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MGT3078 Homework 1 – Solution 1) Your business will pay you the different Cash Flows at the end of each year for 4 year. Assuming the present value of all Cash Flows are equal $9500 today, what is dollar amount of the Cash Flow at the end of the third year. (Consider the discount rate as 10%) 1st year $3,000 nd 2 year $3,500 rd 3 year ?? th 4 year $3,700

Answer: 9500 = (3000/1.1) + (2500/1.1^2) + (x/) 9500 =

3000 1.1

3500

+ (1.1)2 +

x (1.1)3

Solve for x  x= $1801

3700 1.1)4

+(

2) First Simple Bank pays 9.5 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bank over an investment horizon of 8 years? For the same scenario, what would be the APR rate if Bank B pays interest on its accounts compounded monthly? Answer: The total interest paid by First Simple Bank is the interest rate per period times the number of periods. In other words, the interest by First Simple Bank paid over 8 years will be: .0950*(8) = 0.760 First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor of $1 minus the initial investment of $1, or: (1 + r)8 – 1 Setting the two equal, we get: .0950*(8) = (1 + r)8 – 1 r = 1.7601/8 – 1 r = .0732, or 7.32% In the second part: simple interest remains the same as what it was in the first part. It means it is 0.760. But the compound interest is monthly compounded here. So our time interval will be monthly, and we are looking for monthly rate first: 0.760 = (1 + r)8*12 – 1  solve for r. Then you have monthly rate. To calculate APR using monthly rate = APR = r * 12

3) What is the value of an investment that pays $35,000 every other year forever, if the first payment occurs one year from today and the discount rate is 7 percent compounded daily? What is the value today if the first payment occurs four years from today? Answer: This is an example of perpetuity. But we know in perpetuity formula C and R should be both in the same interval. Here C is paid in every two years, but R is given annually. So we need to calculate R for two-year period. Just for a second ignore the question. If I ask you what is the EAR if the nominal rate is 7% per year and the interest will be compounded daily: 𝐸𝐴𝑅 = [1 +

0.07 365 ] 365

-1

Now the only difference is that the time interval is two years. The ratio of 7% over 365 is representing compounding process which is daily. Here, it is the same. Still it is daily compounded. So we keep the ratio. Power of 365 is representing number of compounding during the time interval. Here, it is 2 years. So it should be 2 time 365. It means: 𝑅 = [1 +

0.07 2∗365 ] 365

–1

Notice: do not call the rate above as EAR. EAR is abbreviation of Effective Annual Rate. This is not annual. We are calculating this rate using the same logic of EAR. But this rate is actually two years effective rate, not annual. Now we can calculate the PV of perpetuity using the formula of C/R If you reach up to here, you are done with the first part of the question (and I do not deduct point for missing steps after this – Of course, for the first part of the question 3, not the second part) NOTICE: What we calculated above is not accurate. Because the question says the first payment will be one year from now. Generally, we have two types of annuities and perpetuities: ordinary and Due: In case of Ordinary the payment will be done at the end of each period. In case of Due the payment will be done at the beginning of each period. Our question is none of these. Time period is two years. But the payment will be done in the middle of it. (end of the first year). What we calculated above using C/R formula is PV of ordinary perpetuity. It means we assumed the first payment will be at the end of the second year (while it is at the end of first year) To have an accurate calculation we need to shift it forward using the compound factor (1+r) for one more year to have ordinary perpetuity. In the second part: If the payment occurs four years from now: In the first part we calculated PV of the perpetuity. If we discount it for four years we can have the value of perpetuity in four years: (still this is not accurate but at the level of this class I don’t ask for more) 0.07

You should divide it by [1 + 365 ]4∗365

4) You are making a decision on choosing a credit card. What is your choice? Name Chase Citi Bank Bank of America Discover

APR 19.4% 20% 19.8% 19%

Compounding Daily Quarterly Monthly Infinite

Answer: Here APRs are given. To have an accurate comparison we need EAR. m

 APR  1 EAR  1  m   For the contentious (infinite compounding): EAR = enominal Rate − 1 Then compare EARs. Lower interest rate on Credit card is better. Chase: 21.40% Citi: 21.55% BofA: 21.70% Discover: 20.92%

5) You are planning to save for retirement over the next 25 years. To do this, you will invest $760 per month in a stock account and $360 per month in a bond account. The return of the stock account is expected to be 9.6 percent, and the bond account will pay 5.6 percent. When you retire, you will combine your money into an account with a return of 6.6 percent. How much can you withdraw each month from your account assuming a 20-year withdrawal period? Answer: Here, you have two annuities in the first part when you are saving, and since you want to know how much you can save in 25 years, you want to know the Future Value of these two annuities. Then, when you finish saving you want to start receiving this money for 20 years in monthly order. It means at that time you know how much is your total money, and you only need to know monthly withdrawal. When you find the FV of stock and bond annuities, this amount will be Present Value of your next step annuity (when you receive money), and you just need to solve for monthly Cash Flows (C in the formula) We need to find the FV of annuities payments in retirement. When retirement savings ends, the retirement withdrawals begin. so the FV of stock plus bond annuities will be PV of retirement withdrawals annuity. So, first, we find the FV of the stock account and the FV of the bond account and add the two FVs. Stock account: FV = $760[{[1 + (.096/12)]300 − 1}/(.096/12)] Stock account: FV = $942,249.51 Bond account: FV = $360[{[1 + (.056/12)]300 − 1}/(.056/12)] Bond account: FV = $234,669.77 So, the total amount saved at retirement (it means FV of annuity at 25 years from now) is: $942,249.51 + 234,669.77 = $1,176,919.28 Now this is what you have in your saving account. Assume it is 25 years from now. You have this dollar amount of money in your account and you want to start receiving money each month for 20 years. It means here we have PV of the annuity, we have rate and we have time period. We just need to solve for C (monthly cash flow, or monthly withdrawal) Solving for the withdrawal amount in retirement using the PV of annuity formula gives us: PV = $1,176,919.28 = $C[1 – {1/[1 + (.066/12)]240}/(.066/12)] C = $1,176,919.28/133.07214  C = $8,844.22 withdrawal per month...