Vadim di Pietro
Assignment 1: Solutions
Topic: Time value of money: Retirement savings problem
1) Today is July 1, 2010. You just graduated university. You plan to take a year off to travel and then start work one year from today. Your first monthly salary of $5,000 will be paid on August 1, 2011. Assume your monthly salary will increase by 0.8% each month thereafter, until you retire. Suppose that you plan to retire on July 1, 2041, right after receiving your last paycheck on that same day. For each pay check, you save a fraction of your salary and the rest is used to pay off your bills. You expect to live for another 40 years after the day you retire. Your goal is to save enough of each pay check such that in retirement you can afford to purchase each month the same amount of goods that $1,000 can buy today. Assume that in retirement your purchases are made each month with the first purchase on August 1, 2041, and the last purchase on July 1, 2081.
The inflation rate is 0.5% per month, and the nominal interest rate is 12% APR, with monthly compounding.
a) What is the per-month real interest rate?
The nominal monthly interest rate is given by APR/12 = 12%/12 = 1%. If nominal dollars grow by 1% per month, but prices increase by 0.5% per month, then the purchasing power of a risk free investment increases by the real monthly interest rate of
b) What is the PV of the amount of money you need in retirement? Solve this in two ways: first, by using formulas that involve nominal cash flows and nominal interest rates, and second, by using formulas that involve real cash flows and real interest rates.
The first step is to figure out what the nominal cash flows will have to be in retirement. In order to buy on Aug 1, 2041 what $1,000 can buy today, you will need to have
on Aug 1, 2041. Each subsequent month, you will need 0.5% more than the previous month in order to purchase the same real $1,000. Thus, the required cash flows are a forward starting growing annuity, where the first cash flow C is $6426.01, the growth rate is g = i = 0.005, and there are a total of n = (40)(12) = 480 purchases (note: there are 480 total purchases, not 479). Thus
Note: the term [pic]discounts the value of the growing annuity from July 1, 2041 to July 1, 2010. If you were to just use[pic], this would give you the value of the cash flow stream as of July 1, 2041 (one period before the first cash flow on Aug 1, 2041).
In the real approach, the only differences are that: the first real cash flow is just $1,000, the real cash flows are constant (so we have a forward starting constant real annuity), and you have to use the real discount rate you found in part a. So the PV is given by:
Note: You may have obtained slightly different answers for both approaches due to rounding error.
c) What fraction of your salary must you save each month in order to meet your retirement needs?
You need to save enough each month, such that the PV of your total savings equals the PV of your required future purchases. Your savings are a growing annuity, where the first savings is 5,000(X), where X is the fraction of your salary you save, the first savings occurs on Aug 1, 2011, and the growth rate is 0.8% per month. There will be a total of n = (30*12) savings. Thus you need to solve for X in the following equation
← X = 2.54%
Again, you needed to include the term [pic] because it is a forward starting growing annuity, where the first cash flow is 13 periods from today.
Topic: Time value of money: Lamborghini gift purchase decision
2) After several years of successfully implementing strategies you learned in MGCR 341 you are considering purchasing a new Lamborghini for your old professor as a small token of your appreciation. You can obtain the car by making monthly payments of $20,000 a...