Exam III: Mathematics Summary Material | Lecture notes Advanced Accounting

Exam III Summary Material

•

F.1

–

Interest

is a fee charged for the use of someone

else's money (bank charges you when you borrow;

bank charges themselves when you save)

–

The

simple interest

I=P rt

(

=principal,

or initial amount,

=interest rate as a decimal,

=time)

–

The

future value

F=P(1 + rt)

•

F.2

–

Compound Interest

is interest applied to in-

terest over time (formula:

A=P1 + r

mmt

(

=number of compoundings per year)

–

Continuous Compound Interest

formula:

P ert

–

Eective Yield

is the actual rate of interest

earned per year when compounded

–

TVM Solver

∗

I%=

(as a percent)

∗

PV=

∗

PMT=0 (*in this case, since nothing is added

to or removed from the account throughout);

∗

FV=

∗

P/Y=C/Y=

•

F.3

–

future value annuity

is an investment where

you make given periodic deposits to an account (in-

terest earned at the same time as each deposit).

(EXAMPLES: IRA, 401(k))

–

sinking fund

is set up (often by a company) to

make periodic deposits into an account to achieve

a given nal amount over a given time period.

–

(Both cases are handled similarly: In the TVM

Solver, PV=0 (usually); one of PMT or FV is

given, and we solve for the other)

•

F.4

–

Present Value Annuities-

Given periodic with-

drawals on an account for a given amount of time,

at least how much money must initially be in the

account to accomplish this? (Given PMT with

FV=0, nd PV)

–

Amortization

-how much in periodic payments in

order to kill a given debt over a given amount of

time (Given PV with FV=0, nd PMT)

–

Equity

: The amount you have already paid on

your house (Cost of house

−

Remaining Balance)

–

Closing Costs and Points

(

only needed for Take-

home quiz

): When you buy a house, you pay a lot

of extra legal fees up front called

closing costs

Points

are extra fees you pay up front to lower

your interest rate slightly. (In both cases ex-

tra means they do NOT change the price of your

house!)

•

4.3

–

linear equation

is an equation that can be writ-

ten as

A1x1+A2x2+· · · +Anxn=b

–

The

matrix form

of a linear system is

AX =B

∗A

is the

coecient matrix

∗X

is the

variable matrix

∗B

is the

constant matrix

–

Gaussian Elimination:

a set of operations you

can perform on a system of equations (rows of the

matrix) to create an equivalent system of equa-

tions:

Switch rows (equations)

Multiply or divide a row by a

nonzero

num-

ber

Mult/Div a row by a (nonzero) number and

add/subtract to another row.

–

General Strategy (Goal-1's on diagonal, 0's

rest of the column):

Divide the diagonal row by the given number

(if zero, switch rst)

Multiply the diagonal row by the number in

another row and subtract from that row. (do

this for each of the other rows in the matrix)

–

To use the calculator, put the

augmented matrix

[A|B]

in the calculator, then

rref

the matrix

•

4.4

–

If you get a false statement (

0 = 1

) after rref, the

system of equations has no solution

–

If you get more variables than equations, the sys-

tem of equations has an innite number of solutions

∗

Choose free variables (all columns which do

not have the Goal stated in 4.3 above)

∗

Solve all other variables in terms of the free

variables

•

5.1-5.2

–

m×n

matrix has

rows and

columns.

∗

Row matrix:

m= 1

∗

Column matrix:

n= 1

∗

Square matrix:

m=n

Partial preview of the text

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Exam III Summary Material

• F.

– Interest is a fee charged for the use of someone

else's money (bank charges you when you borrow; bank charges themselves when you save)

– The simple interest I = P rt (P =principal,

or initial amount, r=interest rate as a decimal, t=time)

– The future value F = P (1 + rt)

• F.

– Compound Interest is interest applied to in-

terest over time (formula: A = P

r m

)mt

(m=number of compoundings per year)

– Continuous Compound Interest formula: F =

P ert

– Eective Yield is the actual rate of interest

earned per year when compounded

– TVM Solver:

∗ N=mt

∗ I%=r (as a percent)

∗ PV=P

∗ PMT=0 (*in this case, since nothing is added

to or removed from the account throughout);

∗ FV=A

∗ P/Y=C/Y=m

• F.

– A future value annuity is an investment where

you make given periodic deposits to an account (in- terest earned at the same time as each deposit). (EXAMPLES: IRA, 401(k))

– A sinking fund is set up (often by a company) to

make periodic deposits into an account to achieve a given nal amount over a given time period.

– (Both cases are handled similarly: In the TVM

Solver, PV=0 (usually); one of PMT or FV is given, and we solve for the other)

• F.

– Present Value Annuities- Given periodic with-

drawals on an account for a given amount of time, at least how much money must initially be in the account to accomplish this? (Given PMT with FV=0, nd PV)

– Amortization-how much in periodic payments in

order to kill a given debt over a given amount of time (Given PV with FV=0, nd PMT)

– Equity: The amount you have already paid on

your house (Cost of house − Remaining Balance)

– Closing Costs and Points (only needed for Take-

home quiz ): When you buy a house, you pay a lot of extra legal fees up front called closing costs. Points are extra fees you pay up front to lower your interest rate slightly. (In both cases ex- tra means they do NOT change the price of your house!)

– A linear equation is an equation that can be writ-

ten as

A 1 x 1 + A 2 x 2 + · · · + Anxn = b

– The matrix form of a linear system is AX = B

∗ A is the coecient matrix

∗ X is the variable matrix

∗ B is the constant matrix

– Gaussian Elimination: a set of operations you

can perform on a system of equations (rows of the matrix) to create an equivalent system of equa- tions:

1. Switch rows (equations)

2. Multiply or divide a row by a nonzero num-

ber

3. Mult/Div a row by a (nonzero) number and

add/subtract to another row.

– General Strategy (Goal-1's on diagonal, 0's

rest of the column):

1. Divide the diagonal row by the given number

(if zero, switch rst)

2. Multiply the diagonal row by the number in

another row and subtract from that row. (do this for each of the other rows in the matrix)

– To use the calculator, put the augmented matrix

[A|B] in the calculator, then rref the matrix

– If you get a false statement (0 = 1) after rref, the

system of equations has no solution

– If you get more variables than equations, the sys-

tem of equations has an innite number of solutions

∗ Choose free variables (all columns which do

not have the Goal stated in 4.3 above)

∗ Solve all other variables in terms of the free

variables

– An m × n matrix has m rows and n columns.

∗ Row matrix: m = 1

∗ Column matrix: n = 1

∗ Square matrix: m = n

– The Aij element of a matrix is the number in the

ith row and jth column of A.

– Two matrices A and B are equal if Aij = Bij for

all i and j

– Add and subtract matrices by adding/subtracting

the corresponding elements (matrices must be the same size)

– Multiply a matrix by a scalar by multiplying every

element by the scalar

– The transpose of a matrix is formed by interchang-

ing the rows and columns

– The dot product of a row matrix [a 1 a 2 · · · an]

and a column matrix

b 1 b 2 · · · bn

 is given by a 1 b 1 +

a 2 b 2 + · · · + anbn (must be the same length)

– Multiply matrices: ABij is the dot product of the

ith row of A and the jth column of B (number of columns in A must equal the number of rows in B)

– The inverse of a square matrix A is a matrix B

such that AB = BA = I (where I is the n × n identity matrix). We write B = A−^1.

– If a square matrix A does not have an inverse, it is

a singular matrix.

– Given a matrix equation AX = B, if A has an

inverse, then X = A−^1 B

Exam III Summary of Examples in Class

• F.

1. You borrow money at 9% simple interest, and at

the end of 2 years, you pay back $531. How much did you originally borrow?

2. You've bought a house using a $135,000 30-year

mortgage at 3.75%. Later, we will learn that your monthly payments are $625.21. What is the new balance on your loan after one month?

• F.

1. When you were born, your grandparents deposited

$10,000 in an account which earned about 3.5% per year compounded quarterly. How much money was in the account when you turned 18? How much interest did the account earn?

2. What is the eective yield of the account above?

3. Nineteen years ago, estimated annual tuition to

attend A&M was $1,948. Today it is about $10,400 (Sources: www.collegecalc.org and tu- ition.tamu.edu). To the nearest 0.1%, what is the annual increase in tuition?

• F.

1. When you were born, your grandparents deposited

$140/quarter into an account which earned about 3.5% per year compounded quarterly.

(a) How much money was in the account when

you turned 18?

(b) How much interest did the account earn?

2. (Give up this day our daily Starbucks...) You

forego your 3 lattes/week and contribute the $50/month you save into a high-risk fund which earns 7.5% per year compounded monthly.

(a) How much money will you have in the account

when you retire in 45 years?

(b) How much interest will you have earned total

on the account?

(c) How much interest did the account earn dur-

ing the 8th month of the 42nd year?

• F.

1. On Sept 6, the Lotto Texas advertised jackpot was

$8.25M, to be paid in 30 equal annual payments. Their interest factor is 1.4108%. How much money needs to be in their account initially in order to pay a winner?

2. You decide to purchase a $29,000 new car. The

car dealership oers you a no down payment op- tion, nancing the entire amount at 4.9% per year compounded monthly for 6 years.

(a) What are your monthly payments?

(b) How much do you end up paying in interest?

(c) Construct the start of an Amortization

table for the car you bought in the previous example: Period Payment Interest Principal Balance 0 1 2 3

(d) How much of the 20th payment goes toward

the principal?

3. You start working for SuperMegaConglo-

moMonopolCorp.com at 22 with a monthly salary of $4500.

(a) Assuming an average of 2% per year ination,

what will your monthly salary be when you retire in 48 years?

(b) If you want to draw this monthly salary af-

ter retirement for 15 years, how much money should you have in your 401(k) when you re- tire if it is in a conservative fund which earns 3% per year compounded monthly?

Exam III: Mathematics Summary Material, Lecture notes of Advanced Accounting

Related documents

Partial preview of the text

Download Exam III: Mathematics Summary Material and more Lecture notes Advanced Accounting in PDF only on Docsity!

Exam III Summary Material

• F.

– Interest is a fee charged for the use of someone

– The simple interest I = P rt (P =principal,

– The future value F = P (1 + rt)

• F.

– Compound Interest is interest applied to in-

– Continuous Compound Interest formula: F =

– Eective Yield is the actual rate of interest

– TVM Solver:

∗ N=mt

∗ I%=r (as a percent)

∗ PV=P

∗ PMT=0 (*in this case, since nothing is added

∗ FV=A

∗ P/Y=C/Y=m

• F.

– A future value annuity is an investment where

– A sinking fund is set up (often by a company) to

– (Both cases are handled similarly: In the TVM

• F.

– Present Value Annuities- Given periodic with-

– Amortization-how much in periodic payments in

– Equity: The amount you have already paid on

– Closing Costs and Points (only needed for Take-

– A linear equation is an equation that can be writ-

– The matrix form of a linear system is AX = B

∗ A is the coecient matrix

∗ X is the variable matrix

∗ B is the constant matrix

– Gaussian Elimination: a set of operations you

1. Switch rows (equations)

2. Multiply or divide a row by a nonzero num-

3. Mult/Div a row by a (nonzero) number and

– General Strategy (Goal-1's on diagonal, 0's

1. Divide the diagonal row by the given number

2. Multiply the diagonal row by the number in

– To use the calculator, put the augmented matrix

– If you get a false statement (0 = 1) after rref, the

– If you get more variables than equations, the sys-

∗ Choose free variables (all columns which do

∗ Solve all other variables in terms of the free

– An m × n matrix has m rows and n columns.

∗ Row matrix: m = 1

∗ Column matrix: n = 1

∗ Square matrix: m = n

– The Aij element of a matrix is the number in the

– Two matrices A and B are equal if Aij = Bij for

– Add and subtract matrices by adding/subtracting

– Multiply a matrix by a scalar by multiplying every

– The transpose of a matrix is formed by interchang-

– The dot product of a row matrix [a 1 a 2 · · · an]

– Multiply matrices: ABij is the dot product of the

– The inverse of a square matrix A is a matrix B

– If a square matrix A does not have an inverse, it is

– Given a matrix equation AX = B, if A has an

Exam III Summary of Examples in Class

• F.

1. You borrow money at 9% simple interest, and at

2. You've bought a house using a $135,000 30-year

• F.

1. When you were born, your grandparents deposited

2. What is the eective yield of the account above?

3. Nineteen years ago, estimated annual tuition to

• F.

1. When you were born, your grandparents deposited

(a) How much money was in the account when

(b) How much interest did the account earn?

2. (Give up this day our daily Starbucks...) You

(a) How much money will you have in the account

(b) How much interest will you have earned total

(c) How much interest did the account earn dur-

• F.

1. On Sept 6, the Lotto Texas advertised jackpot was

2. You decide to purchase a $29,000 new car. The

(a) What are your monthly payments?

– Eective Yield is the actual rate of interest

∗ A is the coecient matrix

1. Divide the diagonal row by the given number

2. Multiply the diagonal row by the number in

2. What is the eective yield of the account above?

2. (Give up this day our daily Starbucks...) You

(a) Assuming an average of 2% per year ination,