Reference no: EM131073740
1. Suppose you borrow $4000 to attend college at the beginning of your freshman year. The current federal student loan rate is 4.29% per year. Typically, no payments are due until 6 months after you graduate. If you graduate in 4 years, how much will you owe in 4.5 years when you have to start repaying the loan?
(a) Calculate the amount due in 4.5 years using a simple yearly interest rate.
(b) How much interest do you owe using a simple interest rate?
(c) Calculate the same amount using semi-annual compounding.
(d) How much interest do you owe with compounding?
(e) What is the effective annual percentage rate (APR) for part (c).
2. It is often said that it is a virtue to want to make money. But it is foolish to want to make money fast. Suppose you have $1000 to invest. Money invested in mutual stock funds have a long term rate of return of about 6%.
(a) At this rate, how long will it take to double your money if the interest is compounded continuously?
(b) Use the rule of 72 to estimate the doubling time. (Note: we use the rule of 70 when the interest rate evenly divides into 70. We use the rule of 72 when the interest rate evenly divides into 72. If the interest rate does not evenly divide into 70 or 72, we use 69 = 100*ln(2) and a calculator!)
(c) Do your two answers from (a) and (b) agree? Yes or No? If "no", explain why not.
(d) Calculate the doubling time using the rule of 69.
(e) Which estimate is closer to the time you computed in (a)? Using 69 or 72?
3. Suppose you are living paycheck to paycheck. Your rent is due on the 1st of the month but you don't get paid until the 15th of the month. Your net paycheck is $3000. So you go to a payday loan company which will loan you $3000 for 2 weeks for a $50 fee.
(a) What is the effective yearly interest rate for this 2 week loan? (Use 52 weeks = 1 year).
(b) People who loan at "exorbitant" rates of interest are known as "loan sharks". But since the company is technically charging a "fee" instead of 'interest", payday loans are not illegal. So if you are living paycheck to paycheck, why would you want to get a loan from a payday loan company after calculating the effective interest rate?
4. You need an emergency loan of $10,000 to repair your roof. You ask the bank for a 3month home equity loan at a 4% interest rate per year. Because of the relatively short period of time, you do not have to make monthly payments but will just repay the loan at the end of the 3 months. So the bank will charge you the interest "up front" in the form of a discount loan.
(a) How much interest does the bank plan to make (i.e. the discount)?
(b) If you want the proceeds to be $10,000, how much should you apply to borrow (i.e. what is the maturity value that you will have to repay at the end of the 3 months) ?
5. Companies often face the choice of buying or leasing equipment. Suppose a piece of equipment costs $20,000. Typically, companies depreciate the value of their equipment over 5 years for tax purposes.
(a) Suppose you pay cash for the equipment. The total cost over 5 years is the principal plus 5 years of compound interest. If savings interest rate is 3% per year compounded monthly, what is the total cost of the equipment for the 5 year period?
(b) Suppose instead the company borrows the cost from a bank. The bank will charge 5% interest per year. How much is the future value of the $20,000 that the bank plans to have at the end of the 5 years, again using monthly compounding?
(c) The bank gets its money from depositors. If the bank has to pay off the depositors the amount in (a), then how much profit does the bank make from the loan at the end of 5 years?
(d) Now the company has to make monthly payments in order to re-pay the loan. This can be calculated by equating the future value of the $20,000 at 5% interest to the future value of an ordinary annuity paying 5% interest. Calculate the monthly payment using
P(1 + )nt = m[ (1 +)nt - 1]/(r/n)
(e) Finally, the company could lease the equipment at $650per month for the 5 years. Now if you did the problem right, the cost of plan (e) > plan (b) >plan (a). So why would someone choose plan (b) over plan (a)?
(f) Plan (d) is the most expensive option. So why would someone choose plan (d) over two other cheaper options?
6. On Sat May 7, 2016, someone won the Powerball jackpot. The winner gets a choice between two different payouts. (1) $429.6 million paid out in 30 equal yearly installments. (2) A lump sum payment of $284.1 million. Let us make the following assumptions: (a) the inflation rate is zero, (b) the lottery is well managed so it will actually be able to pay off the 30 installments, and (c) the winner has a life expectancy of more than 30 years so that either choice is a viable option.
What is the estimated average long term interest rate r that the lottery is using to calculate the lump sum payment so that it equals the annuity of yearly installments? You cannot do this problem with one equation. Rather you must iterate answers to the two formulas until both formulas give the same answer (or close to the same answer). (see page 149 of Applied Finite Math)
The compound interest formula is:
Ai = P(1 + )nt
The total value of an ordinary annuity formula is
Aa = m [(1 + )nt -1 ] (r/n)
Let n = 1 and t = 30.
What is P? _______
What is m? _______
Fill in the following table where Ai and Aa are in millions of dollars.
|
Interest rate r
|
Ai
|
Aa
|
|
0.00
|
|
|
|
0.01
|
|
|
|
0.02
|
|
|
|
0.03
|
|
|
|
0.04
|
|
|
|
0.05
|
|
|
Estimate the interest rate r so that Ai = Ar.
r =<>_________%. Fill in the number and circle one of the three operators ("=", "<", or ">").
Look up on the internet what the current 30 year US treasury bond interest rate is. ________%
How does this bond rate compare with the interest rate r that you estimated from the table? Circle one of the following.
Less than Greater than About the same
7. The current Social Security "full" retirement age is 66 for the so-called "Baby Boomer" (post World War 2) generation. The Social Security Administration (SSA) allows for both early and delayed retirement payouts. If one starts to collect retirement benefits before age 66, the benefits are permanently reduced 5% per year of early retirement. But it if one waits and starts to collect retirement benefits after age 66, the benefits are permanently increased by 8% per year of delayed retirement. Below is a table used by the SSA to compute the reduced and increased benefits.
|
Age of retirement
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Fraction of full benefit pay out
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Break Even age with 66
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|
62
|
0.80
|
|
|
63
|
0.85
|
|
|
64
|
0.90
|
|
|
65
|
0.95
|
|
|
66
|
1.00
|
n/a
|
|
67
|
1.08
|
|
|
68
|
1.16
|
|
|
69
|
1.24
|
|
|
70
|
1.32
|
|
Thus, a potential retiree has a trade-off to make. If one expects to die "young", then he will come out ahead if he retiresearly. But if one thinks he is going to have a longer life, then he should delay his retirement and thus collect more money than someone who retires earlier.
The future total SSA payout for a person who retires at age 66 and lives to age X is
A = 1.00(X - 66).
(a) What is the corresponding formula for a person who retires at age 62 and lives to age X?
(b) If one takes early retirement at age 62, at what age will the total SSA payout equal that of a person who retires at age 66?
(c) What is the future total SSA payout for a person who retires at age 70 and lives to age X?
(d) If one delays retirement until age 70, at what age will the total SSA payout equal that of a person who retires at age 66?
(e) Fill in the values for all of the ages in the table.
(f) After making these calculations, what would you recommend to someone who is approaching age 62 about when to start his retirement as far as his lifetime Social Security payout is concerned? Remember, in the above table ages 62-65 is the age when the early retiree will "fall behind" the standard aged retiree. For ages 67-70, it is the age when the late retiree will "pull ahead" of the standard aged retiree.