Chapter 11: Continuous Compounding & Ratios

Types of Compounding

A. Discrete: when interest is earned every year, quarterly, month, day, etc. Will use for comparison but we will explore thoroughly later.

B. Continuous: interest is earned continuously. We will use this for project 3.

Discrete Compounding Formula: F= future value

P=present value

i=interest rate per period

n=total number of compounding periods

F = P(1 + i)n

Find the future value of $1000 in 3 years with annual interest of 5% compounded monthly:

P = $1000

i = 5%/12

n = 36 compounding periods

Find F

Continuous Compounding Formula:

F = future value

P = present value

r= annual interest rate

t = years

F = Pert

Example: find F

What is the future value of an investment of $500 at annual interest of 5.5% for 10 years, compounded continuously?

How to do in Excel?

Example: Find t

Suppose a couple invest $2500 in an account that earns 4.3% compounded continuously. How long will it take before they earn $1000 in interest?

….. 1.4 = e 0.043t (will continue later)

Logarithms:

Recall p = logb n means bp= n

Example: log10 1000 = 3 because

Example: log2 16 = 4 because

Similarly for base e,

loge x = p because ep = x where e is an irrational number

We write loge x = ln x

What is ln 1000?

ln 1000 = some number p such that

ep = 1000.

Hence, e1 = 2.7

e2 = ?

e3=?

etc

Calculator

You can use “ln” key in a scientific calculator and find ln 1000 = 6.908

This means e 6.908 = 1000

Back to a previous problem:

1.4 = e 0.043t

ln 1.4 = ln (e 0.043t)

ln 1.4 = 0.043t ln e

ln 1.4 = 0.043 t

ln 1.4/0.043 = t

7.825 = t

Example:

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Ratios

We are interested in comparing stock prices from one week to the one preceding it.

We can do this by finding the ratio of the future value to the present value.

Ratio R = F/P = growth ratio. Also called…

Weekly ratio = how much the value grow per week.

Monthly ratio = how much the value grow per month

Etc.

Example:

A week ago, the stock of a company was $50.43. This week, the value is $51.62. Find the weekly ratio and explain what it means. Find the % increase.

When the growth ratio is greater than one, the stock has increased in value.

When the growth ratio is smaller than one, the stock had decreased in value.

When the growth ratio is equal to one, nothing had changed.

Example

A week ago, the stock of a company was $50.43. This week, its value is $48.21. What is the weekly ratio and what does it mean? The the % decrease.

Other ways to find ratios:

For continous compounding:

F/P = e rt

For discrete compounding:

F/P = (1 + i)n

Example

An investment is growing at a monthly rate of 0.5%.

What is the monthly ratio?

What is the yearly ratio?

What is the annual percentage yield? (or % increase)

Example

If a bank account compounds interest continuously at an annual interest of 10%, what is the monthly ratio? (1.008368)

Focus on the Project:

The first step to pricing our stock option is to compute the weekly ratios from the data we downloaded.

This can be easily done by dividing next week’s adjusted closing price by the current closing price. Use Excel.

Focus on the Project

Use the fact that our class project risk-free annual interest rate of 4% to compute the weekly risk-free ratio. Keep in mind that there are 52 weeks in a year.

Hence, R = e rt

= e (0.04)(1/52)

= 1.0007695

Focus on the Project

1.0007695 is our weekly risk-free ratio for our class project.

Note that it is a good idea to keep several decimal places at this point of our computations.

Using the risk-free rate and the fact that we have a 20-week option period, we can get a preliminary estimate for the price of the stock option after this 20-week period.

Note that the time 20 weeks has to be converted in years.

Focus on the Project

The closing price of DIS stock at the start of the option period was $21.87. Our preliminary estimate for the value of the option is:

F = Pe rt

= 21.87 e (0.04)(20/52)

= 22.21

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