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Accounting for Money Chapter 24

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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

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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)

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Ordinary Financial Transactions $50,000,000,000,000 Total Money in Universe Initial State Final State Store $1,000,000 Customer $1,000 StoreCustomer Store $1,000,020 Customer $980 $50,000,000,000,000 Shirt $20

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Universal Accounting Equation Final – Initial = Input – Output + Generation – Consumption 0 0 Ordinary Transactions Extraordinary Transactions Final – Initial = Input – Output + Generation – Consumption

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Money Generation (simplified explanation) $50,000,000,000,000 Total Money in Universe Initial State Final State 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

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Money Consumption $50,000,000,000,000 Total Money in Universe Initial State Final State 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

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Inflation – Money is generated FASTER than the economy grows Deflation – Money is generated SLOWER than the economy grows

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BorrowerLender P + I I, Interest – rent paid for the use of money Some definitions… P, Principal – amount of money borrowed t I, Interest period – length of time after which interest is due Number of interest periods BorrowerLender P } Terms set by contract Interest rate Time passes Interest in a single interest period (%) (%/yr)

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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

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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?

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Compound Interest P0P0 Interest Principal I p = P n 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.

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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?

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Compound Interest – Multiple Interest Periods per Year Annual interest rate (yr -1 ) Number of interest periods per year (yr -1 ) Time (yr) Number of interest periods

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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?

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Continuous Compound Interest P0P0 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.

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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?

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Present Worth and Discount $ $ now future P0P0 S Money now is worth more than money in the future. Discount = S – P 0

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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?

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Sinking Fund Annuity R n R S 1 2 3 n = 4 Interest Period Periodic payment Sum of annuity payments Total value of annuity

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Each payment

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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 23 rd birthday and the last deposit on his 30 th 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 65 th birthday?

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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 31 st birthday and the last deposit on his 65 th 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 65 th birthday?

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“Compound interest is the most powerful force in the universe.” Dubiously attributed to Albert Einstein

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Installment Loan R P0P0 1 2 3 n = 4 Interest Period Periodic payment Principal Interest

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Each payment

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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?

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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?

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Perpetuity R P0P0 1 2 3 n = 4 Interest Period Periodic payment

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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?

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Good luck in your exams and have a great summer!!! Good luck in your exams and have a great summer!!!

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Chapter 5 Mathematics of Finance

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