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Math Lecture Notes
Investing

In-Class Activity link to here link to index

The Rule of 72

Doubling While Increasing

investment doubling chart

We have already discussed what interest is. We learned how to use the formulas for simple interest and compound interest.

Those formulas allowed us to find precise amounts of interest. Great!

However, in real life we sometimes want a quick and easy estimate about how much an investment might help us prepare for the future.

For the moment, let's use the compound interest formula to watch an investment grow. We can look for a pattern.

1. Imagine that you invest $1,000 in an account that earns 10% interest every year, compounded annually. How many years will it take to double your money?

1. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.11 = $1,100

After 2 Years = $1,000 × 1.12 = $1,210

After 3 Years = $1,000 × 1.13 = $1,331

After 4 Years = $1,000 × 1.14 = $1,464.10

After 5 Years = $1,000 × 1.15 = $1,610.51

After 6 Years = $1,000 × 1.16 ≈ $1,771.56

After 7 Years = $1,000 × 1.17 ≈ $1,948.72

After 8 Years = $1,000 × 1.18 ≈ $2,143.59

The original investment had almost doubled after 7 years, and has more than doubled after 8 years.

2. Imagine that you invest $1,000 in an account that earns 10% interest every year, compounded annually. How many years will it take to double your money?

2. We can use the compound interest formula to find the final amounts as years go by.

After 1 Year = $1,000 × 1.071 = $1,070

After 2 Years = $1,000 × 1.072 = $1,144.90

After 3 Years = $1,000 × 1.073 ≈ $1,225.04

After 4 Years = $1,000 × 1.074 ≈ $1,310.80

After 5 Years = $1,000 × 1.075 ≈ $1,402.55

After 6 Years = $1,000 × 1.076 ≈ $1,500.73

After 7 Years = $1,000 × 1.077 ≈ $1,605.78

After 8 Years = $1,000 × 1.078 ≈ $1,718.19

After 9 Years = $1,000 × 1.079 ≈ $1,838.46

After 10 Years = $1,000 × 1.0710 ≈ $1,967.15

After 11 Years = $1,000 × 1.0711 ≈ $2,104.85

The original investment had almost doubled after 10 years, and has more than doubled after 11 years.

The Rule of 72 is a way to estimate how long it takes for doubling to happen.

The Rule of 72 (for Doubling)

The Rule of 72 says

annual interest rate × years until doubling ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

In our first example 10 rate × 7 years ≈ 72. The product 10 rate × 7 years was slightly below 72, hinting that the answer would happen soon after the seventh year.

In our second example 7 rate × 10 years ≈ 72. The product 7 rate × 10 years was slightly below 72, hinting that the answer would happen soon after the tenth year

3. Imagine that you invest $1,000 in an account that earns 2% interest every year, compounded annually. Use the Rule of 72 to estimate how many years will it take to double your money.

3. The Rule of 72 says 2 rate × years until doubling ≈ 72

We divide both sized by 2 to find an answer of 36 years.

A very long time! No one can save for retirement with only a 2% annual interest rate.

Halving While Decreasing

Does the Rule of 72 also work when an initial amount of money shrinks?

4. Imagine that you inherit $1,000. Each year you spend 10% of what is left. How many years will it take to halve your inheritance money?

4. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.1, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.1) = 0.9

After 1 Year = $1,000 × 0.91 = $900

After 2 Years = $1,000 × 0.92 = $810

After 3 Years = $1,000 × 0.93 = $729

After 4 Years = $1,000 × 0.94 = $656.10

After 5 Years = $1,000 × 0.95 = $590.49

After 6 Years = $1,000 × 0.96 ≈ $531.44

After 7 Years = $1,000 × 0.97 ≈ $478.30

The original investment had almost halved after 6 years, and has more than halved after 7 years.

Yes, the Rule of 72 again provided a close estimate. 10 rate × 7 years ≈ 72

Let's check again.

5. Imagine that you inherit $1,000. Each year you spend 7% of what is left. How many years will it take to halve your inheritance money?

5. We can use the compound interest formula to find the final amounts as years go by.

Instead of using the One Plus Trick to set (1 + rate) as 1.07, we need to use the One Minus Trick to set (1 + rate) as (1 − 0.07) = 0.93

After 1 Year = $1,000 × 0.931 = $930

After 2 Years = $1,000 × 0.932 = $864.90

After 3 Years = $1,000 × 0.933 ≈ $804.38

After 4 Years = $1,000 × 0.934 ≈ $748.05

After 5 Years = $1,000 × 0.935 ≈ $695.69

After 6 Years = $1,000 × 0.936 ≈ $646.99

After 7 Years = $1,000 × 0.937 ≈ $601.70

After 8 Years = $1,000 × 0.938 ≈ $559.58

After 9 Years = $1,000 × 0.939 ≈ $520.41

After 10 Years = $1,000 × 0.9310 ≈ $483.98

The original investment had almost halved after 9 years, and has more than halved after 10 years.

Yes, the Rule of 72 again provided a close estimate. 7 rate × 10 years ≈ 72

Hooray! The Rule of 72 works both ways.

The Rule of 72 (for Halving)

The Rule of 72 says

annual interest rate × years until halving ≈ 72

Notice that we pretend the annual interest rate is a whole number. We do not turn it into a decimal.

Facts link to here link to index

Mortgages

amortization table

Amortization is the paying off of debt with repeating, schedules repayments.

Imagine that you had a home or car loan, and needed to make the same monthly payment again and again for many years. What would be the proper monthly payment amount? This problem is tricky, because the debt shrinks with payments and grows with interest at the same time. But accountants have solved the problem many times, and saved the answers as an amortization table.

In this class we will use the amortization table to the right. In real life, the amortization table is huge. It has more rows, for interest rates with decimals (usually to the hundredths place). It has more columns, for loans with other durations of months and/or years.

Knowing a Loan, Finding a Payment

The amortization table values show us the correct monthly payment for a $1,000 loan with a certain interest rate and duration.

We need to scale up this number by how many thousands are in the actual loan amount.

6. Find the monthly payment for a $130,000 mortgage that is a twenty-year loan with an interest rate of 6%.

6. The amortization table value for 6% and 20 years is $7.16.

So the monthly payment is $7.16 for each $1,000 of loan.

There are 130 thousands in the loan size of $130,000.

So $7.16 per thousand × 130 thousands = $930.80 monthly payment.

That is easy to do!

Unfortunately, it is not practical. In real life, anyone who gets a loan is told their monthly payment amount. No bank expects customers to figure it out themselves!

Knowing a Payment, Finding a Loan

We can also use the amortization table backwards to find how large a home someone can afford based on their desired monthly payment.

This is more practical for real life.

7. The Wahl family has an annual income of $45,000. They want to spend 20% of that income on a mortgage. How much are they prepared to pay each month?

7. The Wahl family wants to spend each year $45,000 × 0.2 = $9,000 per year.

This becomes $9,000 ÷ 12 = $750 per month.

8. Continuing with the Wahl family, if mortgage interest rates are at 6% then how large a twenty-year loan can they afford?

8. In the previous problem we found that the Wahl family could afford to spend $750 each month.

Imagine that the Wahl family goes to the bank with that $750.

The amortization table value for 6% and 20 years is $7.16. Every time the Wahl family gives the bank $7.16 they can get $1,000 more loan.

So $750 ÷ $7.16 ≈ 104.75.

They can give the bank that $7.16 amortization table amount 104.75 times before they run out of money.

Because each of those 104.75 instances of spending the amortization table amount got the Wahl family $1,000 more loan, our last step is to multiply by $1,000.

The loan size is 104.75 × $1,000 = $104,750 loan.

Often mortgage loans are rounded to the nearest thousand dollars. If you did that, then the answer would be a $105,000 loan.

9. Continuing with the Wahl family, how much will be paid total over the twenty years? How much of that is interest?

9. The monthly payment of $750 happens twelve times per year for twenty years.

$750 × 12 × 20 = $180,000 paid total.

Of that amount, all the money is either loan or interest.

So there is $180,000 − $104,750 = $75,250 interest total.

If you had earlier rounded the loan amount to $105,000 loan, then this problem would have an answer of $75,000 interest total.

Adding Real-Life Complications

A few real-life issues add to the cost of buying a home.

That is a lot of stuff! But it does not actually change the math very much.

We need to reduce the home value by the down payment to find the mortgage size. Then we can use the amortization table to find the monthly payment.

Those other details are usually paid up front by the buyer, but some might be paid by the seller or added onto the mortgage amount. We need to add up the items paid up front to find how much total cash the home buyer is required to have available.

10. A home is purchased for $150,000. The down payment is 20%. The mortgage has a 3% fee. The first month's mortgage payment must be paid up front at closing. Inspections cost $200. The various up front costs for insurance and fees total $950. The loan is for 30 years with a 5% rate. What is the mortgage amount? What is the mortgage monthly payment? How much cash must the buyer have available for all the up front costs?

10. The down payment is $150,000 × 0.2 = $30,000.

So the mortgage amount is $150,000 − $30,000 = $120,000.

The mortgage fee is $120,000 × 0.03 = $3,600.

The amortization table value for 5% and 30 years is $5.36. So the monthly payment is:

$5.36 per thousand × 120 thousands of loan = $643.20 monthly payment

The total up front costs are:

$30,000 mortgage + $3,600 mortgage fee + $643.20 first payment + $200 inspections + $950 other = $35,393.20

Formulas link to here link to index

Sum of Annuity Due

Sum of Annuity Due

One weakness of the simple and compound interest formulas was that the principal was the only deposit. Saving for retirement never works like that! Most households deposit money each year in a retirement account.

If a deposit is made each year the formula changes into the sum of annuity due formula.

Notice that the word "principal" is incorrectly used in the formula (for historic reasons) instead of "annual deposit".

The Sum of Annuity Due Formula

Final Amount = [ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate

This formula might look big, but it is actually much easier to use than the compound interest formula. When using the compound interest formula we had to be alert for needing to divy up the annual interest rate into monthly or quarterly payouts. No longer! The sum of annuity due formula always involves annual deposits and payouts.

Also notice that the inside parenthesis can be avoided when using a calculator: if the rate was 6% we could just key in 1.06 in both places with (1 + rate) to skip the inner parenthesis.

So this formula might look scary, but it is actually less tricky than compound interest.

11. Someone saves $1,200 each year for retirement. If this is invested with 8% annual interest over 45 years, how much will it grow to be worth?

11. Final Amount =

[ Principal × (1 + rate)(years + 1)    −    Principal × (1 + rate)] ÷ rate =

[ $1,200 × 1.0846 − $1,200 × 1.08] ÷ 0.08 ≈ $500,911

Monthly Annuity?

A monthly version of the formula would also work. It would use monthly deposits, monthly interest, and (months + 1) as the exponent.

In our class we will only use an annual sum of annuity due formula. But perhaps a monthly version would help your household plan for retirement or be useful in a math project.

Homework link to here link to index

More Practice

4. Does doubling the income double the size of the numbers? Next consider the Moneybag family that has an annual income of $90,000, with everything else the same as the first problem. How large a loan can they afford? How much will be paid total over the twenty years? How much of that is interest?

5. Does halving the number of years either halve or double the size of the numbers? Next consider Mr. Short, who wants a ten-year loan, with everything else the same as the first problem. How large a loan can he afford? How much will be paid total over the ten years? How much of that is interest?

6. Brenda can afford to spend $900 per month on mortgage payments. Currently mortgage rates are 8% per year. How big a twenty-year mortgage can she afford? How big a thirty-year mortgage can she afford?

7. When the length of the term (years) of a loan is made longer, does the monthly payment increase or decrease? Does the total amount paid increase or decrease?

8. When the length of the term (years) of a loan is made longer, does the affordable mortgage size increase or decrease?

4. Yes, doubling the amount you pay per month makes all the other numbers double. The Moneybag family pays $1,500 per month for a $209,500 loan. Over the twenty years they pay $360,000 total, of which $150,500 is interest.

Again, you could round the loan size to the nearest thousand and those answers change to a $210,000 loan and total interest paid of $150,000.

5. Mr. Short still has a monthly payment of $750. Halving the years to ten years changes the amortization table value to $11.10. (This is not double the $7.16 from before, so the mortage size will not be halved.) The new mortage size is $750 ÷ $11.10 ≈ 68, so a $68,000 loan. The monthly payment is unchanged, so $750 happening twelve times per year but only for ten years. $750 × 12 × 10 = $90,000 paid total (half what was paid before, of course). There is $90,000 − $68,000 = $22,000 interest total (much less than half the interest!).

6. The amortization table value for 8% and 20 years is $8.36, which results in a $107,700 loan. The amortization table value for 8% and 30 years is $7.34, which results in a $122,600 loan. (You could round these estimates differently.)

7. When the length of the term (years) of a loan is made longer, the monthly payment decreases but the total amount paid increases. Each payment is smaller, but they eventually add up to more.

8. When the length of the term (years) of a loan is made longer, the affordable mortgage size increases.

B10. Bradley has heard that it is wise to spend no more than 25% of your income on your mortgage. He earns $39,600 per year. He wants a thirty-year home loan. Mortgage interest rates are at 5%. How large a loan can he afford? How much will he pay total over the thirty years? How much of that is interest? Where is one Lane County neighborhood that Zillow suggests is appropriate for Bradley?

market increase image

B11. The historical return for the stock market is about 11%. If the bank decided not to do business with Bradley, but instead invested the loan amount in the stock market, how much would the bank have after thirty years of 11% annual compound interest each year? Why would a bank choose to offer a mortgage to Bradley considering that stock market gain is so much smaller?

48. Someone saves $1,800 each year for retirement. If this is invested with 8% compound interest over 30 years, how much will it grow to be worth?

49. According to research by Vanguard, the average 25-year-old American has saved $3,865 for retirement. If this single deposit is invested with 9% compound interest for 40 years, how much will it grow to be worth?

average savings chart

50. That same research says the average American who retires at age 65 has saved about $176,000 for retirement. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to that amount, over 40 years and at 9% annual interest.

48. Final Amount = ( $1,800 × 1.0831 − $1,800 × 1.08 ) ÷ 0.08 ≈ $220,223

Compared to the previous problem, these deposits are 50% bigger than before. But the other problem had a time frame that was another 50% longer than this problem. Which helped more?

49. This problem has a single deposit, so we are back to the compound interet formula. Final Amount = Principal × (1 + interest rate per payout)number of payouts = $3,865 × 1.0940$121,397

That is a nice amount of money, but a middle-class retirement costs a lot more, especially for someone who pays for their own health insurance.

50. Final Amount = ( $478 × 1.0941 − $478 × 1.09 ) ÷ 0.09 ≈ $176,044.

For this problem we had to guess-and-check with different principal amounts. The answer is about $478. That is not very much to save each year towards retirement, especially later in life after your career has made some progress!

B15. The general rule for retirement savings is to, at age 65, have saved $20 for each dollar in income your retirement expenses will exceed your retirement income (from Social Security, pensions, etc.). Typical retirement expenses are greater than retirement income by about $20,000 per year—meaning typical Americans should save $400,000 for retirement. Fiddle around using the sum of annuity due formula to find what annual deposit (principal) would grow to that amount, over 40 years and at 8% annual interest.

B16. Another general guideline is to save 10% to 15% of your income each year. Make it personal, and use estimates about how your future income will change in a spreadsheet to predict your retirement savings this way. How does your answer compare to the $400,000 guideline?

B17. How does the long-term financial cost of infant care compare to college? The "big three" child care centers in Eugene (Vivian Olum at UO, Early Learning Childrens Community at LCC, and Child Development Center at EWEB) cost roughly $13,000 per year. (Oregon is an unusually expensive state for child care.) Use the Sum of Annuity Due Formula for two years (when the infant is age 1 and 2), and then the Compound Interest Formula for sixteen years (until the child is 18). If the family insted saved the infant care money, what annual interest rate would be needed for the savings to cover the child's college tuition?

Three Sisters

three sisters retirement problem chart

B13. Cindy, Clara, and Chloe are three sisters. Each has plans to save for retirement, but their plans are somewhat different. Despite their different lives, by the time each was 25 years old she had the ability to set aside $3,000 per year for retirement. All three use a retirement account that earns 8% annual compound interest. How much will each sister have saved when she retires at age 65?

cigarette money

B14. Cliff is Clara's husband. He stops smoking, and decides to devote the money he used to spend on cigarettes to retirement. He used to smoke 1 pack per day, at $5.70 per pack. How much money per year was Cliff spending on cigarettes? If he instead puts his "cigarette money" annually into a retirement account that earns 9% annual compound interest, how much will extra will he have for retirement after thirty years?

Random Homework Problems

Homework

Each time you load the page these problems change!

LCC Math 25 Packet Problems

Percent review problems are in the packet on pages:
  • PF-1 to PF-5
  • MS-28 to MS-29

Charge options problems are in the packet on pages:
  • PF-42 to PF-44

Mortgage problems are in the packet on pages:
  • PF-31 to PF-36

Sum of annuity due problems are in the packet on pages:
  • PF-65 to PF-73
  • PF-60 to PF-64