《金融学(第二版)》讲义大纲及课后习题答案详解 第五章

CHAPTER 5
LIFE-CYCLE FINANCIAL PLANNING
Objectives
In this chapter you will learn how to analyze:
•How much to save for retirement.
•Whether to defer taxes or pay them now.
•Whether to get a professional degree.
•Whether to buy or rent an apartment.
•How to minimize estate taxes.
Outline
5.1 A Life-Cycle Model of Saving
5.2 Taking Account of Social Security
5.3 Deferring Taxes Through Voluntary Retirement Plans
5.4 Should You Invest in a Professional Degree?
5.5 Should You Buy or Rent?
Summary
•In making lifetime saving/consumption decisions:
(1) Do the analysis in real terms (constant dollars) to simplify the calculations and to avoid having to
forecast inflation.
(2) Start by computing the present value of your lifetime resources. The present value of your lifetime
spending cannot exceed this amount.
•Social security or any other forced saving program will offset voluntary saving. It may have a positive or a negative effect on the present value of your total lifetime resources.
•Tax-deferred retirement accounts are advantageous because they allow you to earn a before-tax rate of return until money is withdrawn from the account. They are advantageous if you are in the same tax bracket before and after you retire, and even more so if your tax bracket is lower after you retire. •Getting a professional degree can be evaluated as an investment in human capital. As such, it should be undertaken if the present value of the benefits (such as increase in your earnings) exceeds the present value of the costs (such as tuition and forgone salary.)
•In deciding whether to buy or rent an apartment or a consumer durable, choose the alternative with the lower present value of costs
Solutions to Problems at End of Chapter
Saving for Retirement
1. Assume that you are 40 years old and wish to retire at age 65. You expect to be able to average a 6% annual rate of interest on your savings over your lifetime (both prior to retirement and after retirement). You would like to save enough money to provide $8,000 per year beginning at age 66 in retirement income to supplement other sources (social security, pension plans, etc.). Suppose you decide that the extra income needs to be provided for only 15 years (up to age 80). Assume that your first contribution to the savings plan will take place one year from NOW.
a.How much must you save each year between now and retirement to achieve your goal?
b.If the rate of inflation turns out to be 6% per year between now and retirement, how much will
your first $8000 withdrawal be worth in terms of today’s purchasing power?
SOLUTION:
a. Age 40 41 65 66 80
Time 0 1 25 26 40
. . . .
X X 8,000 8,000
It is a 2 part computation. First compute the amount needed at age 65 to finance the $8,000 per year annuity
in terms of today’s purchasing power.
2. You are saving for retirement and you come across the following table. It shows the percentage of your current salary that you should save for your retirement in order to retire with an annuity equal to 70% of your salary if you have not yet saved anything. It assumes that your annual salary will remain constant in real terms until retirement, and that you will live for 25 years after retiring. For instance, if you have 35 years left before you retire and earn
3.5% per year on your investments, then you should save 17.3% of your current salary.
a. Fill in the missing number in Table A.
SOLUTION:
The method for computing how much saving is needed to reach the desired target (70%) consists of two steps:
First compute the amount you need to have accumulated in your personal retirement account when you reach the retirement age. (We’ll do the calculations as percentages of real salary)
Then compute the annual amount of saving needed to reach that future value
59.79% if you have 15 years to retire
29.62% if you have 25 years to retire
17.30% if you have 35 years to retire
b. Now fill in Table B. It assumes that instead of targeting a 70% replacement rate of preretirement income, your goal is to maintain the same level of consumption spending both before and after retirement.
First compute the amount you need to have accumulated in your personal retirement account when you reach the retirement age. (We’ll do the calculations as percentages of real salary)
Then compute the annual amount of saving needed to reach that future value
85.41% if you have 15 years to retire
42.31% if you have 25 years to retire
24.72% if you have 35 years to retire
3. You are saving for retirement and you come across the following the table. It shows the increase in the annual benefit you can receive in retirement per dollar that you increase your annual retirement saving in the years before retirement. It assumes that you will live for 20 years after retiring. For instance, if you have 30 years left before you retire and earn an interest rate of 3% per year, then you will obtain an increase of $3.20 in your annual retirement benefit for every $1 per year increase in annual saving. Fill in the missing table values.
4. George Thriftless is 45 years old, earns $50,000 per year, and expects that his future earnings will keep pace with inflation, but will not exceed inflation. He has not yet saved anything towards his retirement. His company does not offer any pension plan. George pays Social Security taxes equal to 7.5% of his salary, and he assumes that when he retires at age 65, he will receive $12,000 per year in inflation-adjusted Social Security benefits for the rest of his life. His life expectancy is age 8
5.
George buys a book on retirement planning that recommends saving enough so that when private savings and Social Security are combined, he can replace 80% of his preretirement salary. George buys a financial calculator and goes through the following calculations:
First, he computes the amount he will need to receive in each year of retirement to replace 80% of his salary: .8 x $50,000 = $40,000.
Since he expects to receive $12,000 per year in Social Security benefits, he calculates that he will have to provide the other $28,000 per year from his own retirement fund.
Using the 8% interest rate on long-term default-free bonds, George computes the amount he will need to have at age 65 as $274,908 (the present value of $28,000 for 20 years at 8% per year). Then he computes the amount he will have to save in each of the next 20 years to reach that future accumulation as $6,007 (the annual payment that will produce a future value of $274,908 at an interest rate of 8% per year). George feels pretty confident that he can save 12% of his salary (i.e., $6,007/$50,000) in order to insure a comfortable retirement.
a.If the expected long-term real interest rate is 3% per year, approximately what is the long-
term expected rate of inflation?
b.Has George correctly taken account of inflation in his calculations? If not, how would you
correct him?
c.How much should George save in each of the next 20 years (until age 65) if he wants to
maintain a constant level of consumption over the remaining 40 years of his life (from age 45 to age 85)?
Ignore income taxes.
SOLUTION:
a. The long-term expected rate of inflation can be approximated by subtracting the expected real rate of
interest (3% per year) from the long-term nominal interest rate (8% per year). The expected rate of inflation is therefore approximately 5% per year.
The exact value can be calculated using:
1+nom= (1+real)(1+inf)
hence, the inflation rate is 4.854% per year.
b. George has used the nominal interest rate to discount real cash flows. As a result, he has seriously
underestimated how much saving he must do to achieve an 80% replacement rate. The time line for this problem is:
Age 45 46 65 66 85
Time 0 1 20 21 40
. . . .
X X 28,000 28,000
Using the real rate of 3% per year in the calculations results in a needed accumulation at age 65 of $416,569.30:
The annual saving needed to achieve this accumulation at a 3% real rate is $15,502.92:
So instead of $6,007 per year, George must save $15,502.92 per year. This is 31% of his salary rather than 12%.
c. The time line for this problem is:
Age 45 46 65 66 85
Time 0 1 20 21 40
. . . .
Inflows 46,250 46,250 12,000 12,000
Outflows C C
If George wants to maintain a constant level of consumption both before and after retirement, he must find C , where C is the solution to the following equation:
∑∑∑===+=40140212010310001203125046031t t t
t t t .,.,.C Equation 1 says that the present value of consumption spending over the next 40 years equals the present value of labor income over the next 20 years (after paying Social Security taxes) plus the present value of Social Security benefits received after retirement.
First we find the value on the right side of the equation, the PV of George’s lifetime resources:
Step 4: Find PV of lifetime resources as of age 45:
688,083.21 + 98,847.56 = 786,930.78
So each year between the ages of 46 and 65 George must save $12,205.50, the difference between his income after SS tax ($46,250) and his consumption spending ($34,044.50).
Now let us check to make sure that by saving this amount George will indeed have enough to provide the same constant level of consumption spending after retirement as before.
Thus, by saving $12,205.50 per year for 20 years earning a real rate of 3% per year, he will have at age 65 an accumulation of $327,966.36.
Adding this annuity to the SS benefit we get:
$22,044.50 + $12,000 = $34,044.50
5. George’s company has a defined-benefit pension plan. Suppose the plan pays a benefit equal to 1% of final salary per year of service. George is 40 years old and has worked for the company for 15 years. His last year’s salary was $50,000 and is expected to remain so in real terms until retirement. The expected rate of inflation is 4%.
a.If normal retirement age is 65, the interest rate is 8%, and Geor ge’s life expectancy is 80, what is
the present value of his accrued pension benefit?
b.What effect should his pension benefit have on George’s planned saving assuming he has a 75%
target replacement rate?
SOLUTION:
a.George’s last year salary $50,000 has a r eal growth rate of 0%, hence it will keep up with inflation until
retirement but not beat it. At retirement, he would have worked for the company for a total of forty years. Hence his annual pension benefit will be equal to 1% x (50,000x 1.0425) x 40 = $53,316.73. We must first find the value of those annual payments in the year when he retires, then discount that back to today.
PV65 = $456,363.41
40
b.For a 75% replacement rate, George expects to have an annual income of 0.75 x 50,000 =$37,500 in
real terms after retirement. Since his pension benefit is providing him with part of his financial needs after retirement, he would only need to worry about the difference between his target income and what the pension is providing him, hence decreasing his planned savings before retirement.
6.Analyze the “expert’s” responses to the following questions:
Question: How early do you recommend people begin saving for retirement? Would it be too early for my 14-year-old to start saving?
Expert: It’s never too early.
Question: For a college student , what would you suggest for a savings plan?
Expert: I’d suggest deciding on a specific a mount to set aside each month, then making sure you do it, no matter the temptations not to.
SOLUTION:
a.Because of the time value of money, obviously, the earlier you start saving for retirement the more
value you’ll have for each dollar saved.
Suppose you save $1 at age 15 and you expect to retire at age 65, this dollar will be worth at 8% interest rate $46.9. Of course there’s a trade-off because you’ll also be postponing your spending (enjoying life) to your retirement.
b.Again, because of the time value of money, each dollar you save as a college student will earn you
more than a dollar saved later on in your life. That’s why, you must decide on a certain amount you can afford to save and stick with it.
Investing in Human Capital
7. You are 30 years old and are considering full-time study for an MBA degree. Tuition and other direct costs will be $15,000 per year for two years. In addition you will have to give up a job with a salary of $30,000 per year. Assume tuition is paid and salary received at the end of the year. By how much does your salary have to increase (in real terms) as a result of getting your MBA degree to justify the investment? Assume a real interest rate of 3% per year and ignore taxes. Also assume that the salary increase is a constant real amount that starts after you complete your degree (at the end of the year following graduation) and lasts until retirement at age 65.
SOLUTION:
Buy or Rent?
8. Suppose you currently rent an apartment and have an option to buy it for $200,000. Property taxes are $2,000 per year and are deductible for income tax purposes. Annual maintenance costs on the property are $1,500 per year and are not tax deductible. You expect property taxes and maintenance costs to increase at the rate of inflation. Your income tax rate is 40%, you can earn an after-tax real interest rate of 2% per year, and you plan to keep the apartment forever. What is the “break-even” annual rent such that you would buy it if the rent exceeds this amount? SOLUTION:
The after-tax annual outlay if you buy is:
$1,500 + .6 x $2,000 = $2,700
The PV of this is $2700/.02 = $135,000.
The PV of the costs of owning are $135,000 + $200,000 = $335,000.
The break-even rent is .02 x $335,000 = $6,700 per year.
Buy or lease a car
9. You have decided to acquire a new car which costs $30,000. You are considering whether to lease it for 3 years or to purchase it and finance the purchase with a 3-year installment loan. The lease requires no down payment and lasts for 3 years. Lease payments are $400 monthly starting immediately, whereas the installment loan will require monthly payments starting a month from now at an annual percentage rate (APR) of 8%.
a.If you expect the resale value of the car to be $20,000 3 years from now, should you buy or lease
it?
b.What is the break-even resale price of the car 3 years from now, such that you would be
indifferent between buying and leasing it?
SOLUTION:
a.We have to compare the NPVs of the two alternatives:
Since in this case, the car is costing me more, I should choose the first alternative of leasing the car.
b. In order to be indifferent among the two alternatives, their respective NPV must be equal, i.e. –12,850.
Hence, the PV of the resale price is 30,000-12,850=17,150
10. Analyze the following newspaper column:
“Many of us who started families late share a nightmare image: having to pay huge college bills just as we’re giving up paychecks and shouldering the ste ep costs of retirement. In fact, the double whammy might not be so bad, assuming the parents have prepared properly. On the plus side, older parents are likely to have enjoyed their best earnings years before the college costs begin, allowing them to put a side more than younger parents can. They’ve also had more years for investments to compound. In the ideal situation, older parents can avoid borrowing to meet college costs, taking the preferred route of earning interest on investments rather than paying i t on student loans.” (Excerpted from Jeff Brown’s Personal Finance column in the Philadelphia Inquirer, May 11, 1998.) SOLUTION:
As it is mentioned in the newspaper column, ideally older parents can avoid borrowing to meet college costs provided that they had prepared properly and started saving early in their lives.
However, one can argue that with no children in the household, parents have less incentive to save (and enjoy their younger years) and might in fact be faced with this nightmare image of having to pay huge college bills just as they are giving up paychecks.
11. Analyze the following newspaper column:
“What’s the best age for a person to start collecting Social Security benefits? According to conventional wisdom, retirement starts at age 65. I t’s true that full benefits don’t start until age 65, but 62 year olds can retire and collect 80% of their benefits.
Take the hypothetical cases of John and Mary, who have the same birthday and who are both slated to start drawing $1,000 a month in Social Security benefits at age 65. On his 62nd birthday, John decides to go ahead and start claiming his benefits of $800 a month (80% of $1000). Mary decides to wait until she’s 65, when she can claim the full $1000. Three years later, Mary turns 65 and begins receiving $1,000 a month from the Social Security Administration. John continues to receive $800 a month. But he has already been paid $28,800 while Mary received nothing.
Five years go by, with Mary drawing $1,000 a month and John $800 a month. At 70, John has received $76,800, compared to Mary’s $60,000. When they reach 77, Mary will pull ahead. So, it seems if a person doesn’t live past 76, it would better to start collecting Social Security benefits at 62. For those who reach their upper 70’s, it pays to wait until they are 65 to collect Social Security. (adapted from 1998, Atlanta Business Chronicle, Gary Summer Contributing writer, June 29, 1998.) SOLUTION:
The analysis in this newspaper column ignores the time value of money. The best way to look at this situation is to assume an interest rate, say 5%.
The FV at age 65 of the $800 payments that John was taking is:
At age 65, when Mary decides to start receiving the benefits from SSA, John has been paid $31,003. Not $28,800 as mentioned in the article.
Now, in order to see when Mary will pull ahead, we have to see how many $200 payments (=1000-800) are the $31,003 worth.
After 250 month, i.e. at age 85 and 10 months, Mary will eventually pull ahead.
Personal Balance Sheets
12. Using the finance concepts presented in this chapter, construct a personal balance sheet showing your assets, liabilities and net worth.
a.Did you value your assets at cost or at current market value? Why?
b.Did you include your human capital as an asset? Why?
c.Did you include deferred taxes as a liability? Why?
a.I valued my assets at current market value because their cost is irrelevant to me.
b.Even though human capital is probably anyone’s biggest asset, I didn’t include it in the balance sheet
because it’s uncert ain, hard to quantify and I will need to make too many assumptions concerning the PV of my future earnings.
c.Once again, I haven’t.
Home ownership Over the Life Cycle
13. Suppose you buy a house for $200,000 when you are 35 years old. You make a 20% down payment and borrow the other 80% from a mortgage lender. The mortgage loan is at a fixed interest rate of 8% per year for 30 years and requires level annual payments. At age 65, you plan to take out a “reverse-mortgage” loan which will allow you to borrow a constant annual amount for the rest of your life to be paid off by the sale of your house when you die. Your life expectancy is age 85. The interest rate on both the original mortgage loan and the reverse mortgage will be 8% per year.
a.Suppose that you expect the inflation rate to be 3% per year and you can rent an equivalent
house for $10,000 per year. Is it worth buying the house?
b.Show how buying the house will affect your assets, liabilities and cash flow over the next 50 years.
c.In Making the Most of your money, JB Quinn has written: “ Over the long run, the value of
homes should follow the inflation rate. But over the time that you won your particular house, its value might rise or fall or stall. You can’t predict. But there are reasons other than profit for owning a house.
•Mortgage payments force to save, while rental payments don’t.
•You get tax deductions, and can tax-shelter your capital gains.
•You’re landlord free.
•You know the deep contentment of holding a spot of ground that others can enter by invitation only.
•You won’t lose your lease.
•You can renovate to suit.
• A house is collateral for a loan.
Comment.
SOLUTION:
a.In order to compare the two alternatives, we have to compare their NPVs.
Buy Alternative:
Basically, this alternative consists of buying the house now at $200,000 and selling it 50 years from now at its market value, accounting for inflation:
The FV of the house is:
The house will be worth : $876,781 when you are 85.
Now, to calculate the NPV of this alternative, we have to discount at 8% to account for the mortgage and the reverse mortgage.
Hence the NPV of this option is: -200,000 + 18,694 = -181,306
Rent Alternative:
We assume that rent will be $10,000 in real terms and hence must be discounted at the real interest rate = 4.854%
Hence, it is more economical to buy the house.
c. Most of the points mentioned by JB Quinn were discussed in this exercise, namely the fact that
mortgage payments force you to save and that a house can serve as collateral for a loan. She also presents in his discussion some “intangible” benefits from owning a house such as not having a landlord and renovating to suit.。