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William F. Bentz1 Session 11 - Interest Cost

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William F. Bentz2 Interest A.Interest is the compensation that must be paid by a borrower for the use of a lender’s money. The amount of the total compensation (total interest) is a function of the amount borrowed, the period of the loan, and the timing of the required payments.

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William F. Bentz3 Total Interest B.The total amount of interest = total interest payments + (total principal repaid - loan proceeds - fees), or simply = total payments to the lender minus total cash received from the lender and any fees

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William F. Bentz4 Interest Specification C.Interest may be stated explicitly in an agreement (e.g., a car loan contract), or it may be an implicit component of an agreement (e.g., car sales or lease contracts).

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William F. Bentz5 Interest Allocation D.In any case, the amount of the total interest is known. The accounting issue is how best to allocate the total interest over the accounting periods that include the loan period.

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William F. Bentz6 Principal VS. Interest E.Any agreement involving interest includes an amount on which interest is being earned-- the principal, a rate at which interest is earned, a method of determining interest, and the timing of all payments.

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William F. Bentz7 Principal u The amount on which one is earning interest is known as the principal. The original amount of a car loan would be called the principal. At any date thereafter, the principal normally would be the unpaid balance of the loan.

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William F. Bentz8 Interest Rates u Different types of loans require the calculation of interest for different loan periods (e.g., monthly, quarterly, semi- annually, etc.), but a rate for any period can be converted to an equivalent annual rate.

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William F. Bentz9 Annual Interest Rates u By determining effective annual interest rates, we have a common measure of interest costs. The effective annual rate allows us to compare different loans with different payment patterns and terms.

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William F. Bentz10 Annual Interest Rates u Without a common basis with which to compare loans, the task would be much more difficult. The usual approach is to determine the effective annual yield of a loan or investment (internal rate of return).

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William F. Bentz11 Interest Method 3.There are two basic methods of computing interest: u simple interest, and u compound interest.

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William F. Bentz12 Method: Simple Interest Total amount of simple interest I = principal x rate x number of interest periods I = P x r x N

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William F. Bentz13 Simple Interest Example A loan of $1,000 for two years earning simple interest of 10% per year would result in total interest of $200.00 = $1,000 x 0.10 x 2 = $200

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William F. Bentz14 Method: Compound Interest Total compound interest amount I = principal x (1 + rate) n - principal or I = principal x [(1 + rate) n - 1]

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William F. Bentz15 Method Difference u The difference between the two methods is that with compound interest, additional interest is earned on any unpaid interest. No further interest is earned on unpaid simple interest.

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William F. Bentz16 Method: Compound Interest u Interest amount for two years (with annual compounding) I = (principal x rate) x 2 + rate x (rate x principal) I = simple interest + interest on unpaid interest I = P(1 + r) 2 - P

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William F. Bentz17 Method: Compound Interest u I = P(1 + 2r + r 2 ) - P u I = [P x r x 2] + r[rP] u Thus, compound interest is simple interest [P x r x 2] plus interest on the accumulated interest r[rP].

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William F. Bentz18 Interest As A Growth Rate u Think of interest as growth in wealth over time. u With simple interest, the growth in wealth is linear. u With compound interest, the growth in wealth is exponential.

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William F. Bentz19 The Key Accounting Issues u Measuring and reporting interest income or expense u Separating the interest component from other operating activities (revenue or cost). u Valuing selected assets and liabilities at their present values.

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William F. Bentz20 Cash Flows Determining the amount and timing of the cash flows is crucial to all but the most elementary decisions or accounting problems involving interest. To analyze about any case or problem, diagram the cash flows first. Diagram the cash flows first!!!

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William F. Bentz21 Cash Flows Cash flows are assumed to take place at points in time, or uniformly over periods of time. In this course, we are going to assume that cash flows take place at discrete points in time.

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William F. Bentz22 Scenario 1 u Suppose you invest $1,000 on 1/1/01 and receive $100 as of 12/31/01, and your $1,000 investment, for a total of $1,100 What is you net income before tax? What is the rate of return on investment?

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William F. Bentz23 Scenario 2 What is the rate of return on investment? Suppose you invest $1,000 on 1/1/01 and receive $100 as of 1/1/01, and your $1,000 investment on 12/31/01, for a total of $1,100 What is you net income before tax?

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William F. Bentz24 u Suppose you invest $1,000 on 1/1/01 and receive $210 and your $1,000 investment as of 12/31/02, for a total of $1,210. What is you net income before tax? What is the rate of return on investment?

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William F. Bentz25 u Suppose you invest $1,000 on 1/1/01 and receive $210 and your $1,000 investment as of 1/1/02, for a total of $1,210. What is you net income before tax? What is the rate of return on investment?

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William F. Bentz26 u Suppose you invest $1,000 on 1/1/01 and receive $100 on 12/31/01 and $100 on 12/31/02, plus your $1,000 investment as of 12/31/02, for a total of $1,200. What is you net income before tax? What is the rate of return on investment?

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William F. Bentz27 u Suppose you invest $1,000 on 1/1/01 and receive $100 and your $1,000 investment as of 1/1/01, another $100 as of 1/1/02, and your investment of $1,000 on 12/31/02, for a total of $1,200. What is you net income before tax? What is the rate of return on investment?

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William F. Bentz28

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William F. Bentz29 Compound Interest Let amounts be the principal and interest that have accumulated to some point in time. Thus, amounts represent wealth at a point in time. Interest is the growth factor in changes in wealth over time. 1 1

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William F. Bentz30 Compound Interest An amount of wealth A 0 will grow to an amount A 1 in one period in accordance with the model A 1 = (1 + r)A 0 where r is the growth (interest) rate (a.k.a. internal rate of return)

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William F. Bentz31 Compound Interest If we know two of the three variables in the model (r, A 0, or A 1 ), the we can determine the value of the third variable.

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William F. Bentz32 Fundamental Equations

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William F. Bentz33 Fundamental Equations For a given interest rate, and an amount a time t = 0, we can find an unknown amount at t = 1. If the present amount A 0 is $10,000, and the interest rate is 11%, then the future amount at t = 1, denoted A 1, = $10,000 x (1 +.11) = $11,000.

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William F. Bentz34 Fundamental Relationships On the other hand, if the future amount is known to be $10,000, and the interest rate is 11%, then the present value (amount at t = 0) is

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William F. Bentz35 Generalizing The relationship between an amount at t = 0 and an amount at t = 1 can be generalized to any times N - 1 and N. An amount at time t = N - 1 will grow in one period to an amount A N given by

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William F. Bentz36 Further Generalizing By repeated substitutions for the amounts at t = N -1, N - 2, N - 3, etc. we get the expression for the present value of an amount at t = N and thus

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William F. Bentz37 Annuities u Annuities are nothing more than sums of present value amounts or of future amounts at some interest rate r. where all the payments equal A

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William F. Bentz38 Annuities The A term is a constant amount, so the expression can be simplified to In this case The sum is the present value, at time t = 0, of an ordinary annuity of $A per period.

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William F. Bentz39 Annuities If we let PVOA represent the present value of an ordinary annuity, then we have that

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William F. Bentz40 Annuity Formulas This is a series that can be summed (see notes for derivation) as follows:

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William F. Bentz41 Finding the Effective Rate(r) u Many times in accounting and finance, we want to know the expected rate of return of a proposed project, or the actual rate of return earned by a project or investment. We want a universal measure of profitability.

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William F. Bentz42 Examples u What is the effective interest rate of the bond issue we just sold? u Is a transaction fairly priced given the market rates available? u What is the expected rate of return on a proposed project or investment?

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William F. Bentz43

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William F. Bentz44 The Answer To answer the above questions, we need to find the internal rate of return of the loans or investments under consideration. The internal rate of return is that rate which discounts a stream of cash flows, including at least one negative cash flow, to zero.

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William F. Bentz45 The Internal Rate The rate that solves the following equation is called the internal rate of return: where C 0,C 1,…,C N are cash flows, at least one of which is negative.

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William F. Bentz46 An Alternative Form u For many investment transactions, the first cash flow is negative and the subsequent cash flows are positive. The above formula can be rewritten (equivalently) in a more intuitive form as

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William F. Bentz47 Equivalent Expression

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William F. Bentz48 Implementation of IRR u For every change in sign from positive to negative, or negative to positive, there will be another solution to the above equation. However, we can ignore the negative rates if total net cash flows are positive. For most realistic cases, IRR calculations work fine.

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William F. Bentz49 Use of the IRRWHIZ u Use the IRR calculator on the web in my directory to make internal rate of return (IRR) calculations. By looking at the formulas and instructions, I think you will be able to see how to make the whiz fit your problem.

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William F. Bentz50 Some Application Areas u Simple interest is used in the discounting of notes receivable and other transactions involving simple interest over periods typically less than one year.

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William F. Bentz51 Compound Interest u The compound interest model (effective yield method) is used to account for long-term receivables, investments held to maturity, bonds payable, leases, notes payable, and pension computations, to name a few important areas. (GAAP)

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William F. Bentz52 To Summarize u Estimating cash flows and using the compound interest model to discount those cash flows to analyze the profitability of business transactions is fundamental to the study of accounting and finance. Few concepts are more important.

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William F. Bentz53 THE END

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Copyright © 2011 Pearson Prentice Hall. All rights reserved. The Time Value of Money: Annuities and Other Topics Chapter 6.

Copyright © 2011 Pearson Prentice Hall. All rights reserved. The Time Value of Money: Annuities and Other Topics Chapter 6.

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