a year. a. $500 per year for 8 years at 14% b. $250 per year for 4 years at 7% c. $700 per year for 4 years at 0% d. Rework parts a, b, and assuming they are annuities due.

FVoa=PMT ((1+r/n)^(t)(n)) -1/(r/n) PMT=500 r=14% t=8 n=annually 1 =500 * ((1+.14/1)^(8)(1))-1/(.14/1) 500 * (2.8525864-1)/.14 500 * (1.8525864/.14) =6616.38 .b. $1,109.99. c. Future value will be sum of the annuity as there is no interest on annuity= $700+ $700+ $700+ $700 = $2,800 d. (1). 500*1.14*((1.14^8) -1)/.14 =$7,542.67. d(2). 250*1.07*((1.07^4)-1)/.07 =$1,187.68. d(3). same as C above $2,800

You borrow $130,000 and must make annual loan payments of $12,271.09 for 20 years, starting at the end of the year. What interest rate are you being charged?

In calculator Ti-83 hit App hit finance hit TVM solver N=20 I%=0 PV=130,000 PMT=-12,271.09 FV=0 P/Y=1 C/Y=1 click on I% then hit ALPHA enter 7%

5-3 FINDING THE REQUIRED INTEREST RATE Your parents will retire in 19 years. they currently have $350,000 saved, and they think they will need $800,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds?

No of years =19 Present value = $350,000 Future value = $800,000 annual interest rate=(FV/PR)^1/n)-1 (800,000/350,000)^1/19)-1 =0.0445 or x100= 4.45

You want to buy a house, and a mortgage company will lend you $165,000. The loan would be fully amortized over 30 years (360 months), and the nominal interest rate would be fixed at 9.75 percent, compounded monthly. What would be the monthly mortgage payment?

PMT= PV(r/n)/1-(1+r/n)^-(t)(n) 165,000(.0975/12)/1-(1+.0975/12^-(30)(12) 165,000(.008125)/1-.054302709 1340.625/0.9457 =1417.60

What is the present value of a security that will pay $16,000 in 15 years if securities of equal risk pay 7 percent annually?

PV = FV/(1+r)^n =16000/(1+.07)^15 =5799.14

5-8 LOAN AMORTIZATION AND EAR You want to buy a car, and a local bank will lend you $40,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest rate will be 8% with interest paid monthly. What will be the monthly loan payment? What will be the loan's EAR?

PV= 40,000 r=8% t= 5yrs n=12 PMT= PV(r/n)/1-(1+r/n)^-(t)(n) = 40,000(.08/12)/1-(1+.08/12)^-(5)(12) =40,000(.0066667)/1-0.6712104 =266.668/0.3287896 =811.06 EFF%= (1+r/n)^n -1 =(1+.08/12)^12 -1 =1.083-1 =8.3

An investment will pay $900 at the end of each of the next 3 years, $1,000 at the end of Year 4, and $1,300 at the end of Year 5. What is its present value if other investments of equal risk earn 11 percent annually?

PV=(CF1/(1+r) + CF2/(1+r)^2 + CF3/ (1+r)^3 +CF4/(1+r)^4 + CF5/(1+r)^5 900/(1+.11)+900/(1+.11)^2 +900/(1+.11)^3 +1,000/(1+.11)^4 +1,300/(1+.11)^5 =3629.56

5-7 PRESENT AND FUTURE VALUES OF A CASH FLOW STREAM an investment will pay $150 at the end of each of the next 3 years, $250 at the end of year 4, $300 at the end of year 5, and $500 at the end of year 6. If other investments of equal risk earn 11% annually, what is its present value? its future value?

PV=(CF1/(1+r) + CF2/(1+r)^2 + CF3/ (1+r)^3 +CF4/(1+r)^4 + CF5/(1+r)^5 + CF6/(1+r)^6) =150/1+.11 + 150/(1+.11)^2 + 150/(1+.11)^3 +250/(1+.11)^4 + 300/(1+.11)^5 +500/(1+.11)^6 =976.60 FV=(CF1x(1+r)^5 + CF2x(1+r)^4 + CF3x(1+r)^3 + CF4x(1+r)^2 + CF5x(1+r) +CF6 =150x(1+.11)^5 + 150x(1+.11)^4 + 150x(1+.11)^3 +250x(1+.11)^2 +300x(1+.11)^1 + 500 =1826.64

Six years from today, you will need $35,000 for a down payment on a new home. You believe you can earn 5% annually in an investment account. How much must you deposit today to ensure that you will have the necessary funds?

PV=FV/(1+r)^n

5-24 PRESENT VALUE FOR VARIOUS DISCOUNTING PERIODS Find the present value of $500 due in the future under each of these conditions: a. 12% nominal rate, semiannual compounding, discounted back 5 years b. 12% nominal rate, quarterly compounding, discounted back 5 years c. 12% nominal rate, monthly compounding, discounted back 1 year d. Why do the differences in the PVs occur?

PV=FV/1(1+r/n)^(n*t) a. PV = 500 / (1 + 0.12/2)^(2*5) PV ≈ $279.20 b. PV = 500 / (1 + 0.12/4)^(4*5) PV ≈ $276.84 c. PV = 500 / (1 + 0.12/12)^(12*1) PV ≈ $443.72 d. The differences arise due to the increased frequency of compounding, which reduces the present value in each case.

What is the present value of a perpetuity that pays $3,400 per year if the discount rate is 7 percent?

PV=Payment/percent 3400/.07=48,571.43

5-1 FUTURE VALUE If you deposit $2,000 in a bank account that pays 6% interest annually, how much will be in your account after 5 years?

Present Value = $2,000 Interest rate = 6% Periods =5 FV=PV x (1+r)^n FV= 2000 x (1+6%)^5 =2000 x 1.3382255776 = $2,676.45

5-4 TIME FOR A LUMP SUM TO DOUBLE if you deposit money today in an account that pays 4% annual interest, how long will it take to double your money?

Since the money is to be doubled, let us treat the present value (P) as 1 and the future value (F)as 2. number of compounding periods (n)= ln(f/p)/ln(1+r) ln(2/1)/ln(1+.04) =17.67

5-18 UNEVEN CASH FLOW STREAM a. Find the present values of the following cash flow streams at a 5% discount rate. b. What are the PVs of the streams at a 0% discount rate?

a. Stream a Present Value = [ $ 150 * 1/(1.05) ^ 1 + 450* 1/(1.05) ^2+450* 1/(1.05) ^3+450* 1/(1.05) ^4+ 250* 1/(1.05) ^5 ]= 1,505.84 stream b Present Value = [ $ 250 * 1/(1.05) ^ 1 + 450* 1/(1.05) ^2+450* 1/(1.05) ^3+450* 1/(1.05) ^4+ 150* 1/(1.05) ^5 ]= 1,522.73 b. same for both Present Value = [ $ 150 * 1/(1.0) ^ 1 + 450* 1/(1.0) ^2+450* 1/(1.0) ^3+450* 1/(1.0) ^4+ 250* 1/(1.0) ^5 ] = $ 1,750

5-34 AMORTIZATION SCHEDULE a. Set up an amortization schedule for a $19,000 loan to be repaid in equal installments at the end of each of the next 3 years. The interest rate is 8% compounded annually. b. What percentage of the payment represents interest and what percentage represents principal for each of the 3 years? Why do these percentages change over time?

a. PMT= ((PV(r/n))/(1-(1+r/n)^-(t)(n)) 19,000(.08/1)/1-(1+.08/1)^-(3)(1) 19000(.08)/1-0.7938322 =1520/0.2061678 =7,372.64

5-12 EFFECTIVE RATE OF INTEREST Find the interest rates earned on each of the following: a. You borrow $720 and promise to pay back $792 at the end of 1 year. b. you lend $720 and the borrower promises to pay you $792 at the end of 1 year c. you borrow $65,000 and promise to pay back $98,319 at the end of 14 years d. you borrow $15,000 and promise to make payments of $4,058.60 at the end of each year for 5 years.

a. r=(FV-PV)^(1/n) *100 ((792-720)^(1/1) *100 1.1-1=0.1 0.1*100 =10% b. ((792-720)^(1/1) *100 1.1-1=0.1 0.1*100 =10% c. [(FV/PV)^(1/n) -1]*100 (98,319/65,000)^(1/14) -1]*100 [(1.5126)^(1/14) -1]*100 (1.0300004998-1)*100 =3% d. 11%

5-21 EVALUATING LUMP SUMS AND ANNUITIES Kristina just won the lottery, and she must choose among three award options. She can elect to receive a lump sum today of $62 million, to receive 10 end-of-year payments of $9.5 million, or to receive 30 end-of-year payments of $5.6 million. a. If she thinks she can earn 7% annually, which should she choose? b. If she expects to earn 8% annually, which is the best choice? c. If she expects to earn 9% annually, which option would you recommend? d. Explain how interest rates influence her choice.

a. 30-year payment plan; PV = $69,490,630.63 b. PVoa=PMT(1-(1+r/n)^-(t)(n)/(r/n) 9,500,000(1-(1+0.08/1)^-(10)(1)/(0.08/1) 9,500,000(1-0.4631935)/.08) 9,500,000(0.5368065/0.08) 63,745,771.88 c. Lump sum; PV = $62,000,000 d. This also holds true between options 2 and 3, at 7% interest rate, option 3 is favorable, but at 8% interest rate, option 2 with the shorter time period is now more favorable.

5-25 FUTURE VALUE OF AN ANNUITY Find the future values of the following ordinary annuities: a. FV of $400 paid each 6 months for 5 years at a nominal rate of 12% compounded semiannually b. FV of $200 paid each 3 months for 5 years at a nominal rate of 12% compounded quarterly

a. FV=PMT * (((1+r)^n)-1)/r) 400 * (((1+.06)^10)-1)/.06) =5,272.32 b. FV=PMT * (((1+r)^n)-1)/r) 200 * (((1+.03)^20)-1)/.03) =5,374.07

5-19 FUTURE VALUE OF AN ANNUITY Your client is 26 years old. She wants to begin saving for retirement, with the first payment to come one year from now. She can save 58,000 per year, and you advise her to invest it in the stock market, which you expect to provide an average return of 10% in the future. a. If she follows your advice, how much money will she have at 65? b. How much will she have at 70? c. She expects to live for 20 years if she retires at 65 and for 15 years if she retires at 70. If her investments continue to earn the same rate, how much will she be able to withdraw at the end of each year after retirement at each retirement age?

a. FV=PMT x (1+r)^n)-1)/r 8000 x (1+.10)^39)-1)/.10 =3,211,582.22 b. FV=PMT x (1+r)^n)-1)/r 8000x (1+.10)^44)-1)/.10 =5,211,126.22 c. 3,211,582.22/((1+.10)^-20)/.10)= 377,231.24 5,221,126.09/((1+.10)^-20)/.10)= 686,441.17

5-10 PRESENT AND FUTURE VALUES FOR DIFFERENT INTEREST RATES Find the following values. Compounding/discounting occurs annually. a. An initial $200 compounded for 10 years at 4% b. An initial $200 compounded fat 10 years at 8% c. The present value of $200 due in 10 years at 4% d. The present value of $1,870 due in 10 years at 8% and at 4% e. Define present value and illustrate it using a time line with data from part d. How are present values affected by interest rates?

a. FV=PV(1+r)^n 200(1+.04)^10 200(1.4802) =296.05 b.FV=PV(1+r)^n 200(1+.08)^10 200(2.1589) =431.78 c. PV=FV/(1+r)^n =200/(1+.04)^10 =200/1.4802 =135.11 d. PV=FV/(1+r)^n =1870/(1+.08)^10 =1870/2.1589 =866.17 1870/(1+.04)^10 1870/1.4802 =1,263.30 e. PV refers to the value of an investment which is in the current. On this value the future value of the annuity is determined. the PV for the given cash flows in calculated at the specified rate of interest. The time line is a visual representation of the cash flows. It shows the PV, the rate of interest, the time period and the future value.

5-31 REQUIRED LUMP SUM PAYMENT Starting next year, you will need $5,000 annually for 4 years to complete your education. (One year from today you will withdraw the first 55,000.) Your uncle deposits an amount today in a bank paying 6% annual interest, which will provide the needed $5,000 payments. a. How large must the deposit be? b. How much will be in the account immediately after you make the first withdrawal?

a. PVoa=PMT(1-(1+r/n)^-(t)(n)/(r/n) PVoa=PMT x (1-(1+.06/1)^-(4)(1) =5000(1-.7920937/.06) =5000(.2079063/.06) =17,325.53 b. FV=P(1+r/n)^(t)(n) =17,325.53(1+.06/1)^(1)(1) =17,325.53(1.060^1 =18,365.06

5-15 PRESENT VALUE OF AN ANNUITY Find the present values of these ordinary annuities. Discounting occurs once a year. a. $600 per year for 12 years at 8% b. $300 per year for 6 years at 4% c. $500 per year for 6 years at 0% d. Rework parts a, b, and c assuming they are annuities due.

a. PVoa=PMT* (1-(1+r)^-n)/r 600*(1-(1+.08)^-12)/.08 4,521.65 b.300*(1-(1+.04)^-6)/.04 1,572.64 c. PVoa=N x Payment 6x500= 3,000 d. PVad=(PMT x (1-(1+r)^-n)/r) x (1+r) d.a. (600 x (1-(1+.08)^-12)/.08) x (1+.08) 4883.38 d.b. (300 x (1-(1+.04)^-6)/.04) x (1+.04) 1,635.55 c. same as above

Questions 3 If a firm's earnings per share grew from $1 to $2 over a 10 year period, the total growth would be 100%, but the annual growth rate would be less than 10%. TF

annual growth rate=[(ending value/beginning value)^1/number of years)-1] ending value= $2 beginning value= $1 number of years =10 annual growth rate %= [(2/1)^1/10)-1 =0.07177 or x100 = 7.18 True

Joe plans to retire in 15 years. He currently has saved up $450,000, and he believes he will need $1,000,000 at retirement. What annual interest rate must Joe earn to reach his goal, assuming he does not save any additional funds between now and retirement?

annual interest rate=(FV/PR)^1/n)-1 (1,000,000/450,000)^1/15)-1 then times by 100 =1.0546762485-1 =0.0546762485 *100 =5.47

Mary recently received a credit card with a nominal interest rate of 17 percent. With the card, she purchased some new clothes for $400. The minimum payment on the card is only $20 per month. If Mary makes the minimum monthly payment and makes no other charges, how long will it be before she pays off the card?

to get the nominal interest rate you divide the percent by yearly months 17/12=1.42 23.7

If you deposit $5,000 today into an account that earns a compounded annual return of 11.9 percent, how long will it take for your money to grow to $20,000?

total amount=principal amount(1+r/100)^n 20,000=5,000(1+11.9/100)^n 20,000/5,000=(1.119)^n 4=(1.119)^n Ln on both sides ln4=ln(1.119)^n 1.3862943611=n(0.1124354293) 1.3862943611/0.1124354293 n=12.33

5-17 EFFECTIVE INTEREST RATE You borrow $230,000; the annual loan payments are $20,430.31 for 30 years. What interest rate are you being charged?

with calculator go to apps hit finance hit TVM solver N=30 I=0 PV=230,000 PMT=-20430.31 PV=0 P/Y=1 C/Y=1 then put clicker over I then press ALPHA enter 8%

As a fixed-rate, amortized loan is paid off over time...

more and more of each payment goes toward reducing principal, with less and less goes toward interest

Suppose you invest $2,600 today in an account that earns a nominal annual rate (inom) of 16 percent, with interest compounded monthly. How much money will you have after 8 years?

FV=PV *(1+r/n)^n*t =2600*(1+.16/12)^8*12 =2600*1.01333333^96 +2600*3.566345679 9.272.50

Congratulations! You have just won $600,000 in the state lottery. Before you get too excited, realize that you don't get all the money right now. Your prize will be paid out as an annuity, with annual payments of $20,000 at the beginning of each year for 30 years. In lieu of this, a finance company has offered to pay you one lump sum now. How large should that lump sum be if a fair rate of return is 8% per year?

N=30 I=8% PMT=20,000 FV=0 P/y=1 C/y=1 PMT=begin 518,773.15

5-37 PAYING OFF CREDIT CARDS Simon recently received a credit card with an 18% nominal interest rate. With the card, he purchased an Apple iPhone 7 for $372.71. The minimum payment on the card is only $10 per month. a. If Simon makes the minimum monthly payment and makes no other charges, how many months will it be before he pays off the card? Round to the nearest month. b. If Simon makes monthly payments of $35, how many months will it be before he pays off the debt? Round to the nearest month. c. How much more in total payments will Simon make under the $10-a-month plan than under the $35-a-month plan? Make sure you use three decimal places for N.

formula in picture a. =log[1-(372.71(.18/12)/10]/(-12log(1+.18/12) =log[1-0.559065]/-12log(1.015 =log[0.440935]/-12(0.0064660422) =-0.3556254269/-0.077592507 =4.58324 yrs 4.58324*12=55 months b. (372.71(.18/12)/35]/-12log(1+.18/12) =log[1-.1597328571]/-12log(1.015) =log[0.8402671429]/-12(0.0064660422 =-0.0755826184/-0.077592507 =0.9741 =0.9741*12=12 months(rounded) c. number of years * months per year=number of payments * monthly payment=total payment Plan1 ($10 per month) 4.58324*12=55 55*10=550 plan2($35 per month) 0.9741*12=11.689 11.689*35=409.12 550-409.12=140.88

5-6 FUTURE VALUE:ANNUITY VERSUS ANNUITY DUE what's the future value of a 5%, 5 year ordinary annuity that pays $800 each year? if this was an annuity due, what would its future value be?

future value of ordinary annuity=Annual payment((1+r)^n)-1)/r) = 800((1+.05)^5)-1)/.05) =4,420.51 FV of annuity due= annual payment((1+r)^n)-1) x (1+r)/r =800[(1+.05)^5)-1) x (1+.05)/.05) =4,641.53

5-2 PRESENT VALUE What is the present value of a security that will pay $29,000 in 20 years if securities of equal risk pay 5% annually?

Future cash inflow (FV) = $29,000 Period (n) = 20 Years Risk free rate (r) = 5% PV=FV/(1+r)^n PV=29,000/(1+.05)^20 PV=10,929.80

5-33 FV OF UNEVEN CASH FLOW You want to buy a house within 3 years, and you are currently saving for the down payment. You plan to save $9,000 at the end of the first year, and you anticipate that your annual savings will increase by 5% annually thereafter. Your expected annual return is 8%. How much will you have for a down payment at the end of Year 3?

yr1=9,000(1.08)^2=10,497.60 yr2=9450.00(1.08)^1=10,206.00 yr3=9,922.50(1.08)^0=9,922.50 10,497.60+10,206.00+9,922.50=30,626.10 Notes 1. 9,000(1.05)=9,450 2. 9,450(1.05)=9,922.50 3. future value of the cash flows at the end of year 3 is 30,626.10 another way in image