Accounting for Money

Chapter 24

Objectives

• Understand how to apply the Universal Accounting Equation to money• Do calculations using simple and compound interest• Do present worth and discount calculations• Do sinking fund annuity calculations• Do installment loan calculations• Do perpetuity calculations

Engineering – “the art of doing…well with one dollar which any bungler can do with two.”

Attributed to A.M. Wellington (1847-1895)

Engineering – the profession in which a knowledge of the mathematical and natural sciences gained by study, experience, and practice is applied with judgment to develop ways to utilize, environmentally friendly and economically, the materials and forces of nature for the benefit of all humanity.

Adapted from ABET (1985)

Ordinary Financial Transactions

$50,000,000,000,000

Total Money in Universe

InitialState

FinalState

Store$1,000,000

Customer$1,000

Store Customer

Store$1,000,020

Customer$980

$50,000,000,000,000

$50,000,000,000,000

Shirt

$20

Universal Accounting Equation

Final – Initial = Input – Output + Generation – Consumption0 0

Ordinary Transactions

Extraordinary Transactions

Final – Initial = Input – Output + Generation – Consumption

Money Generation (simplified explanation)

$50,000,000,000,000

Total Money in Universe

InitialState

FinalState

Bank$100,000,000

Federal Reserve$1,000,000,000

$50,000,000,000,000

$50,000,001,000,000

$1,000,000

Bank$100,000,000

Federal Reserve$1,000,000,000

Bank$101,000,000

Federal Reserve$1,000,000,000

Money Consumption

$50,000,000,000,000

Total Money in Universe

InitialState

FinalState

Bank$100,000,000

Furnace

$50,000,000,000,000

$49,999,999,990,000

$10,000 old bills

Bank$100,000,000

Furnace

Bank$99,990,000

Furnace

Inflation – Money is generated FASTER than the economy grows

Deflation – Money is generated SLOWER than the economy grows

Borrower LenderP + I

I, Interest – rent paid for the use of money

Some definitions…

P, Principal – amount of money borrowed

tI, Interest period – length of time after which interest is due

Number of interest periods

Borrower LenderP

}Terms set by contract

It

tn

Interest rateP

Ii p

Time passes

I

p

Pt

Ii ˆ

Interest in a single interest period

(%) (%/yr)

Simple Interest

P

S = P+I =P(1+ i n) Interest

Principal P

I = P i n

0 1 2 3 4 n

Interest Periods (dimensionless)

Sum to be repaid

Pairs Exercise #1

To purchase a car, you borrow $10,000 from your Dad. The contract with your Dad states that after 7 years, you must repay the entire principal plus interest. The interest is calculated at a rate of 5% per interest period; the interest period is defined as 1 year. How much money must you repay your Dad at the termination of the loan?

Compound Interest

P0

Interest

Principal

Ip = Pn i

0 1 2 3 4 n

Interest Periods (dimensionless)

Interest is due at the end of each interest period, rather than the termination of the loan. If the interest is NOT paid on time, it is rolled into the principal so that interest can be charged on the unpaid interest.

10 )1( n

n iPP

niPS )1(0

Pairs Exercise #2

To purchase a car, you borrow $10,000 from your Dad. The contract with your Dad states that after 7 years, you must repay the entire principal plus compound interest. The interest is calculated at a rate of 5% per interest period; the interest period is defined as 1 year. How much money must you repay your Dad at the termination of the loan?

Compound Interest – Multiple Interest Periods per Year

niPS )1(0

m

ii

ˆ

Annual interest rate (yr-1)

Number of interest periods per year (yr-1)

mtn Time (yr)

Number of interest periods

mt

m

iPS

ˆ10

Pairs Exercise #3

To purchase a car, you borrow $10,000 from your Dad. The contract with your Dad states that after 7 years, you must repay the entire principal plus compound interest. The interest rate is 5% per year and the interest period is one month. How much money must you repay your Dad at the termination of the loan?

Continuous Compound Interest

P0

0 1 2 3 4 t

Time (yr)

Interest is due at the end of each interest period, rather than the termination of the loan. If the interest is NOT paid on time, it is rolled into the principal so that interest can be charged on the unpaid interest. The interest period is differentially small.

tiePSˆ

0

Pairs Exercise #4

To purchase a car, you borrow $10,000 from your Dad. The contract with your Dad states that after 7 years, you must repay the entire principal plus continuous compound interest. The interest is calculated at a rate of 5% per year. How much money must you repay your Dad at the termination of the loan?

Present Worth and Discount

$ $now futureP0

S

Money now is worth more than money in the future.

Interest) Simple( 1

10 in

SP

Interest) Compound (

1

10 ni

SP

)periods/yrinterest :Interest Compound ( ˆ

1

10 m

mi

SPmt

Interest) Compound Continuous( 1ˆ0 tie

SP

Discount = S – P0

Pairs Exercise #5

A company issues a bond that promises to pay $10,000 in 5 years. The interest rate is 5% per year with continuous compounding. What should you pay for this bond?

Sinking Fund Annuity

R

n RS

1 2 3 n = 4

Interest PeriodPeriodic payment

Sum ofannuitypayments

Total value of annuity

Interest) (Compound

11

i

iRS

n

)periods/yrinterest :Interest (Compound ˆ

1ˆ

1

mi

mi

mrS

mt

g)Compoundin s(Continuou ˆ

1ˆ

i

emrS

ti

Each payment

Pairs Exercise #6a

Andy starts saving for retirement as soon as he starts work. He invests $5000 per year, making the first deposit on his 23rd birthday and the last deposit on his 30th birthday. He lets his investment accrue earnings, but makes no further deposits after age 30. How much does he deposit into his retirement account? If he earns an interest rate of 10%, compounded continuously, how much does he have on his 65th birthday?

Pairs Exercise #6b

When Brad starts work, he spends all his disposable income on loan payments for a huge house, a sports car, a ski boat and several exotic vacations. He decides to start investing for retirement after paying these off (except the house). He invests $5000 per year, making the first deposit on his 31st birthday and the last deposit on his 65th birthday. How much does he deposit into his retirement account? If he earns an interest rate of 10%, compounded continuously, how much does he have on his 65th birthday?

“Compound interest is the most powerful force in the universe.”

Dubiously attributed to Albert Einstein

Installment Loan

R

P0

1 2 3 n = 4

Interest PeriodPeriodic payment

Principal

Interest

Interest) (Compound

110 i

iRP

n

)periods/yrinterest :Interest (Compound ˆ

ˆ11

0 mi

mi

mrP

mt

g)Compoundin s(Continuou ˆ

1ˆ

0 i

emrP

ti

Each payment

Pairs Exercise #7

To purchase a car, you borrow $10,000. The bank charges 5% interest and uses continuous compounding. The loan must be completely repaid after 5 years. What is your monthly payment?

How much do you pay for use of this $10,000 for 5 years?

Pairs Exercise #7b

To purchase a house, you borrow $100,000. The bank charges 5% interest and uses continuous compounding. The loan must be completely repaid after 30 years. What is your monthly payment?

How much do you pay for use of this $100,000 for 30 years?

Perpetuity

R

P0

1 2 3 n = 4

Interest PeriodPeriodic payment

Interest) (Compound

11

10

niRP

)periods/yrinterest :Interest (Compound

1ˆ

1

10 m

mi

RPmt

g)Compoundin s(Continuou 1

1ˆ0

tieRP

Pairs Exercise #8

After becoming a multimillionaire, you decide to create an endowment for a scholarship that pays $5,000 per year. The bank gives 5% interest compounded continuously. How much money must you give the bank to set up the scholarship?

Good luck in your exams and have a

great summer!!!

Good luck in your exams and have a

great summer!!!