SDSU BA 323 CH 5 math problems Part 1
If you deposit money today in an account that pays 4% annual interest, how long will it take to double your money?
t = ln(FV / PV) / ln(1 + r) t = ln($2 / $1) / ln 1.04 t = 17.67 years
Find the amount to which $500 will grow under each of these conditions. a. 12% compounded annually for 5 years b. 12% compounded semiannually for 5 years c. 12% compounded quarterly for 5 years d. 12% compounded monthly for 5 years
Time in years, t = 5 Rate of Interest, r = 12% a) Compounded Annually Amount when rate is compounded annually, A = P{ 1 + r/100 }t Therefore, A = 500(1 + 12/100)^5 A = 500 ( 1.12 )^5 A = 881.17 $ 500$ will amount to 881.17$ when compounded annually at 12% for 5 years b) Compounded semi-annually or half yearly Amount when rate is compounded semi-annually, A = P{ 1 + r/200 }2t Therefore, A = 500 [ 1 + 12/200 ]^2x5 A = 500 (1.06)^10 A = 895.42 $ 500$ will amount to 895.42$ when compounded semi-annually at 12% for 5 years c) Compounded Quarterly Amount when rate is compounded quarterly, A = P{ 1 + r/400 }4t Therefore, A = 500 [ 1 + 12/400 ]^4x5 A = 500 (1.03)^20 A = 903.055 $ 500$ will amount to 903.055 $ when compounded quarterly at 12% for 5 years d) Compounded monthly Amount when rate is compounded monthly, A = P{ 1 + r/1200 }12t Therefore, A = 500 ( 1 + 12/1200)^12x5 A = 500 (1.01)60 A = 908.35 $ 500$ will amount to 908.35 $ when compounded monthly at 12% for 5 years
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?
We use the formula: A=P(1+r/100)^n =$5000[(1.12)^n-1]/0.12 220,000=33556.25*(1.12)^n+$5000[(1.12)^n-1]/0.12 220,000=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 Taking log on both sides; n*log 1.12=log 3.478549866 n=log 3.478549866/log 1.12 =11 years(Approx).
Sawyer Corporation's 2018 sales were $5 million. Its 2013 sales were $2.5 million. a. At what rate have sales been growing? b. The calculation described in the quotation fails to consider the compounding effect of interest.
A) The formula for calculating the annual average growth rate is = ( Ending value of Investment / Initial value of Investment ) ( 1/ t ) - 1 T = Period of investment = 5, Ending value of Investment in 2018 = 5 million Initial value of Investment in 2013 = 2.5 million EV = (5 mill / 2.5 mill) ^ (1/5) - 1 = 14.87% B) $2,500,000(1.20)^5 = $2,500,000(2.48832) = $6,220,800,
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?
CAGR = (Ending balance / Beginning balance)(1/n) - 1 CAGR = (800000 / 350000)^(1/19) -1 = CAGR = 4.45% (Approximation)
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%
Case A: Amount Invested = $300 Desired Sum = $600 Interest Rate = 6% Let Period be n years Amount Invested * (1 + Interest Rate)^Period = Desired Sum $300 * 1.06^n = $600 1.06^n = 2 n * ln(1.06) = ln(2) n = 11.90 years Case B: Amount Invested = $300 Desired Sum = $600 Interest Rate = 13% Let Period be n years Amount Invested * (1 + Interest Rate)^Period = Desired Sum $300 * 1.13^n = $600 1.13^n = 2 n * ln(1.13) = ln(2) n = 5.67 years Case C: Amount Invested = $300 Desired Sum = $600 Interest Rate = 21% Let Period be n years Amount Invested * (1 + Interest Rate)^Period = Desired Sum $300 * 1.21^n = $600 1.21^n = 2 n * ln(1.21) = ln(2) n = 3.64 years. Case D: Amount Invested = $300 Desired Sum = $600 Interest Rate = 100% Let Period be n years Amount Invested * (1 + Interest Rate)^Period = Desired Sum $300 * 2.00^n = $600 2^n = 2 n * ln(2) = ln(2) n = 1 year
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 guaranteed, and they would be made at the end of each year. Terms of each contract are as follows. As his advisor, which contract would you recommend that he accept?
Contract 1 (3 million / 1.07)^1 + (3 million / 1.07)^2 + (3 million / 1.07)^3 +(3 million / 1.07)^4 = 10161633.77 Contract 2 (2 million / 1.07)^1 + (3 million / 1.07)^2 + (4.5 million / 1.07)^3 +(5.5 million / 1.07)^4 = 12,358,739.18. Contract 3 (7 million / 1.07)^1 + (1 million / 1.07)^2 + (1 million / 1.07)^3 +(1 million / 1.07)^4 = 8994687.90
Find the present value of $500 due in the future under each of these conditions: a. 12% nominal rate, semiannual compounding, discounted back 5 years b. 12% nominal rate quarterly compounding, discounted back 5 years c. 12% nominal rate, monthly compounding, discounted back 1 year d. Why do the differences in the PVs occur?
Future value =$ 500 a) Rate of interest :12% ,Compounding frequency :Semi Annual ,Time period =5 Years We know that Future Value = Present value (1+i)^n Here I = Interest rate per period n=No of Compounding Periods No. of Compounding periods in 5 Years time =5*2= 10 ( 1 Year= 2 semi annual payments) Interest rate per six months =12% *6/12 =6% Future Value = Present value (1+i)^n $ 500= Present value ( 1+0.06)^10 $ 500= Present value( 1.06)^10 $500= Present value* 1.790848 $ 500/1.790848=Present value Present value =$ 279.20 b) Rate of interest :12% ,Compounding frequency :Quarterly ,Time period =5 Years We know that Future Value = Present value (1+i)^n Here I = Interest rate per period n=No. of Compounding Periods No .of Compounding periods in 5 Years time =5*4= 20 ( 1 Year= 4 Quarterly payments) Interest rate per six months =12% *4/12 =3% Future Value = Present value (1+i)^n $ 500= Present vlue ( 1+0.03)^20 $ 500= Present value( 1.03)^20 $500= Present value* 1.806111 $ 500/1.806111=Present value Present value =$ 276.84 c) Rate of interest :12% ,Compounding frequency :Annual ,Time period =1 Year We know that Future Value = Present value (1+i)^n Here I = Interest rate per period n=No. of Compounding Periods Future Value = Present value (1+i)^n $ 500= Present value ( 1+0.12)^1 $ 500= Present value( 1.12) $500/1.12= Present value $ 446.4286=Present value Present value =$ 446.43 d) Difference in PV is due to differences arising in Compounding freuency and time Period
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?
Present value = CF1/(1+R)^1 + CF2/(1+R)^2 + CF3/(1+R)^3 + CF4/(1+R)^4 + CF5/(1+R)^5 + CF6/(1+R)^6 = 150/1.11^1 + 150/1.11^2 + 150/1.11^3 + 250/1.11^4 + 300/1.11^5 + 500/1.11^6 = 976.60 Future value = Present value * (1 + 11%)^6 = 976.60*1.11^6 = 1826.65 Use a Graphic Calculator to find the Present Value. Press the app button, then press the "npv (number 7 on keypad)" npv(11,0, {150,250,300,500},{3,1,1,1} = 976.6
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.
Solution A). % of Interest =( Amount to be pay back - Amount borrowed) / Amount borrowed * 100 = (792 - 720) / 720 * 100 = 10% (interest to be paid) B). % of Interest =( Amount going to be received - Amount given as borrowing) / Amount given as borrowing * 100 = (792 - 720) / 720 * 100 = 10% (interest earned) C). Future Value = Amount (1+r)n 98319 = 65000 (1+r)^14 (1+r)^14 = 1.5126 r = 3% (interest to be paid)
Find the present values of the following cash flow streams at a 5% discount rate. CF0 CF1 CF2 F=33 4 CF3 A 0 150 450 450 450 250 B 0 250 450 450 450 150
The present value is computed as shown below: Present value = Future value / (1 + r )n The present value of Stream A is computed as shown below: = $ 0 + $ 150 / 1.05 + $ 450 / 1.05^2 + $ 450 / 1.05^3 + $ 450 / 1.05^4 + $ 250 / 1.05^5 = $ 1,505.84 Approximately The present value of Stream A is computed as shown below: = $ 0 + $ 250 / 1.05 + $ 450 / 1.05^2 + $ 450 / 1.05^3 + $ 450 / 1.05^4 + $ 150 / 1.05^5 = $ 1,522.73 Approximately b. The PV of stream A is computed as follows: = $ 0 + $ 150 + $ 450 + $ 450 + $ 450 + $ 250 = $ 1,750 The PV of stream B is computed as follows: = $ 0 + $ 250 + $ 450 + $ 450 + $ 450 + $ 150 = $ 1,750
Find the following values. Compounding /discounting occurs annually. a. An initial $200 compounded for 10 years at 4% b. An initial $200 compounded for 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%
A) $200(1.04)^10 = $296.05 B) $200(1.08)^10 = $431.78. C) $200/(1.04)^10 = $135.11. D) $1,870.0/(1.08)^10 = $866.17. Solve for PV $1,870.00/(1.04)^10 = $1,263.30.
Find the future values of these ordinary annuities. Compounding occurs once a year. a. $500 per year for 12 years at 8% b. $250 per year for 4 years at 7% c. $700 per year for 4 years at 0% d. Rework parts a, b, and c assuming they are annuities due
A) Ordinary Annuity = 500 Time = 12 Annual Interest Rate = 8 Number of payments = 1 (500/0.08) x [( 1 + 0.08)^12 - 1 ] = 9488.56 B) Ordinary Annuity = 250 Time = 4 Annual Interest Rate = 7% Number of payments = 1 (250/.07) x [ (1+.07)^4-1 ] = 1109.99 C) 700 x 4 = 2800 D) Find the Annuity Annuity 1: AP = 500 Payments = 12 Interest Rate = 8% FV = (500/ .08) x 1.08 x (1.08^12 - 1) = 10247.65 Annuity 2: AP = 250 Payments = 4 Interest Rate = 7% FV = 1187.69 Annuity 3: AP = 700 Payments = 4 Interest Rate = 0 FV = 700 x 4 = 2800
Find the following values using the equations and then a financial calculator. Compounding/discounting occurs annually. Do not round intermediate calculations. Round your answers to the nearest cent. 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) 600*(1+6%) = 636.00 B) 600*(1+6%)^2 = 674.16 C) 600/(1+6%) = 566.04 D) 600/(1+6%)^2 = 534.00
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?
FV of ordinary annuity = Deposit*((1+R)^n-1)/R = 800*((1+10%)^5-1)/10% = 800*6.1051 = 4884.08 FV of annuity due = FV of ordinary annuity * (1+Rate) = 4884.08*(1+10%) = 5372.49
If you deposit $2,000 in a bank account that pays 6% interest annually, how much will be in your account after 5 years?
FV^5 = $2,000(1.06)^5= $2,000(1.338226) = $2,676.45.
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?
Formula: PV = PMT/Interest Rate 600/.05 = 12,000 600/.10 = 6,000
You borrow $230,000, the annual loan payments are $20,430.31 for 30 years. What interest rate are you being charged?
N = 30; PV = 230,000; PMT = -20,430.31; I = 8%
What is the present value of a security that will pay $29,000 in 20 years if securities of equal risk pay 5% annually?
PV5 = $29,000/(1.05)^20= $29,000/2.6532977 = $10,929.80.
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
Part A 600 [ (1/0.08) - 1 / 0.08(1.08)^12 ] = 4521.65 Part B 300 [ ( 1 / 0/04) - 1 / 0.04 ( 1.04) ^6] = 1572.64 Part C 500(6) = 300 Part D 600 [ (1/0.08) - 1 / 0.08(1.08)^12 ] x 1.08 =4883.38 300 [ ( 1 / 0/04) - 1 / 0.04 ( 1.04) ^6] x 1.04 = 1635.55 500(6)(1) = 300
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 $8,000 per year, and you advise her to invest 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 the 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?
Part A ACF = 8000 Rate = 10% N = 65 - 26 = 39 8000 [(1.1)^39 - 1 / .10] = 3,211,582.22 Part B ACF = 8000 Rate = 10% N = 70 - 26 = 44 8000 [(1.1)^44 - 1 / .10] = 5,221,126.09 Part C Retirement Period = 20 Rate = 10% Present Val = 3,211,582.22 3,211,582.22 ( .10) / 1-(1.1) ^ -20 = 377,231.24
Kristina just won 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?
d. As the interest rate is increasing then she must choose to elect the option in which there is less time to receive the money.
