Financial Curiosities

M 110 Modeling with Elementary FunctionsV.J. Motto

Annual Percentage Yield

As an officer of a financial firm, you have to invest $1,000,000 for a year. Your choices

1. 5% compounded monthly

2. 4.95% compounded daily

Which would you choose?

APY

The indicator you should use is called annual percentage yield (APY). It measures the percentage of money in a compound account that is returned at the end of each year. The formula is

Interest Earned for One YearAPY

Original Investment

Example 1

You are interested in investing $100 in an account paying 5.08% interest compounded monthly?

a. What is the APY for this account?

b. If $1,000 were invested what should be the APY?

Solution

Using TVM Solver we find that the account (FV) is actually worth $105.20 at the end of the year. Thus, $5.20 of interest was earned on $100 investment. Hence, the

APY = $5.20/100 = 5.20 %

Solution (cont)

To answer the second question, change PV to – 1000 and solve for FV. This time we find that the interest earned is $52, which represents a percentage earning of 52/1000 = 0.0520 or 5.2%

Comments

APY depends only on the annual interest rate and the number of times the money is compounded, not on the amount involved.

When computing APY, set N = 1 since we want the interest generated in one year. Also set PMT = 0, and P/Y = 1 because we are computing compound interest.

The advertised interest rate is called the nominal rate of the account.

We will round all APY’s to the nearest hundredth of a percent

More comments

If an account is compounded annually, the nominal rate is the same as the APY. Otherwise, the nominal rate is less than the APY.

APY is sometimes referred to as Effective Annual Yield (EAY)

Example 2

You have the option of investing $1,000,000,000 for a year in an account paying 5% compounded semi-annually or in an account paying 4.95% compounded daily.

Solution

5%, semi-annually 4.95%, monthly

Hence, the interest of $50,625,000was earned. The APY is

50625000/100000000 = 5.06%

Hence, the interest earned is$50,742,066. So the APY is

50742066/1000000000 = 5.07%

Example 3

What is the doubling time of an investment of $2,000 deposited in an account paying 4.3% interest compounded monthly?

It takes 16.15 years.

Example 4

Under the same conditions as the previous example, what is the doubling time for an investment of $2,000,000?

It is 16.15 too! Thus, doubling time does not depend on the investment.

Doubling Time Discussion

A second topic of interest is the doubling time of an account; that is, the time that it takes an investment to double in value.

Let’s get some data. We show

of the calculations

Rate 0.01 0.03 0.05 0.08 0.11 0.14 0.20

Doubling Time 69.34 23.13 13.89 8.69 6.33 4.98 3.49

The Graph: Linear Relationship?

At the right you see the data graph and the calculation to discover if there is a linear relationship.

Graph: Quadratic Relationship?

At the right you see the data graph and the calculation to discover if there is a quadratic relationship.

Graph: Power Relationship?

You are welcome to explore the Exponential Model on your own. Here we search for a Power Function Model --- y = axb .

The Model – Rule of 70

So we have the following model

This is often called the “Rule of 70” and is expressed as

1 0.7.70y x

x

70y

Interest Rate

Continuous Compounding

Suppose that you were able to invest $1 at 100% interest for a year. Chart the value of your account as it is compounded more and more frequently.

PeriodsPer Year

Balance

Semi-annually 2 2.25

Quarterly 4 2.44140625

Monthly 12 2.613035290

Weekly 52 2.692596954

Daily 365 2.714567482

Hour 8760 2.718126692

Minute 525600 2.718279243

second 31536000 2.718281785

Comments

The value of the account is increasing, but tapers off to approximately 2.71828. Although, theoretically, the account grows larger and larger with time, it never exceeds$2.72!

The number 2.71828… plays an imporant role in mathematics. It is the base of an exponential function used heavily in calculus and is denoted by “e” in honor of the great Swiss mathematician Leonard Euler.

Continuous Compounding

The formula for continuous compounding is

whereA = Future value of the account

P = Present value of the account

R = Annual interest rate (as a decimal)

T = Term of the account (in year)

rtA Pe

Example 5

A sum of $14,500 is invested in a continuously compounded account praying 5% annual interest. Find the value of the account after 10 years.

Using the formula:

Using the calculator:

Since the TVM Solve does not have a built in formula for continuously compounded interest, we use 1,000,000,000 to help approximate the value.