... Sections 4B,4C,4D ...

for interest payed once a year: \(A = P \times (1+APR)^Y \)

for interest paid \(n\) times a year: \(A = P \times (1+\frac{APR}{n})^{(nY)} \)

for continuous compounding: \(A = P\times e^{(APR \times Y)} \)

\(A\) = accumulated balance

\(P\) = starting principal

\(APR\) = annual percentage rate (in decimal)

\(Y\) = number of years

\(n\) = number of compounding periods per year

\(e \approx 2.71828\) is a special number

An account with an APR of 4% and quarterly compounding increases in value every three months by

(a) 1% (b) 1/4% (c) 4%

Solution. An APR of 4% means an increase by that 4% every year, which is four quarters. Which means 1% every quarter, that is 1% every three months (a quarter).

(a) is the right choice.

You deposit $3,200 in an account with an annual interest rate of 3.5%. Calculate the amount of money you will have after 5 years assuming that you earn simple interest.

Solution.

__ Simple interest __ means that you earn the same amount of

\(3200 \times 0.035 = 112\) yearly.

Over a period of 5 years you earn

\(112 \times 5 = 560\)

After 5 years, you have

\(3200+560=\$3,760\) <=== Answer

Use the compound interest formula to compute the balance if \(\$ 15,000 \) is invested for 25 years in an account with an APR of 3.2% compounded annually

Solution.
The formula

\(A = P \times (1+APR)^Y \)
gives us

\(A=15000 \times (1+0.032)^{25} = \$ 32,967.32 \)

What is the accumulated balance if \(\$2000 \) is invested for 15 years with an APR of 5% and monthly compounding?

Solution.
The compound interest formula for interest paid \(n\) times a year is

\(A = P \times (1+\frac{APR}{n})^{(nY)} \)

In this case, n=12, P=2000, Y=15, APR=0.05, and

\(A = 2000 \times (1+\frac{0.05}{12})^{(12 \times 15)} = \$ 4227.41\) <=== answer

Find the APY if a bank offers an APR of 1.23% compounded monthly

Solution.
Invest \(\$100 \), and after 1 year you will have

\(A = P \times (1+\frac{APR}{n})^{(nY)} = 100 \times (1+ \frac{0.0123}{12})^{(12 \times 1)} = \$ 101.237\)

The relative difference is the APY

\(APY= \frac{101.237-100}{100} =0.01237 =1.237\% \approx 1.24\% \)

Compute the balance after 20 years for a \(\$3000 \) deposit in an account with APR of 6%, and continuous compounding

Solution.
Continuous compounding formula

\(A = P\times e^{(APR \times Y)} \)

gives us

\(A = 3000\times e^{(0.06 \times 20)} = \$ 9,960.35\)

How much must you deposit today into an account with monthly compounding and an APR of 6% in order to have \(\$ 25,000 \) in 8 years?

Solution.
Compound interest formula for interest paid \(n\) times a year

\(A = P \times (1+\frac{APR}{n})^{(nY)} \)

solved for the principal

\(P= \frac{A}{(1+\frac{APR}{n})^{(nY)} }\)

gives us

\(P= \frac{25000}{(1+\frac{0.06}{12})^{(12\times 8)} } \approx \$15,488.10\)

\( A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)} \)

\(A\) = accumulated savings plan balance (FV -- future value)

\(PMT\) = regular payment (deposit) amount

\(APR\) = annual percentage rate (in decimal)

\(n\) = number of payment periods per year

\(Y\) = number of years

Savings Plan Formula solved for payments:

\(
PMT= A \times \frac {\left( \frac{APR}{n} \right)} { \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }
\)

Exercise 21, p. 233

You put $300 per month in an investment plan that pays an APR of 3.5%. How much money will you have in 18 years?

Solution. Use Savings Plan Formula \( A = PMT \times \frac{ \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }{\left( \frac{APR}{n} \right)} \) with \(APR=0.035,\ \ \ PMT=300, \ \ \ n=12, \ \ \ Y=18 \): \( A = 300 \times \frac{ \left[ \left( 1 + \frac{0.035}{12} \right)^{(12 \times 18)} -1 \right] }{\left( \frac{0.035}{12} \right)} = \$ 90,091.51 \)

Exercise 21, p. 233 ... continued

You put $300 per month in an investment plan that pays an APR of 3.5%. What is the total deposit made over the period of 18 years?

Solution.
You deposit $300 every month over a period of 18 years.

Total deposit \( = 300 \times 12 \times 18 = \$ 64,800 \)

Exercise 21, p. 233 ... continued

You put $300 per month in an investment plan that pays an APR of 3.5%.

Summary.

While the total deposit is \( \$ 64,000\),

the accumulated balance is \( \$90,091.51\)

Exercise 28, p. 233

Suppose that you are 25 years old, and would like to retire at the age 65. Furthermore, you would like to have a retirement fund from which you can draw an income of $200,000 per year -- forever. How can you do it? Assume constant APR of 6%.

Solution. We start with calculating the lump sum necessary for the indicated income. $200,000 should be 6% of this amount. The necessary amount is thus \( \frac{200000}{0.06} = \$ 3,333,333.33 \)

The problem becomes: find the monthly payment for a saving plan with an APR of 6% which accumulates a balance of $3,333,333.33 over the period of 65-25=40 yeas.

We apply saving plan formula solved for payments

\(
PMT= A \times \frac {\left( \frac{APR}{n} \right)} { \left[ \left( 1 + \frac{APR}{n} \right)^{(nY)} -1 \right] }
\)

with \(A=\$ 3,333,333.33 \), $APR=0.06$, $Y=40$, and $n=12$ to find

\( PMT= 3,333,333.33 \times \frac {\left( \frac{0.06}{12} \right)} { \left[ \left( 1 + \frac{0.06}{12} \right)^{(12*40)} -1 \right]} = \$1,673.79 \)

Exercise 31, p. 233

Twenty years after purchasing shares in a mutual fund for \(\$6,500\), you sell them for \(\$11,300\). Compute total and annual return.

Solution.
For the total return, we use the formula

\(
\text{total return} = \frac{A-P}{P} \times 100 \%
\)

with \(A=11,300 \ \ \ \text{and} \ \ \ P=6500 \):

\(
\text{total return} = \frac{11300-6500}{6500} \times 100 \% =73.8\%
\)

Exercise 31, p. 233 ... continued

For the annual return, we use the formula\( \text{annual return} = \left( \frac{A}{P} \right) ^{(1/Y)} -1 \)

with \(A=11300, \ \ \ P=6500, \text{and} \ \ \ n=20\):

\( \text{annual return} = \left( \frac{11300}{6500} \right) ^{(1/20)} -1 \approx 0.0280 = 2.8 \% \)

\( PMT= \frac{ P \times \left( \frac{APR}{n} \right) } {1 - \left( 1 + \frac{APR}{n} \right)^{ \left( -n Y \right) } } \)

\(PMT =\) regular payment amount

\(P=\) staring loan principal

\(APR =\) annual percentage rate (in decimal)

\(n=\) number of payment periods per year

\(Y=\) loan term in years

Consider a home mortgage of \(\$ 150,000\) at a fixed APR of 4% for 15 years.

(a) Calculate the monthly payment

Solution.
We use the Loan Payment formula
\( PMT= \frac{ P \times \left( \frac{APR}{n} \right) } {1 - \left( 1 + \frac{APR}{n} \right)^{ \left( -n Y \right) } } \)

with \(P=150000\),
\(APR =0.04\),
\(n=12\),
\(Y=15\):

\( PMT= \frac{ 150000 \times \left( \frac{0.04}{12} \right) } {1 - \left( 1 + \frac{0.04}{12} \right)^{ \left( -12 \times 15 \right) } } \approx \$ 1109.53\)

Consider a home mortgage of \(\$ 150,000\) at a fixed APR of 4% for 15 years.

(b) Determine the total amount paid over the term of the loan.

Solution. Over a period of 15 years, every month, an amount of $1,109.53 was paid.

All together, an amount of

\(n \times Y \times PMT = 12 \times 15 \times 1109.53 = \$ 199,715\)

was paid.

Consider a home mortgage of \(\$ 150,000\) at a fixed APR of 4% for 15 years.

(c) Of the total amount paid, what percentage is paid toward the principal, and what percentage is paid for interest?

Solution.

The total amount paid is \(\$ 199,715 \), while the loan principal was \(\$ 150,000 \).
Thus the percentage paid towards the principal is
\( \frac{150000}{199,715} \approx 0.751 = 75.1 \% \).

The rest \(100\% - 75.1\%=24.9\%\) is paid for interest.

Consider a home mortgage of \(\$ 150,000\) at a fixed APR of 4% for 15 years.

(d) Of the first monthly payment, what percentage is paid toward the principal, and what percentage is paid for interest?

Solution. The first payment (in fact, every payment) is $1109.53.

By the end of the first month, the new principal is

\( 150,000 + \frac{0.04}{12} \times 150,000 = 150,500 \)

The payment of \(\$1109.53\) splits into \(\$500 \) for interest and

\( 1109.53 - 500 = \$600.53 \) toward the principal.

Consider a home mortgage of \(\$ 150,000\) at a fixed APR of 4% for 15 years.

(d) Of the first monthly payment, what percentage is paid toward the principal, and what percentage is paid for interest?

Solution (continued):
We have just found that out of $1109.53

$500 is for interest.

That is \(\frac{500}{1109.53} \times 100\% = 45.06\% \) <==== answer

__ Observation.__
Of the total amount paid over 15 years, 24.9% is for interest, while
of the first payment, 45.06% is for interest.