Mathematics: MATH650 section 02
Wendy Forbes
April 27, 2010

We often hear people say that we should let our money work for us. Using money or capital for income or profit is called an investment.

An accountant manages a company’s money. Then, managers or company investors review their reports to find out the financial status. The demand for accountants increases as more private companies are established. In addition, there are always new and changing laws that increase the need for a person with these skills.

Continuously Compounded interest is what banks normally use to calculate the interest on investments. This method is the equal of continually recalculating the interest based on the current principal amount.

There are different types of Compound Interest. Interest is the amount of money earned in a saving account. * Simple Interest: Interest calculated once on the principal. * Compound Interest: Interest calculated for a given segment of time of the investment. For example, compounded monthly, means the interest is calculated each month and combined with the principal. So, for the next month, interest is earned on the interest.

Continuously Compound Interest Formula

P: ending principal
P0: initial principal
e: constant e
k: interest
t: time as measured in years

Example problem:
Let’s say your accountant invests $5,000.00 in a saving account at a bank for you that earns 5% interest compounded continuously. How would we know what is earned after 10 years?

We start with our formula.

* We determine the values for each of the variables in the formula: * P0 = $5000.00Initial principal
* k = 5% but use the decimal equivalent 0.05
* t = 10Number of years
* e is a special number used to get the natural exponential growth. It is used throughout math and is approximately 2.7183.
Also, it is usually...

...Logarithms are used in a lot of places to scale numbers when there's a big range between the smallest and the largest numbers of interest, which makes them easier to talk about.
y=yi x e^-kt
where:
y - different between temprature of body and the constant temp of room
yi - initial temprature difference of body and room
e - eulers number (2.718...)
t - time in mins
k - constant for that particular body (usually what u are trying to find out in class tasks)
using logarithms, newtons law can predict how how a body (such as cup of coffee) will be after any given period of time.
Example 1: A $1,000 deposit is made at a bank that pays 12% compounded annually. How much will you have in your account at the end of 10 years?
Explanation and Solution:
At the end of the first year, you will have the $1,000 you had at the beginning of the year plus the interest on the $1,000 or . At the end of the year you will have . This can also be written .
At the end of the second year, you will have the you had at the beginning of the year plus the 12% interest on the . At the end of the second year you will have
This can also be written . Another way of writing this is to write the balance at the end of the second year as .
At the end of the third year, you will have the you had at the beginning of the year plus the 12% interest on the . At the end of the third year you will have...

...20
Plan 2:
V0 * (1.06)17 = 3,000,000
V0 = 3,000,000/(1.06)17
V0 = $1,114,093.26
Plan 3:
C * Annuity Compound Factor (6%, 37) = 3,000,000
C * [((1.06)37 – 1)/0.06] = 3,000,000
C *127.27 = 3,000,000
C = $23,572.28
Plan 4:
C * Annuity Compounding Factor (6%,17) = 3,000,000
C * 28.21 = 3,000,000
C = $106,334.41
Plan 5:
C * Annuity Compounding Factor (6%,8) = 3,000,000
C * 9.90= 3,000,000
C = $303,107.83
2. You have just taken out a mortgage for $575,000, at a fixed rate of 4.75% per year, compounded monthly, and a term of 30 years.
a) Calculate the monthly payments
The payments must discount to a value that is equivalent to $575,000 today, assuming a monthly rate of (4.75%/12), or 0.39583% per month, for 360 months.
C * Annuity discount factor (0.39583%,360) = 575,000
C * 191.70 = 575,000
C = $2,999.47
b) For the first six months’ payments, calculate the portion that is interest and the portion that is principal
Each month, the interest portion can be calculated as the monthly rate of 0.39583% times the beginning mortgage balance. The principal portion is simply the total payment less the interest portion. For the following month, the beginning principal will decline by the principal amount which was paid in the previous month.
Month | Beginning Principal | Interest | Principal Paid | Ending Principal |
1 |...

...13.1 CompoundInterest
• Simple interest – interest is paid only on the
principal
• Compoundinterest – interest is paid on both
principal and interest, compounded at regular
intervals
• Example: a $1000 principal paying 10% simple
interest after 3 years pays .1 3 $1000 = $300
If interest is compounded annually, it pays .1
$1000 = $100 the first year, .1 $1100 = $110
the second year and .1 $1210 = $121 the third
year totaling $100 + $110 + $121 = $331 interest
13.1 CompoundInterest
Period
Interest
Credited
Times
Credited
per year
Rate per
compounding
period
Annual
Semiannual
year
1
6 months 2
Quarterly
quarter
4
R
4
Monthly
month
12
R
12
R
R
2
13.1 CompoundInterest
• Compoundinterest formula:
M P (1 i )
n
and
I M P
M = the compound amount or future value
P = principal
i = interest rate per period of compounding
n = number of periods
I = interest earned
13.1 CompoundInterest
• Time Value of Money – with interest of 5%
compounded annually.
2000
$1000 $1000
n
(1 i )
(1.05)10
2010
$1000
2020
$1000(1 i ) n
$1000(1.05)10
13.1 CompoundInterest
• Example: $800 is...

...Chapter 5 : Interet rates
Page161
Interest rate quotes and adjustments
5-1. Your bank is offering you an account that will pay 20% interest in total for a two-year deposit. Determine the equivalent discount rate for a period length of
a. Six months.
b. One year.
c. One month.
a. Since 6 months is [pic] of 2 years, using our rule [pic]
So the equivalent 6 month rate is 4.66%.
b. Since one year is half of 2 years [pic]
So the equivalent 1 year rate is 9.54%.
c. Since one month is [pic] of 2 years, using our rule [pic]
So the equivalent 1 month rate is 0.763%.
5-2. Which do you prefer: a bank account that pays 5% per year (EAR) for three years or
a. An account that pays 2[pic] every six months for three years?
b. An account that pays 7[pic] every 18 months for three years?
c. An account that pays [pic] per month for three years?
If you deposit $1 into a bank account that pays 5% per year for 3 years you will have [pic] after 3 years.
a. If the account pays [pic] per 6 months then you will have [pic] after 3 years, so you prefer [pic] every 6 months.
b. If the account pays [pic] per 18 months then you will have [pic] after 3 years, so you prefer 5% per year.
c. If the account pays [pic] per month then you will have [pic] after 3 years, so you prefer [pic] every month.
5-3. Many academic institutions offer a sabbatical policy. Every seven years a professor is given a year free of...

...solutions to the questions below should be typed (Word or Excel file) and show all steps of calculations. (No points will be given without the steps or work to the final answers)
Please submit your file in Moodle.
1. Discuss the difference between book values and market values and explain which one is more important to the financial manager and why.
2. Bonner Collision has shareholders' equity of $141,800. The firm owes a total of $126,000 of which 60 percent is payable within the next year. The firm net fixed assets of $161,900. What is the amount of the net working capital?
3. Kaylor Equipment Rental paid $75 in dividends and $511 in interest expense. The addition to retained earnings is $418 and net new equity is $500. The tax rate is 35 percent. Sales are $15,900 and depreciation is $680. What are the earnings before interest and taxes?
4. The Widget Co. purchased new machinery three years ago for $4 million. The machinery can be sold to the Roman Co. today for $2 million. The Widget Co.'s current balance sheet shows net fixed assets of $2,500,000, current liabilities of $1,375,000, and net working capital of $725,000. If all the current assets were liquidated today, the company would receive $1.9 million in cash. The book value of the Widget Co.'s assets today is _____ and the market value of those assets is _____.
5. What is the amount of dividends paid in 2011?
6. Al's Sport Store has sales of $897,400, costs...

...COMPOUNDINTEREST
Making or Spending Money
SIMPLE INTEREST FORMULA
If a principal of P dollars is borrowed for a
period of t years at a per annum interest rate
r, expressed as a decimal, then interest I
charged is
I Pr t
This interest is not used very often. Interest is
usually compounded which means interest
is charged or given on the interest and the
principal.
Simple Interest Example
COMPOUNDINTEREST
Payment Periods:
Annually
Once per year
Semiannually
Twice per year
Quarterly
Four times per year
Monthly
Twelve times per year
Weekly
Fifty two times per year
Daily
365 (360 by banks) per year
COMPOUNDINTEREST FORMULA
The amount A after t years due to a principal
P invested at an annual interest rate r
compounded n times per year is
r
A P 1
n
nt
A is commonly referred to as the
accumulated value or future value of the
account. P is called the present value.
COMPOUNDINTEREST
Example:
Investing $1000 at an annual rate of 8%
compounded annually, quarterly, monthly,
and daily will yield the following amounts
after 1 year:
Annually
Quarterly
Monthly
Daily
COMPOUNDINTEREST
On-line example
More on-line examples
COMPOUND...

...until retirement to meet your
objectives? Assume interest remains at 9%. [Rs.1254]
2. You can deposit Rs.4000 per year into an account that pays 12% interest. If you
deposit such amounts for 15 years and start drawing money out of the account in
equal annual installments, how much could you draw out each year for 20 years?
[Rs.19964.12]
3. What is the value of a Rs.100 perpetuity if interest is 7%? [Rs.1428.57]
4. You deposit Rs.13,000 at the beginning of every year for 10 years. If interest is
being paid at 8%, how much will you have in 10 years? [Rs.203391.33]
5. You are getting payments of Rs.8000 at the beginning of every year and they are
to last another five years. At 6%, what is the value of this annuity? [35720.84]
6. How much would you have to deposit today to have Rs.10,000 in five years at
6% interest compounded semiannually? [Rs.7440.94]
7. Construct an amortization schedule for a 3-year loan of Rs.20,000 if interest is
9%.
8. If you get payments of Rs.15,000 per year for the next ten years and interest is
4%, how much would that stream of income be worth in present value terms?
[Rs.121663.50]
9. Your company must deposit equal annual beginning of year payments into a
sinking fund for an obligation of Rs.800,000 which matures in 15 years. Assuming
you can earn 4% interest on the sinking fund, how much must the payments be?...

...show All of Your Work. Remember, I do not believe in magic!!!
A) B) Answer C) D)
1. What is the simple interest for a principal of $620 invested at a rate of 7% for 3 years? $173.60 $130.20 $172.60 $129.20
A) B) C) Answer D) E)
2. If you borrow $1100 for 5 years at 14% annual simple interest, how much must you repay at the end of the 5 years? $770.00 $2215.13 $2117.96 $77,000 $1870.00
A)Answer B) C) D)
3. How much interest is earned in 5 years on $2,900 deposited in an account paying 7.1% interest, compounded quarterly? $1,223.07 $1,186.44 $266.68 $1,029.50
A)Answer B) C) D) E)
4. Suppose Emily Yu deposited $1300 in an account that earned simple interest at an annual rate of 8% and left it there for 4 years. At the end of the 4 years, Emily deposited the entire amount from that account into a new account that earned 8% compounded quarterly. She left the money in this account for 4 years. How much did she have after the 8 years? $2355.70 $2427.96 $3233.87 $2457.65 $4850.81
A) B) C) Answer D)
5. If $1,390 is invested in an account which earns 9% interest compounded annually, which will be the balance of the account at the end of 14 years? $11,106,193 $3141 $4645 $21,211
A) B) C) Answer D)
6. Susan bought a 6-month $1100 certificate of deposit. At the end of 6 months, she received $99 simple interest. Find the annual rate of simple...