Understanding the Mathematics of Personal Finance

Solutions

16.1 CHAPTER 1

1.

(a) 7 - (12 - 5) = 7 - 7 = 0

(b) 12(14 - 6) = 12(8) = 96

16-(3 + 7) _ 16-10 _ 6 _ 1 C 3 (7 - 5) _ 3 (2) _ 6 _

(d) (12 - 2)(7 + 3) = (10)(10) = 100

(e) 12 - 2(7 + 3) = 12 - 2(10) = 12 - 20 = -8

(f) (12 - 2)7 + 3 = (10)7 + 3 = 70 + 3 = 73

(g) 6.2 + 1/3 = 6.2 + 0.333 = 6.533 ~ 6.53

2.

(a) x + y + z = 6 + 2 + 3 = 9

(b) z(x - 3)(y + 2) = 3(6 - 3)(2 + 2) = 6(3)(4) = 72

x + 2 6 + 2 8

(c) + 2.25 _ -2-3 + 2.25 _ ^ + 2.25 _ — + 2.25 _ 8 + 2.25 _ 10.25

z - 4 3 - 4 -1 -1

(d) x(x - 1)(x + 2) = 6(6 - 1)(6 + 2) = 6(5)(8) = 240

3.

(a) T1 = 0 P3 = 0 N4 = 4 T5 = 8

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

(b) At more than $20 per wallet, £ Ni = 6 + 2 + 2 + 4 = 14.

6 i=1

All together, £ Ni = 6 + 2 + 2 + 4 + 8 +16 = 38..

i=1

(c) Total cost of wallets sold is 38($12) = $456.

6

Total revenues is £ NiPi = 6 (25) + 2 (24) + 2 (23) + 4 (18) +16 (14) = 696.

i=1

(Total revenues) - (Cost of wallets sold) = $696 - $456 = $240. Subtracting overhead, $240 - $100 = $140. Money was made for the day. 4.

(b) The revenue for the second day was $939.60. The storekeeper’s cost for the wallets was 57($12) = $684.00. The total profit for the day, including over­head, was then $939.60 - $684.00 - $100.00 = $155.60.

5. See Figure 16.1. [33] [34]

6.

7.

A number raised to the 0th power is always equal to 1.

A number raised to the 1st power is always equal to the number itself.

7. $13, $22, $54, and $1,720

8. See Figure 16.3.

The average speed is the total distance divided by the total time: 3.0/2.0 = 1.5mph. This walk was taken at two steady speeds (two straight line segments on the graph).

For the first hour, 1mi was traveled. The instantaneous speed any time in the first hour was therefore 1/1 = 1mph. For the second hour, (3 - 1) = 2mi was traveled. The instantaneous speed any time in the second hour was therefore 2/1 = 2 mph.1

16.2 CHAPTER 2

1.

(a) Interest = 10%($6,700.00) = 0.1($6,700.00) = $670.00

(b) Interest = 3(0.06)($500.00) = $90.00

7

(c) Interest = —(0.08)($1,000) = $46.67

(d) Interest = $15,600.00 - $12,000.00 = $3,600.00 Interest per year = $3,600.00/3 = $1,200.00

Interest rate = $1,200.00/12,000.00 = 12/120 = 1/10 = 0.1 = 10%

1. 18.00%, 9.00%, 6.00%, and 4.50%

2. By hand (or pocket calculator):

After 1 year, $1,250.00 + (0.075)1,250.00 = 1.075(1,250.00) = $1,343.75. After 2 years, 1.075(1,343.75) = $1,444.53.

After 3 years, 1.075(1,444.53) = $1,552.87.

Alternate approach (by pocket calculator):

After 1 year, ($1,250X1.075)' = $1,343.75.

After 2 years, ($1,250)(1.075)2 = $1,444.53.

After 3 years, ($1,250)(1.075)3 = $1,552.87.

1 At exactly 1 hour, the slope of the graph changes abruptly. Rigorously speaking, we cannot define an instantaneous speed at exactly this time. In the terminology of differential calculus, the derivative is discontinuous at Time = 1.00 and does not have a unique value.

The balance at the end of 18 months has increased from $11,186.81 to $11,466.48. Going back to the spreadsheet, I will return Init (the up-front cost) to 0 and then tweak the Rate until I again get this new balance (or as close as I can by changing the interest rate to two decimal places):

7.

16.3 CHAPTER 3

1.

(a) $843.86

(b) $2,048.18

(c) $332.14

(d) Look at solution 1a above. This is the same loan except for the principal. Therefore, scaling will work. The amount $250,000 is 2.5($100,000), so the payment is 2.5($843.86) = $2,109.65.

1. 2005 $2,482.72

2010 $4,601.21

2015 $2,673.79

2020 $116.39

2. After payment number 110, in September of 2014.

3. May of 2013 is payment number 94. Replacing the regular payment of $843.86 with $10,000, the regular payment for the rest of the loan falls to $712.60:

4. This requires a little planning. For the amortization table from problem 2, note that the balance after payment number 3 is $98,963.26, after payment number 6 is $97,910.10, and so on. The first month’s interest is $500.00, so change payment number 1 to $500.00. Again, everything readjusts and the second month’s interest is now $500. Set the second month’s payment to $500.

Now “tinker with” the third payment until the balance is again $98,963.26. You’ll find that the third payment must be $1,536.74. Continuing, at the fourth payment, the interest is $494.82, so set the fourth payment to $494.82 and so on: [35]

Figure 16.4

and then adjust the principal until the payment gets very close to $1,000. This problem illustrates the importance of getting as low an interest rate as you can to maximize your borrowing power for a fixed payment amount.

See Figure 16.4.

7. The easiest way to look at this is to say that you’re paying the $250 up-front costs and therefore only putting $4,750 down on the car. The principal on your loan is therefore $31,800 - $4,750 = $27,050.

16.4 CHAPTER 4

1. Three points on a $325,000 loan is 3%($325,000) = $9,750. The total up­front cost is $9,750 + $450 = $10,200. The monthly payment on a loan of
$325,000 + $10,200 = $335,200 at 5.8% is $2,362.96. The interest rate on $325,000 to match this payment, to the nearest 0.01%, is 6.18%, yielding a monthly payment of $2,362.28. The effective interest rate is therefore 6.18%.

2. The payments are exactly the first interest calculation: $1,673.75. Since these are interest-only payments, your balance never changes. It remains at $325,000.00. At the end of 3 years (the thirty-seventh payment), your new regular monthly payment is $2,577.66.

3. The balance after 60 months of the mortgage of the previous problem is $301,971.35. A 5% mortgage with a principal of $390,580.00 and a 20-year payment period has a monthly payment of $2,577.66. This is therefore the desired new mortgage. The amount pulled out in cash at refinancing is $390,580 - $301,971 = $88,609 (to the nearest dollar).

4.

(a) The first mortgage is for 80% of $425,000 = $340,000. The regular monthly payment is $1,825.19. After 7 years, the outstanding balance is $299,013.05.

(b) The 3% a year appreciation is a compounding situation with 3% annual compounding. After 7 years, therefore, the house is appraised at $425,000 (1 + 0.03)7 = $522,696.40. (You can do this calculation on a pocket calculator or a blank spreadsheet, or by treating it as a compound interest problem using any compound interest calculator.)

(c) Eighty percent of the new house value is 80%(522.696.40) = $418,157.10. The second mortgage is a 30 - 7 = 23-year mortgage.

(d) The second mortgage principal is the difference between 80% of the new value and the balance of the first loan, which is $418,157.10 - $299,013.05 = $119,144.05. A 23-year mortgage at 6.2% annual percentage rate (APR) has monthly payments of $811.20.

(e) The total monthly payment is $1,825.19 + $911.20 = $2,736.39.

16.5 CHAPTER 5

1.

(a) Using my spreadsheet, the worst case penalty occurs after monthly payment 21. It is $93.58.

(b) It would cost the payoff amount of $17,303.98 plus the monthly payment of $502.14 plus the penalty of $93.58 = $17,899.69.

(c) Before: $93.58/($502.14 + $17,303.98) = 0.53%.

After: $93.58/17,303.98 = 0.54%.

(d) After payment 21, the remaining balance is $17,303.98. One month’s interest on this balance is $112.06, so 3 month ’ - interest is $336.18. Compared with the rule of 78 penalty, this is 336.18/93.58 = 3.59, which is 259% larger.

1. The strategy here will be to consider all the different options and compare the savings account balance at a time when all of the options have the loan fully paid off, that is, at the end of 60 months.

First, consider the case where there is no prepayment penalty. If the loan is paid off after, say, the third payment, the savings account balance at that time is calculated by considering a $100,000, 7.00% APR account, which is compounded monthly and has $502.14 withdrawn monthly:

At that time, the loan balance is $23,962.04. After paying off the loan, the savings account balance is $100,245.00 - $23,962.04 = $76,282.96. This balance then earns 7.00% interest for the remaining 57 months:

($76,282.96)(1 + 0.07/12)57 = $106,270.06.

Showing the results for a few of these calculations:

The last row of the table, payoff month 60, is identically the case when the loan is not prepaid.

The best case is to pay the loan off as quickly as possible—each month that you delay leaves you with a little less money at the end of the 5-year period.

If you must pay a prepayment penalty, then your only option from the above list is to never take the loan (payoff month = 0) in the first place or, having taken the loan, not to pay it off early. Since the former case is better for you, the bench­mark of $106,321.89 is what you’re trying to beat.

The following is generated in the same manner as the previous table. The only difference is that in each case, the loan balance is augmented by the prepay­ment penalty when the loan is paid off.

Regardless of whether you’re dealing with the rule of 78 or the 3 months’ interest on the remaining balance prepayment penalties, once you’ve signed the loan, you want to pay off the loan as quickly as you can get down to the bank.

2. In this case, I don’t have to show any calculations. If I’m earning more than my loan is costing, I just make my loan payment every month. There is no point to paying off the loan. While I have the loan, I still have the remaining loan balance earning 8.0% while I’m paying 7.6% on the loan. Prepayment penalties never come into the picture because I have no intention of paying off the loan early.

This situation can be generalized: If you can earn a higher interest rate on borrowed money than you are paying on this borrowed money, borrow as much as you can and use it to start earning. Whether you pay off your loan month by month or all at the end of some period, you are making money every month.

16.6 CHAPTER 6

1.

(a) $150.00

(b) In the month of your purchase, your daily balance every day equals your average daily balance, which is $150.00. Your interest for the month is then ($150)(0.000333)(30) = $1.50. Your statement bill is for $150.00 if you pay before the due date, but since you don’t, your new daily balance on the first day of the next billing period is $151.50. On the second day, your payment is credited and your daily balance drops to $51.50.

(c) Your daily balance will remain at $51.50 until the end of this billing period. Your average daily balance at that time will be

The interest on this average daily balance is ($54.73)(0.000333)(31) = $0.57, and your purchase balance is then $51.50 + $0.57 = $52.07.

2. I’m assuming that there are no other balances on your card and that you’re not using your card for anything else.

Starting with the first month that you do this, from the first through the nineteenth of this month, your daily balance is 0. From the twentieth to the thirty-first of this month your daily balance is $250.00. Your average daily balance is

19 ($0.00) +12 ($250.00)
31

and your interest is $96.77(0.0005)(31) = $1.50 . Your bill for the first month is $250.00 + $1.50 = $251.50.

For the first 2 days of the second month, your daily balance is $251.50. On the third day, your payment for this amount reaches the bank and your daily balance drops to 0, where it remains until the twentieth of the month when it jumps to $250.00 and stays there for the rest of the month. Your average daily balance for the second month is then

Your interest for the second month is $113.00(0.0005)(31) = $1.75, and your bill for the second month is $250.00 + $1.75 = $251.75. This can be continued for as many months as you wish.

3. Starting on the first day of the month, you have a purchase daily balance (PDB) of $150.00 and a cash advance daily balance (CDB) of 0. On the twentieth of the month, the CDB jumps to $250.00. There are no other changes in the daily bal­ances for the rest of the first month.

On the last day of the first month, the average daily balances are

Purchases (PADB) = 31= $150.00

and

Cash advance (CADB) = 19(I0 00)+J2<250 00) = $96.77.

Your finance charges (interest) are the following:

Purchase interest = $150.00(0.0003)(31) = $1.40.

Cash advance interest = $96.77(0.0006)(31) = $1.80.

Your end of first month balances are the following:

Purchases: $150.00 + $1.40 = $151.40.

Cash advance: $250 + $1.80 = $251.80.

Total: $151.40 + $251.80 + $403.21. This is the amount on your statement.

For the first 2 days of the second month, PDB is equal to $151.40 and CDB is equal to $251.80. On the third day of the second month, your payment of $50
of your first month’s statement, which is 50%($403.21) = $201.61, arrives. This amount is first applied to your PDB, reducing it to 0, and then the remainder of your payment, $206.61 - $151.40 = $55.21, is applied to your CDB, reducing it to $251.80 - $55.21 = $196.59. These daily balances remain the same until the end of the second month.

The average daily balances for the second month are

PADB = 2 ($15140)+ 29 (0) = $9.77
31

and

CADB = 2 ($25L80) + 29 ($196.59)= $193.67.

31

The finance charges for the second month are

Purchase interest = $9.77(0.0003)(31) = $0.09 and Cash advance interest = $193.67(0.0006)(31) = $3.60, and for the end of the month balances are the following:

Purchases: $0.00 + $0.09 = $0.09.

Cash advance: $196.59 + $3.60 = $200.19.

Total: $200.38. This is the amount on your statement.

16.7 CHAPTER 7

1. $22,000

2. The present value of 24 monthly payments of $100 at different APRs is

If I can save or invest my money at greater than approximately 4%, it’s a good deal for me. In this problem, practical considerations might outweigh numerical results. If I can only get a 2% APR on my savings, then I am “over­paying” about $55 on a $2,300 purchase. This is annoying but probably won’t change my life. If this purchase was for, say, a recreational vehicle costing $230,000 and the monthly payment was $10,000, I would probably be more concerned about the present value.

16.8 CHAPTER 8

1. Generate this information using the spreadsheet Ch8LoanPV. xls. Set an interest rate; set the savings rate (PV Rate on the spreadsheet) to one-half that value; and tweak the principal until the Total PV is about $400,000:

2.

From the given data, it’s pretty clear that the ratio of principal to payment goes down as the loan APR goes up. In other words, even though all of the loans in the example have the same present value, you get a lot more loan for your dollar payment when the interest rate is low than when it is high.

3.

Discussion: Using these data, the lowest figure of merit (FM) is 98. This is for the loan with the lowest payment of the group. The loan with the lowest PV is at the top of the table, with an FM of 153. The arguments made in this chapter for preferring loans with low present values are solid. On the other hand, multi­plying together two numbers that we want to minimize and looking at the product are arbitrary procedures with no real merit. The loan with this FM of 98 has very low payments because there are many of them. It’s still an expensive loan.

3. Eight years into the mortgage is the ninety - sixth payment. At that time, your balance is $195,825.35 (I’m using the ARM tab of Ch8LoanPV. xls). If a lender is offering you a 4.2% mortgage at that time, you can assume that savings interest rates have also dropped and are now at about 2.1%. Your two possible scenarios are the following:

(a)

The new loan is a slightly better deal (PV of $215,377 as compared with $216,036). The new payments are lower, but you’re paying for an extra year.

4. Before going to the spreadsheet, you have to decide what the savings APR should be. It doesn’- make sense to estimate it to be one-half of the loan APR when
you’re looking at three different loan APRs. Just to cover our bets, I’ll set it up for a few different savings APRs:

Regardless of the savings APR chosen, loan C has a higher present value than the sum of the APRs of loans A and B. The differences are extreme enough that up-front cost considerations won’t change the conclusion—take loans A and B.

16.9 CHAPTER 9

1. For each of you:

Single: $4,481.25 + 0.25($50,000 - $32,550) = $8,843.75.

Married filing separately: same.

For both of you, with a total taxable income of $100,000:

Married filing jointly: $8,962.50 + 0.25($100,000 - $65,100) = $17,687.50, which is exactly twice $8,843.75. It would appear that there is no advantage to either situation.

2. Married filing jointly, the taxable income is $100,000 and, as above, the tax is $17,687.50.

Single: $10,000 income is $802.50 + 0.1($10,000 - $8,025) = $1,000.00.

$90,000 income is $16,056.25 + 0.28($90,000 - $78,850) = $19,178.25. Total tax: $1,000 + $19,178.25 = $20,178.25.

Married filing separately:

$10,000 income is $802.50 + 0.15($10,000 - $8,025) = $1,098.75. $90,000 income is $12,775 + 0.28($90,000 - $65,725) = $19,572.00. Total tax: $20,670.25.

3. I’ll start with the loan information. I’m not interested in the payments right now, but in the interest totals for the year:

Year 6.0%

15 years 20 years

Also, the total present value of these loans is $427,681 and $470,522 for the 6.0% and 6.5% loans, respectively.

Next, I need to calculate your tax bill for the next 20 years with the interest deductions in the two scenarios. Then, I’ll need the present value of your bills:

The total present value of all of your tax bills is lower for the 6.5% loan. However, it is not lower enough to outweigh the significantly lower present value of the 6.0% loan. Being able to deduct interest payments from your tax bill closes the gap between the present values of the two loans but not enough; the 6% loan is still the better deal.

4. I ’ ll simplify a little and just do end-of-year accountings. In the first year, the $500,000 savings account grows by 5% to $525,000 and then loses $40,000 to withdrawals, leaving $485,000. This process repeats each year, with the cost of living growing every year while the balance falls every year until you run out of money in the fifteenth year:

5. An amount of $100,000 taxable income is pretty close to the middle of the 25% tax bracket for married couples filing jointly. Since your interest is going to be in the several thousand dollar range, you will remain in this bracket. This means that your incremental tax rate is 25%; that is, 25% of all taxable savings or investment income goes to the IRS. Mathematically, you are keeping 100% - 25% = 75% = 3/4 = 0.75 of your taxable incremental income. This means that when you have a target number for what you want to keep, you have to earn 1/0.75 = 1.33 times that amount.

If inflation is running at 2.5% then a tax-free investment must earn 4.5% in order for you to have 2% actual growth. A taxed investment or savings must earn 1.33(4.5%) = 6.0%.

16.10 CHAPTER 10

Since these are two separate policies, the cost of each is just the amount of the policy times the probability of death during that year (q). The present value of the first policy is just the cost 0.5 year earlier (on the man’s fiftieth birthday). The present value of the second policy is just the cost 1.5 years earlier. The total present value is just the sum of the two individual present values, which is $11,309.

(a) The first year of this policy is identical to the 1-year policy above. However, since this is being written as a 2-year policy with the premium collected up front (on the man’s fiftieth birthday), the second year of the policy costs a little less than it did above because we must account for the fact that the man might die during his fiftieth year:

In this table, we start with 100,000 people so that 150 = 100,000. The number of people that die that year is just l50q50 = 565. The cost of a $1,000,000 policy is then (565)$1,000,000/100,000 = $5,648, which is iden­tical to the cost in solution 1a. Continuing, l51 = l50 - d50 = 99,435 and then d51 = l5Jq5i = 609, leading to a cost of $6,087. The present value of the second year ’s cost is $5,739.

2. In this situation, the person’s age doesn’t matter; we just have year 1 and year 2. Since the probability of living through year 1 is 2/3, the probability of dying during year 1 is q1 = 0.3333333. Assuming the person survives to the second year, death is a certainty during this second year, so q2 = 1. The 2-year Life Table is then

As with any life policy that is in effect until the person dies, the sum of the costs must be the value of the policy. The present values are of course lower because of the time value of money—the insurance company is getting the money up front, before it has to pay out.

3. This problem can be solved in either of two ways. First, we use Table 10.6 exactly as we used the original Life Tables. Remember that in this situation, the first present value is one-fourth reflected one-fourth year back and so on:

The total cost is the sum of the present values, which is $374.83 An alternate approach is to use the first line of the table above for the half year policy from 48.5 to 49.0 years back and then to use the original Life Table for a single-year term policy bought at age 48.5 for the year 49.0-50.0, noting that we are not starting with 100,000 people but with 99,752. In this case, the present value of this latter policy is the cost reflected back one full year:

The total of the present values is $249.69 + $122.64 = $372.33.

The slight discrepancy in the two answers is analogous to the difference between APR and effective annual percentage rate (EAPR).

4. In the following table, for each group of 5 years, the first l value is the number of people alive at the start of that year. The d values for the 5-year group are the number of people who die during that 5 years. The sum of these five d values, divided by the first l value, is the probability of dying during those 5 years, assuming you reached the first of those 5 years alive:

5.

(a) The monthly payment is $5,311.76. At the beginning of each year, the bal­ances are

This table starts with the age, then the q values, for men age 35-39. 0 starts at 100,000 (l35 = 100,000) and then d35 = q35l35 = 165; l36=d35 -135 = 99,835; d36 = q36l36; and so on.

Since I started with 100,000 people, the cost of a $100,000 for any age (i) is just d($100,000)/100,000 = d. To get the cost of any other size policy, say, a $250,000 policy, just scale the numbers: cost for $250,000 policy at age 35 = 165($250,000)/100,000 = $413.16.

The present values are the values at the start of the policy, so the age 35 PV is the age 35 cost half a year earlier, which is $413.16/(1.05)05 and so on.

The total cost of the policy is the sum of all the present values, which is $1,333.46, and the insurance company price will be 35% large, which is (1.35)($1,333.46) = $1,800.17.

(b) Following the procedures of earlier chapters, folding this cost into the loan raises the payments to $5,350.01, which is equivalent to an effective interest rate on the original loan of 10.31%.

16.11 CHAPTER 11

1. From Table 10.1, the 2004 U. S. Life Table for all men, your life expectancy is 10.7 years. From the Life tab of the Ch11fixedannuities. xls spreadsheet, your annuity will pay about $3,950 each month. Your exclusion ratio, up until (and if) you pass your expected date of demise, is 0.690, which means that 31% of your annuity income is taxable. This amounts to 0.31(12)(3,950) = $14,700.

The $150,000 in your savings bank pays 0.05($150,000) = $7,500 in income.

Your total income for tax purposes is therefore $35,000 + $14,700 + $7,500 = $57,200. Your taxable income is $47,200. From Figure 9.1, Schedule X (single filing status), your federal tax is $4,481.25 + 0.25($57,200 - $32,550) = $10,640.

Your after-tax income is therefore $35,000 + $47,400 + $7,500 - $10,640 = $79,260 a year or $6,600 a month.

If you live longer than your expected death date, the entire annuity income becomes taxable: 12(3,950) = $47,400. Your total income, now all taxable, is $35,000 + $47,400 + $7,500 = $89,900 and your taxable income is $79,900.

From the tax table, your tax is $16,056.25 + 0.28($79,900 - $78,850) = $16,350. This leaves an after-tax income of $89,900 - $16,350 = $73,550 or $6,130 a month.

Note that while I used the Life tab of the spreadsheet to calculate the annuity properties, I didn’t use its tax calculations. The spreadsheet can only calculate the incremental tax in a given bracket due to the annuity. It doesn’t know your whole story.

2. From Table 11.4, for a 75-year-old woman and a 70-year-old man, the expected first death is in 8.5 years, the expected second death is in 16.9 years, or 8.4 years after the first death. Going to the Life tab on Ch11fixedannuities. xls, first note that the value put in for Age doesn’t matter here—tt’s just used to predict the expected age at death and doesn’t affect the premium calculation. Also, we’re not interested in the tax calculations:

The premium is approximately $200,400 for the first part of this package. We need a second annuity 8.5 years from now, for $2,000 a month with an expected term of 8.4 years. Using the same spreadsheet, the premium on this annuity is approximately $158,800. However, since we are paying the total premium today, this part has 8.5 years to grow to this value, at 6% APR. The present value is therefore $158,800/(1.06)6, approximately $111,900.

Adding the two amounts that are to be spent today, $200,400 + $111,900 = $312,300.

3. From the Life Tables, an 82-year-old man has a life expectancy of about another 7 years. For all reasonable calculations, 18 years is the longest life expectancy to worry about. Using the Save tab of the Ch3Amortization. xls spreadsheet, think of having your daughter advance you regular monthly payments as if you were a savings bank. The interest rate should be the rate she’s giving up by taking the money from her savings or investments. If you take $500 a month at 6% interest, then after 7 years, the balance is about $52,000. After 18 years, the balance is about $177,000. At the time of your death, your daughter can repay herself from your life insurance policy and still have an inheritance. At 10% interest, after 18 years, the balance is about $266,000, slightly more than the value of your life insurance policy. However, if the interest rate when you start out is down around 5% and slowly drifts up to 10% over the 18-year period, then it’s unlikely that the balance will exceed $250,000.

4. The first thing we have to do here is to estimate how many payments you will receive. Unless you know of some particular health or hereditary issues, the best you can do is to go to the Life Tables. Using the woman tables, we have

Since the Life Table expresses ages in tenths of a year, I assumed that social security payments come 10 times a year rather than monthly. This doesn’t affect the results here because we’re only interested in comparing the numbers, not the actual numbers. I could elect to receive $0.75 per payment starting when I’m 62 and expect to receive 224 of such payments, or to receive $1.00 starting when I’m 66 and expect to receive 192 payments, or $1.32 when I’m 70 and expect to receive 162 payments.

In this table, I’m calculating the present value of every payment on my sixty - second birthday, using an APR of 4.5%, compounded 10 times a year:

In the column for the payments shown starting at age 66, at age 66 is the present value of $1.00 calculated at age 62. Similarly, in the column for the pay­ments starting at age 70, at age 70 is the payment of the present value of $1.32 calculated at age 62. The sums of each of the three columns are $106.45, $108.11, and $106.78.

According to this calculation, it hardly matters at all which choice you make. However, I just guessed at an APR and I ignored inflation.

A more interesting calculation is to assume a rate of inflation, an interest rate that stays about 2% above inflation, and a social security payment cost of living adjustment that tracks with inflation. Without showing the table here, for an infla­tion rate of 2.5% and an interest rate of 4.5%, the present values are (rounded a bit) $136 for starting at age 62, $148 for starting at age 66, and $157 for starting at age 70.

If we assume that the social security cost of living adjustment won’t really keep up with inflation but will only account for about half the inflation rate, then the numbers become $120, $126, and $129, respectively. From these numbers you would conclude that you’re better off waiting until age 70 to start collecting your social security, but this conclusion isn’t compelling.

Before making a decision based on the above numbers, I recommend that you get the latest Life Tables that are appropriate for you and factor in any rel­evant family history and personal health factors that you are aware of.

16.12 CHAPTER 12

1. The present value of $1 10 years from now, at 7% interest (compounded annu­ally), is about $0.50. In other words, for every dollar you loan to your customer today, he or she will owe $2 in 10 years. If you loan him or her half the appraised value of his or her house, and housing prices drop in half over the next 10 years, there will still be enough money to pay back the loan by selling the house. This means that the loan can be for $225,000. Since you are charging an up-front fee of $25,000, the customer can actually receive a check for $200,000.

What if the customer lives longer than age 87? This is where some historical perspective is useful. Home values have historically climbed. If home values climb by an average of 2% for 20 years and then suddenly plummet to two-third of their value, the home value returns to about today’s price. If the customer lives to age 97, his or her outstanding loan balance will be about $450,000 on a $225,000 loan, and there is still enough value in the house to cover the loan.

A real business plan would need a deeper look at this, with real statistics covering many clients and various scenarios of housing values, but for a first pass, these numbers give a feeling of how well the loan is secured by the home.

Now, what about your company’s cash flow? You had to borrow $225,000 today at 5% APR. You’d also like to make some profits over the life of the loan.

This table is very approximate but shows the business plan. On the first year of the loan, you borrow an additional $8,000, which you contribute to company profits (perhaps given out as dividends?). You are paying 5% on this money. At the end of 10 years, you owe about $450,000—the due amount of the loan. At the end of 20 years, you owe only about $786,000, much less than the due amount of the loan. It looks like you have about a $114,000 buffer here to further protect your company against unanticipated dips in home values.

2. Over the course of these years, it’s pretty certain that interest rates will vary. This means that you might have to pay more on your loan waiting while waiting for your customer’s loan to be repaid. You handle this by writing the original loan as an adjustable rate loan; when the rate you have to pay goes up, so does the rate your customer has to pay (and vice versa).

16.13 CHAPTER 13

1. The first thing to do is to normalize the prices of all the stocks, that is, divide each stock price by that same stock’s price at the beginning of the year. In the table below, I created three idealized variations of ±5%, ±10%, and ±20% for the year:

Then the table shows what a $1 purchase will buy in terms of shares of each stock for each month. At the bottom of the table is the average amount of stock I got over the year for each dollar spent. Stock C is clearly the winner here. Also, since the average of a list of different numbers cannot be as large as the largest number in the list, spending all of my money on stock C is both better than spending it on any other stock and better than spending it on partial investments in each stock. Keep in mind that this idealized example ignores volatility and the advantages of diversification.

2. The correlation coefficient is 0.770. These stocks are moderately well correlated. The slope of the best-fit line is positive, indicating that these stocks tend to move together from day to day—perhaps, they’re stocks for the same or related indus­tries. The normalized standard deviation for stock #2 is almost twice that of stock #1, indicating that stock #2 is more volatile than stock #1. Both numbers are rela­tively small, however, indicating a relatively low level of volatility.

When the stock #1 numbers are replaced with the numbers 1-25, all informa­tion about stock #1 is lost. What we have left is a list of stock #2 prices versus the (relative) date when the number was collected. The slope of the best-fit line is a small positive number, indicating that this stock’s price has, on the average, climbed over the time period represented. The correlation coefficient is very high, indicating that the stock’s growth over this time period grew almost as a straight line graph. All other information generated is spurious.

3. Since you were given the options, your cost is 0. There is no reason to exercise these options unless the stock price is higher than the strike price. Figure 16.5 shows that your profit will be $1 per share for every $1 the stock price is higher than the strike price, and 0 otherwise. Exactly when you exercise the option is of course a guessing game. You’d like to wait until the stock price is at its highest, but you don’t know when that will be, and at some time, the options will expire.

4. Figure 16.6 shows your strategy for this situation. You can only make a profit when the stock price is above $14, unlike in problem 1. However, if you’ re

convinced that the stock price will never exceed $14 before your options expire, then you’ll still exercise the option when the stock price is above $13. In the stock price region between $13 and $14, you will lose money, but you’ll lose less than $1 per share, so you’re at least minimizing your losses. If the stock price never gets above $13, then you freeze your losses at $1 per share by letting the option expire.

5. If the stock price never gets above $13, then nobody exercises their calls. You get to keep your $1 per share. If the stock price gets above $18, then you can expect your buyer to exercise. You exercise your calls, buy the stock at $13 a share, and sell it to your buyer at $18 a share. You earn $5 a share ($18 - $13) plus the $1 you already collected, making it $6.

For the stock price between $13 and $18, different things could happen. Your buyer won’t exercise his or her shares. If the stock price reaches, say, $15 a few hours before everything expires, you could take a chance: Exercise your options and sell the stock. You’re betting that the stock won’t cross $18 in the short time remaining. You earn the difference between the stock price and $13 plus the $1 you already have. If you bet wrong and the stock price shoots up past $18, you have to buy back the stock and deliver it to your call buyer. If, for example, the stock suddenly shoots up to $20, you have to buy back the stock that you sold for $15. Your loss is $5 a share minus the $1 you already collected, or $4 a share. If you’re not a gambler, you just sit back and wait. When both sets of options expire, you still have the $1 you’ve already collected.

6. Figure 16.7 shows what you’d do with the call alone, what you’d do with the put alone, and with the combination of these two, which is simply a sum of the call and put graphs. For a stock price above $90, you would exercise the call and ignore the put. For a stock price below $85, you would exercise the put and ignore the call. You only make money if the stock price falls below $82 or climbs above $94 dollars. With this strategy, you are hoping for high volatility in the stock price but you don’t really care which way the stock moves. If the stock stays between $82 and $94, you lose money, but the amount you lose can never be more than the sum of the put and call costs.

16.14 CHAPTER 14

The problems for Chapter 14 are just for fun. There’s nothing to learn about games of chance other than that you can ’ t beat the house. Games like poker combine chance, actual skill at playing hands, and psychological effects such as bluffing; they’re very hard to quantify.

1. Figure 16.8 shows six runs of this spreadsheet “program” starting at +1 (or with $1 to play). If you were actually playing for money, as soon as you crossed below 0, you’d be wiped out. With a random walk, you just keep bouncing around.

If you want an idea of what these graphs would be like starting at +5, or +10, or any number whatsoever, you don’t have to recreate sample runs. All you

have to do is slide the vertical axis up or down until you have what you want. Figure 16.9 shows these results. There are vertical axes for starting out at +1, +5, and +10. Note that in all cases, it’s not so unlikely that you get wiped out.

2.

(a) The state needs to sell 500 million tickets, so the probability of winning is 1/500 million. The probability of losing is 4,999,999/500,000,000, which is extremely close to one:

Another way of looking at this is that if you bought all 500 million lottery tickets, you would win the lottery and have lost $100 million (the state’s profit). Each ticket would therefore have lost $100million/$500 million = 20 cents.

(b) About 1 second in 16 years.

[1] Computer programming languages usually resolve this kind of ambiguity by having a default procedure such as “when there are no explicit instructions (parentheses), work from left to right.” I won’t assume any such default procedures in this book.

[2] You might ask why not be concerned about the car price change—it’s the same $2. As I see it, the difference is that you’ll have your new car about 5 or 8 years, but you buy a pound of coffee every few weeks.

[3] Єі = e5 + e6 + e7 + e8.

i=5

To finish the job, we have to look at the appropriate e values in the table:

8

Єі = e5 + e6 + e7 + e8 = 70.8 + 69.8 + 68.9 + 67.9.

i=5

I won’ t bother actually adding these numbers up because I don’t really care about the answer; I’m just showing how the notation works.

But why bother with such an esoteric notation just to show that I want to add up four numbers? The convenience is that I can represent huge sums of numbers concisely.

[4] The term “10% loan,” unless expressly explained otherwise, is synonymous with a “10% APR loan.”

[5] In this example, 8.333% is an approximation, because of a “repeating decimal.”

[6] When there is a possibility for confusion, the APR is sometimes called the NAPR, or nominal annual percentage rate, to emphasize the distinction between it and the EAPR.

[7] I’m assuming that there are no prepayment penalties associated with this loan. This isn’t always the case. How these penalties are often calculated will be shown in a later chapter.

[8] State tax laws vary from state to state, and both state and federal tax laws can and do change. You’ll have to find out yourself what interest is deductible in any given year and what the rules for calculating the deduction are.

[9] This is the same story as the above footnote. You’ll have to find out what interest is taxable as income.

[10] = Pr (1 + r) .

(1 + r )n -1

r, the interest payment per period, is always a positive number. Therefore, when we add 1 + r, we must always get a number greater than 1. n, the number of payments, is a positive integer such as 1, 2, 5, 20, or 240.

Since (1 + r)n just means (1 + r)(1 + r)(1 + r) ... n times and 1 + r is bigger than 1, we can always be sure that (1 + r) < (1 + r)2 < (1 + r)3, and so on. In other words, as n (the number of payments to pay off the loan) gets larger, so does (1 + r)n.

[11] Even if you learn nothing else from this book, please memorize this paragraph: No two loan

contracts are the same. In many cases, there are government regulations that guide the process, but

there is still often quite a bit of room for creativity on the part of the lender. Never sign a loan contract until you have read and thoroughly understand both everything that you’re agreeing to and all the implications of what you’re agreeing to.

[12] Tax brackets are discussed in Chapter 9. Very quickly for now, it’s the tax rate you’re paying on the top few dollars of your income and is not a bad estimate of how much you’ll get back from the IRS when you report deductible expenses.

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

[14] I should actually use the term “billing cycle” rather than month here. Although the period is 1-month long, it did not start at the beginning of the month.

[15] In this situation, all financial logic says to take the money from the savings account and pay off your credit card balance. My purpose here is to compare transferring your balance to not transferring your balance, so I’ll pass on this choice.

[16] On the other hand, there are some transfer offers that combine 0% interest on transfers and 0% interest on purchases for a year. Since there are some things that you must buy, for example, food for you and your family, this is essentially an offer of extra money for you—if you handle it responsibly.

[17] The daily interest numbers are 0.03011% for purchases and 0.06299% for cash advances.

2. At the end of the previous month, my purchase daily balance was $335.25, and my cash advance daily balance was $480.20.

3. For the previous month, my purchase average daily balance (ADB) was $253.77, and my cash advance ADB was $435.90. Don’t worry yet about how to calculate the ADB—you’ll see it done several times below.

4. The interest is calculated as the (ADB)(daily interest rate)(number of days in the month). For my purchases, this was

($253.77)(0.03011%)(30) = $2.29, and for my cash advances, it was

($176.50)(0.06299)(30) = $3.34.

Adding the interests to the end of the month daily balances, I got a total purchase balance of $337.54 and a total cash advance balance of $483.54, adding up to a total balance of $821.08. These are the numbers that appear on my statement at the end of the month.

I received my statement a few days later. I made my payment and it was received and credited by the bank on the twelfth day of the following month, well before the statement due date.

[18] didn’t create an online spreadsheet to do the work shown in these tables. Your credit card company does this for you and sends it to you every month.

The very busy interweaving of dates on which money comes into and goes out of your account every day is what necessitates the system of daily tallying of charges and payments. It’s complicated but not unnecessarily complicated.

[19] I’m being a little sloppy here. Actually, as your balance approaches $10,000, the minimum payment might get larger than $200. This is a small correction and doesn’t affect the point(s) that I want to make.

[20] Virtually, all modern spreadsheet programs can sort. Where you find the command and just how you use it, however, varies from spreadsheet to spreadsheet so I can’t give generic instructions. The Help function (usually the F1 key if it isn’t on the menu) should get you going.

[21] Your equity in your home is the resale value of the home minus all outstanding debts. In other words, if you were to sell your home and pay off your mortgage loan(s), your equity is the amount of cash left in your hand.

[22] You have a 15-year, $350,000 mortgage with a 5.00% APR. Eight years into the mortgage, you get the opportunity to refinance with a new 8-year loan for the remaining balance of your loan at 4.2%. The up-front cost for this refinancing is $3,000. Is this a good deal or not?

[23] I want to buy a new home and I need to borrow $500,000. Lender A has a very attractive APR of 5.10% on a fixed 30-year mortgage, but he or she will only lend me $350,000. Lender B will give me a second mortgage for the remaining $150,000, but he or she wants 7.20% for a fixed 30-year mortgage. Lender C will loan me the entire $500,000, but he or she wants 6.5% for a fixed 30-year mortgage. What should I do?

Your first instinct should be to treat this as a trick question. The best financial deal is to take Lender A’s loan for $350,000 and then take only the remaining $150,000 from lender C. I’m going to force you to do some work here: Lender C handles “jumbo” mort­gages only; $500,000 is his or her minimum mortgage—take it or leave it.

[24] I’m neglecting a phenomenon called “bracket creep.” Even though your salary might just be tracking inflation, if the tax laws aren’t updated, then you slowly move into higher tax brackets and your tax goes up. You are, in effect, getting poorer.

[25] The link http://www. cdc. gov/nchs/products/life-tables. htm will provide you with tables and a lot of information about how the tables are generated. The tables in this book reprinted from this document are courtesy of the National Center for Health Statistics, E. Arias, United States life tables, 2004. In National Vital Statistics Reports, vol. 56, no. 9 (Hyattsville, MD: National Center for Health Statistics, 2007).

[26] Relative probability means that the probability data are being presented without numbers on the vertical axis. My reason for doing this is that the numbers would not contribute to the discussion but would take a long time to explain. The intent is just to be able to compare different values of probability on the graph, that is, relative probability.

[27] Looking at the table as shown, the multiplication yields 747.5, which is properly rounded to 748.

This is because I chose to show the table to only six figures after the decimal point. Had I shown it to seven figures, you would have seen that the probability is really 0.0074748. The multiplication now yields 747.48 that, to the nearest full person, is 747. Careless rounding of numbers can lead to misleading results, and you could go as far as creating criminal schemes that might pass unnoticed by deliberately engaging in this type of activity.

[28] Again, I’m just showing rounded numbers here. The spreadsheet program is internally storing and using numbers to many more decimal places than shown. This creates some roundoff error in the results shown.

payments), not on the interest earned while the money is still in the account. This is called “tax-deferred growth.”

In the case of the savings account, calculating the tax was easy. I just added up the interest earned each month for a year and then multiplied the result by the tax rate. In the case of this annuity, however, how do I calculate the amount of each

[30] I could have linked the Deferred tab sheet to the IAWPC tab sheet and had it pick up input information automatically. I decided, however, that this would be an endless cause of problems if/when users start modifying these sheets.

[31] IRS. General rule for pensions and annuities. Publication 939. http://www. irs. gov/publications/p939/ ar02.html (accessed February 2009).

[32] Fitting of lines or curves to data is a fascinating topic that unfortunately is far outside the scope of this book. You can find references on the Web or in statistics texts under “fitting of curves to data” or “linear regression.” Most spreadsheet programs offer a best-fit line generation option, usually as part of the graph generation commands.

18

16

14

12

10

8

6

[34]

2

0

Figure 16.1

[35] Many of the calculators on the Web will let you solve this problem directly: entering what you know and getting what you want. My spreadsheet is set up to calculate the payment from the other variables. To solve this problem on my spreadsheet, you’ll have to enter the number of monthly payments and the rate,