# TVM Applications Chapter

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1 Chapter 6 Time of Money UPS, Walgreens, Costco, American Air, Dreamworks Intel (note 10 page 28) TVM Applications Accounting issue Chapter Notes receivable (long-term receivables) 7 Long-term assets 10 Long-term intangibles (patents, copyrights) 12 Notes payable (long-term liabilities) 14 Investments 17 Installment contracts 18 Pensions and other postretirement benefits (OPEB) 20 Leases 21 TVM calculations used in many fair value calculations Use of fair value increasing, TVM more important 1 2 Ch 6: TVM Calculator Chapter Overview TI-83 Plus, TI-84 Plus (Int. Algebra) TI BA II Plus HP 10B, HP 17B, HP 12C Casio FC-200V, Sharp EL733A Rite-Aid FV ( ) and PV ( ) 3 Explain time value of money concept Differentiate simple, compound interest Solve future and present value of \$1 problems Solve future and present value of annuities (ordinary, annuity due) Solve deferred annuities, bonds, and expected cash flows problems 4 Learning Objectives Learning Objectives Identify topics using time value of money (TVM) Distinguish between simple and compound interest Use appropriate compound interest tables. Identify variables needed to solve TVM problems Solve future and present value of \$1 problems Solve future value of ordinary, annuity due problems Solve present value ordinary, annuity due problems Solve deferred annuities and bonds problems Solve expected cash flows problems Identify topics using time value of money (TVM) Distinguish between simple and compound interest Identify variables needed to solve TVM problems 5 6 1

2 Time Of Money Required by GAAP Many applications in accounting assets and liabilities Any amount more due > 1 year Long-term budgeting Basis for all finance TVM Applications In life Saving for retirement: 401(k), IRA, 529 Mortgage payments See WebAccess sample problems and website In general, time value of money calculations must be made whenever a dollar amount will change hands more than one year from today. 7 8 Time Of Money All money earns interest over time \$1 today \$1 tomorrow Every dollar in the future is part principal and part interest Time Of Money Amount of cash is small = Principal Rate Time = \$10 3% 30/360 = \$ Time Of Money Amount of cash is large = Principal Rate Time = \$10,000,000,000 3% 30/360 = \$25,000,000 = \$10,000,000,000 3% 1 = \$300,000,000 Payment to rent money Compensation to lender for use of \$ Compensation for risk Inflation Risk of default Compensation for profit See Intel Annual Report, note 10, page

3 Difference between Beginning balance = \$100 Ending balance = \$106 To lender, interest revenue To borrower, interest expense Simple = Principal Rate Time rate per period Time is number of periods Rate and time must be same periods Year, semi-annual, quarter, month 13 rate usually stated as rate per year, must convert to rate per period 14 Rate Per Period Number of Periods Compounding Annual Rate Periods per Year Rate per Period Annual 12% 1 = 12% Semiannual 12% 2 = 6% Quarterly 12% 4 = 3% Monthly 12% 12 = 1% Compounding Years Periods per Year Periods Annual 20 1 = 20 Semiannual 20 2 = 40 Quarterly 20 4 = 80 Monthly = rate per period = rate per year / periods per year Compounding Annual Rate Periods per Year Rate per Period Annual 12% 1 = 12% Simple : Yearly Semiannual 12% 2 = 6% Quarterly 12% 4 = 3% Monthly 12% 12 = 1% Number of periods = Number of years periods per year Compounding Years Periods per Year Periods Annual 20 1 = 20 Semiannual 20 2 = 40 Quarterly 20 4 = 80 Monthly = Borrow \$100,000 For 20 years Annual interest rate of 12% = Principal Rate Time = \$100,000 12% 20 = \$240,000 calculated on original principal only 18 3

4 Simple : Monthly Borrow \$100,000 For 20 years Annual interest rate of 12% = Principal Rate Time = \$100,000 1% 240 = \$240,000 Same total interest 19 Simple value (PV) \$1,000 rate per year 10% Number of years 3 Compounding periods per year None Period Principal Ending 1 1, , , , , ,300 calculated on original principal only 20 Compound Earn interest on Initial investment accumulated in previous periods 21 One compounding interval per year value (PV) \$1,000 rate per year 10% Number of years 3 Compounding periods per year 1 rate period 10% Number of periods 3 Period Beg Ending 1 1, , , , , , Two compounding intervals per year value (PV) \$1,000 rate per year 10% Number of years 3 Compounding periods per year 2 rate period 5% Number of periods 6 Simple and Compound value (PV) \$1,000 rate per year 10% Number of years 3 Period Beg Ending 1 1, , , , , , , , , , , , Calculation Amount Simple \$1,300 Compounded annually \$1,331 Compounded semiannually \$1,

5 Simple interest compared to compound interest Principal \$100,000 Rate 12% per period Time 3 periods Simple and Compound Simple (\$100,000, 12% per period, 3 periods) Period Beg Bal Rate End Bal 1 100,000 12% 12, , ,000 12% 12, , ,000 12% 12, ,000 Principal Rate Time Simple \$100,000 12% 20 \$240,000 Simple \$100,000 1% 240 \$240,000 same regardless of time periods Total interest 36,000 Compound Period Beg Bal Rate End Bal 1 100,000 12% 12, , ,000 12% 13, , ,440 12% 15, ,493 Total interest 40, Compounding Interval Annually \$864,629 Semiannually \$928,572 Quarterly \$964,089 Monthly \$989,255 different for each compounding interval 26 Five Tables in Textbook Time Of Money 1. of \$1 2. of \$1 3. : Ordinary Annuity of \$1 4. : Ordinary Annuity of \$1 5. : Annuity Due of \$1 Money variables value value Annuity Other variables rate (per period) Time (number of periods) Annuity timing (beginning or end of period) Problem Solving Memorize These Formulas What are you given? What do you need to compute? Draw a timeline Carefully count periods Write down formulas (FV=PV FV\$1) Solve for unknowns Double check what you need to calc Ask: Does answer make sense? 29 value PV\$1 factor = value Annuity PVAnnuity\$1 factor = value value FV\$1 factor = value Annuity FVAnnuity\$1 factor = value 30 5

6 Learning Objectives Time Of Money Compute future value of single amount Time Of Money Given PV Calculate FV Single amount value Single amount value value Principal Original investment value Principal + interest Maturity value Principal 33 Accumulating interest 34 Given PV Calculate FV If we make an investment today, how much will it grow to in the future? Given PV Calculate FV Invest \$10,000 today and earn 20% compounded quarterly for three years Calculate future value compounding periods Today \$10,000 Unknown Unknown

7 Given PV Calculate FV Invest \$10,000 today and earn 20% compounded quarterly for three years Calculate future value Data Given value \$10,000 rate per year 20% Number of years 3 Compounding periods per year 4 rate per period 5% Number of periods Data Given value \$10,000 rate per year 20% Number of years 3 Compounding periods per year 4 rate per period 5% Number of periods 12 of \$1 Periods 4% 5% 6% See TVM tables on WebAccess 38 of \$1 Periods 4% 5% 6% value value FV\$1 factor = Calculation of PV FV\$1 = FV \$10, = FV \$17,960 = FV value Given PV Calculate FV How Does it Work? Invest \$10,000 today and earn 20% compounded quarterly for three years Calculate future value Given present value calculate future value Given: value (PV) \$4,000 rate per year (R) 10% Years of investment (Y) 3 Compounding periods per year (c) 2 Calculate: rate per period (i = R / c) 5% Number of periods (n = Y c) 6 value of \$1 factor \$10,000 \$7,960 \$17,960 value \$5,

8 Period Given present value calculate future value Given: value (PV) \$4,000 rate per year (R) 10% Years of investment (Y) 3 Compounding periods per year (c) 2 Calculate: rate per period (i = R / c) 5% Number of periods (n = Y c) 6 value of \$1 factor value \$5,360 Beginning Ending 1 4, , , , , , , , , , , ,360 Given PV Calculate FV value FV\$1 factor = value value = value FV\$1 factor FV\$1 factor = value value 44 Learning Objectives Compute present value of single amount Given FV Calculate PV How much of future amount is original investment (principal)? Today Principal compounding periods 45 Discounting interest 46 Given FV Calculate PV Need \$90,000 at end of five years, earn 12% compounded semi-annually Calculate present value Given FV Calculate PV Need \$90,000 at end of five years, earn 12% compounded semi-annually Calculate present value Unknown Unknown \$90, Data Given value \$90,000 rate per year 12% Number of years 5 Compounding periods per year 2 rate per period 6% Number of periods

9 Data Given value \$90,000 rate per year 12% Number of years 5 Compounding periods per year 2 rate per period 6% Number of periods 10 of \$1 Periods 5% 6% 7% of \$1 Periods 5% 6% 7% value PV\$1 factor = value Calculation of FV PV\$1 = PV \$90, = PV \$50,220 = PV Given FV Calculate PV value Need \$90,000 at end of five years, earn 12% compounded semi-annually Calculate present value value \$50,220 \$39,780 \$90, How Does it Work? Given future value calculate present value Given: value (FV) \$100,000 rate per year (R) 14% Years of investment (Y) 6 Compounding periods per year (c) 1 Calculate: rate per period (i = R / c) 14% Number of periods (n = Y c) 6 value of \$1 factor value \$45, Period Given future value calculate present value Given: value (FV) \$100,000 rate per year (R) 14% Years of investment (Y) 6 Compounding periods per year (c) 1 Calculate: rate per period (i = R / c) 14% Number of periods (n = Y c) 6 value of \$1 factor value \$45,600 Beginning Ending 1 45,600 6,384 51, ,984 7,278 59, ,262 8,297 67, ,559 9,458 77, ,017 10,782 87, ,799 12, ,000 9

10 Given FV Calculate PV Learning Objectives PV\$1 factor = value value = PV\$1 factor Given present value and future value, solve for interest rate or number of periods PV\$1 factor = value Solving for Other s Four variables in time value of money Given three calculate fourth Manipulating Equation value FV\$1 factor = value FV = PV (1 + i) n Rate Number of Compounding Periods value FV\$1 factor = = value FV\$1 factor value value Calculate Rate: FV\$1 Table Borrow \$1,000 today and repay \$1,082 at end of two periods Calculate interest rate per period \$1,000 \$82 \$1, Calculate Rate: FV\$1 Table Borrow \$1,000 today and repay \$1,082 at end of two periods Calculate interest rate per period Calculation of FV\$1 Factor PV FV\$1 = FV \$1,000 FV \$1 = \$1,082 FV\$1 = \$1,082 \$1,000 FV\$1 = See row 2 of FV\$1 table 60 10

11 of \$1 Table PV FV\$1 = FV \$1,000 FV \$1 = \$1,082 FV\$1 = \$1,082 \$1,000 FV\$1 = See row 2 of FV\$1 table of \$1 Periods 3% 4% 5% Manipulating Equation PV\$1 factor = value value = PV\$1 factor PV\$1 factor = value Solve this question using TVM calculator, not TVM table Calculate Rate: PV\$1 Table Borrow \$1,000 today and repay \$1,082 at end of two periods Calculate interest rate per period of \$1 Table FV PV\$1 = PV \$1,082 PV \$1 = \$1,000 PV\$1 = \$1,000 \$1,082 PV\$1 = See row 2 of PV\$1 table 63 of \$1 Table FV PV\$1 = PV \$1,092 PV \$1 = \$1,000 PV\$1 = \$1,000 \$1,082 PV\$1 = See row 2 of PV\$1 table of \$1 Periods 3% 4% 5% Solve this question using TVM calculator, not TVM table 64 Calculate Periods: FV\$1 Table Calculate Periods: FV\$1 Table Deposit \$47,811 today and accumulate \$70,000 at 10% compounded annually Calculate number of periods \$47,811 \$22,189 \$70, Deposit \$47,811 today and accumulate \$70,000 at 10% compounded annually Calculate number of periods using FV of \$1 Table PV FV\$1 = FV \$47,811 FV \$1 = \$70,000 FV\$1 = \$70,000 \$47,811 FV\$1 = See 10% column of FV\$1 table 66 11

12 of \$1 Table PV FV\$1 = FV \$47,811 FV \$1 = \$70,000 FV\$1 = \$70,000 \$47,811 FV\$1 = See 10% column of FV\$1 table of \$1 Periods 9% 10% 11% Solve this question using TVM calculator, not TVM table 67 Calculate Periods: PV\$1 Table Deposit \$47,811 today and accumulate \$70,000 at 10% compounded annually Calculate number of periods using PV of \$1 Table FV PV\$1 = PV \$70,000 PV \$1 = \$47,811 PV\$1 = \$47,811 \$70,000 PV\$1 = See 10% column of PV\$1 table 68 of \$1 Table FV PV\$1 = PV \$70,000 PV \$1 = \$47,811 PV\$1 = \$47,811 \$70,000 PV\$1 = See 10% column of PV\$1 table of \$1 Periods 9% 10% 11% Solve this question using TVM calculator, not TVM table 69 Learning Objectives Explain the difference between an ordinary annuity and an annuity due Compute the future value of both an ordinary annuity and an annuity due 70 Annuities Ordinary Annuity Series of equal periodic payments Equal amounts Equal time periods Defined period of time Payments made at end of period compounding periods Pay \$2,500 at the end of each quarter for five years Financial calculators: PMT key, specify END or BEG Payments are called Rents No payment Payment 1 Payment 2 Payment By default use ordinary annuity unless told otherwise 72 12

13 Annuity Due (in Advance) Payments made at beginning of period compounding periods Payment 1 Payment 2 Payment No payment Only use annuity due when specifically stated 73 Annuity amount: \$10,000 rate per period: 4% Ordinary Annuity Period Beginning Payment Ending ,000 10, , ,000 20, , ,000 31,216 Annuity Due Period Payment Beginning Ending 1 10,000 10, , ,000 20, , ,000 31,216 1,249 32, Ordinary Annuity Make regular principal investments Calculate future value Ordinary Annuity Equal payments made each period Payments, interest accumulate compounding periods Principal 75 Today Payment 1 Payment 2 Payment of Ordinary Annuity of \$1 Periods 14% 15% 16% Given Annuity Calculate FV Invest \$5,000 at end of each quarter, at 16% compounded quarterly, for 5 years Calculate future value Annuity FVAnnuity\$1 factor = value Calculation of FV of Ordinary Annuity Annuity FVAnnuity\$1 factor = FV \$1, = FV \$4,993 = FV 77 Data Given Annuity \$5,000 rate per year 16% Number of years 5 Compounding periods per year 4 rate per period 4% Number of periods

14 Data Given Annuity \$5,000 rate per year 16% Number of years 5 Compounding periods per year 4 rate per period 4% Number of periods 20 of Ordinary Annuity of \$1 Periods 3% 4% 5% of Ordinary Annuity of \$1 Periods 3% 4% 5% Annuity FVAnnuity\$1 factor = value Calculation of FV of Ordinary Annuity Annuity FVAnnuity\$1 factor = FV \$5, = FV \$148,890 = FV 80 Ordinary Annuity How Does it Work? value Annuity (Amount number) 81 Given ordinary annuity calculate future value Given: Annuity [also called PMT] \$10,000 rate per year (R) 8% Years of investment (Y) 3 Payments / compounding periods per year (c) 2 Calculate: rate per period (i = R / c) 4% Number of periods (n = Y c) 6 value of ordinary annuity of \$1 factor value of ordinary annuity \$66, Period Given ordinary annuity calculate future value Given: Annuity [also called PMT] \$10,000 rate per year (R) 8% Years of investment (Y) 3 Payments / compounding periods per year (c) 2 Calculate: rate per period (i = R / c) 4% Number of periods (n = Y c) 6 value of ordinary annuity of \$1 factor value of ordinary annuity \$66,330 Beginning Payment Ending ,000 10, , ,000 20, , ,000 31, ,216 1,249 10,000 42, ,465 1,699 10,000 54, ,164 2,166 10,000 66,330 Ordinary Annuity Annuity FVAnnuity\$1 factor = value Annuity = value FVAnnuity\$1 factor FVAnnuity\$1 factor = value Annuity 84 14

15 Annuity Due Similar calculations Use FV annuity due table Use FV ordinary ann table (1 + rate) of Ordinary Annuity of \$1 Periods 3% 4% 5% Calculation of FV of Ordinary Annuity Annuity FVAnnuity\$1 factor = FV \$5, = FV \$148,890 = FV Calculation of FV of Annuity Due FV Ordinary Annuity (1 + rate) = FV Annuity Due \$148,890 ( ) = FV Annuity Due 85 \$154,846 = FV Annuity Due 86 Annuity amount: \$10,000 rate per period: 4% Ordinary Annuity Period Beginning Payment Ending ,000 10, , ,000 20, , ,000 31,216 Annuity Due Period Payment Beginning Ending 1 10,000 10, , ,000 20, , ,000 31,216 1,249 32, Learning Objectives Compute the present value of an ordinary annuity and an annuity due 88 PV Ordinary Annuity What amount today is equivalent to a series of payments in the future? PV Ordinary Annuity Withdraw \$10,000 at end of each year For 4 years Earn 10% compounded annually How much do you need to invest today? Today Payment 1 Payment 2 Payment 3 Principal

16 Given Annuity Calculate PV Pay \$7,000 at end of each six months, 10% compounded semi-annually,7years Calculate present value Data Given Annuity \$7,000 rate per year 10% Number of years 7 Compounding periods per year 2 rate per period 5% Number of periods Data Given Annuity \$7,000 rate per year 10% Number of years 7 Compounding periods per year 2 rate per period 5% Number of periods 14 of Ordinary Annuity of \$1 Periods 4% 5% 6% of Ordinary Annuity of \$1 Periods 4% 5% 6% PV Ordinary Annuity Annuity (Amount number) Annuity PVAnnuity\$1 factor = value value Calculation of PV of Ordinary Annuity Annuity PVAnnuity\$1 factor = PV \$7, = PV \$69,293 = PV How Does it Work? Given ordinary annuity calculate present value Given: Annuity [also called PMT] \$2,500 rate per year (R) 7% Years of investment (Y) 6 Payments / compounding periods per year (c) 1 Calculate: rate per period (i = R / c) 7% Number of periods (n = Y c) 6 value of ordinary annuity of \$1 factor value of ordinary annuity \$11, Period Given ordinary annuity calculate present value Given: Annuity [also called PMT] \$2,500 rate per year (R) 7% Years of investment (Y) 6 Payments / compounding periods per year (c) 1 Calculate: rate per period (i = R / c) 7% Number of periods (n = Y c) 6 value of ordinary annuity of \$1 factor value of ordinary annuity \$11,918 Beginning Payment Reduction Ending 1 11, ,500 1,666 10, , ,500 1,782 8, , ,500 1,907 6, , ,500 2,041 4, , ,500 2,183 2, , ,500 2,

17 PV Ordinary Annuity Annuity Due Annuity PVAnnuity\$1 factor = value value Annuity = PVAnnuity\$1 factor Similar calculations Use PV annuity due table Use PV ordinary ann table (1 + rate) PVAnnuity\$1 factor = value Annuity Learning Objectives Annuity problems: Solving for annuity amount, interest rate, number of periods Manipulating Equation Annuity PVAnnuity\$1 factor = value value of ordinary annuity used as example Annuity = value PVAnnuity\$1 factor PVAnnuity\$1 factor = value Annuity Calculate Annuity Borrow \$39,550 for 5 years at 24% interest, compounded semi-annually Calculate semi-annual annuity amount of Annuity of \$1 Periods 11% 12% 13% of Annuity of \$1 Periods 11% 12% 13% of Annuity of \$1 Annuity PVAnnuity\$1 factor = PV Annuity = \$39,550 Annuity = \$39, Annuity = \$7,

18 Calculate Rate Borrow \$20,442 today and pay \$3,000 at end of each period for 12 periods Calculate interest rate per period of Annuity of \$1 Annuity PVAnnuity\$1 factor = PV \$3,000 PVAnnuity\$1 = \$20,442 PVAnnuity\$1 = \$20,442 \$3,000 PVAnnuity\$1 = See row 12 of PVAnnuity\$1 table 103 of Annuity of \$1 Annuity PVAnnuity\$1 factor = PV \$3,000 PVAnnuity\$1 = \$20,442 PVAnnuity\$1 = \$20,442 \$3,000 PVAnnuity\$1 = See row 12 of PVAnnuity\$1 table of Annuity of \$1 Periods 9% 10% 11% Solve this question using TVM calculator, not TVM table 104 Calculate Periods Borrow \$17,118 today and pay \$2,000 at end of each period at 8% per period Calculate number of periods of Annuity of \$1 Annuity PVAnnuity\$1 factor = PV \$2,000 PVAnnuity\$1 = \$17,118 PVAnnuity\$1 = \$17,118 \$2,000 PVAnnuity\$1 = See 8% column of PVAnnuity\$1 table 105 of Annuity of \$1 Annuity PVAnnuity\$1 factor = PV \$2,000 PVAnnuity\$1 = \$17,118 PVAnnuity\$1 = \$17,118 \$2,000 PVAnnuity\$1 = See 8% column of PVAnnuity\$1 table of Annuity of \$1 Periods 7% 8% 9% Solve this question using TVM calculator, not TVM table 106 Learning Objectives Compute the present value of a deferred annuity PV of Deferred Annuity First cash flow of annuity occurs more than one period in future

19 PV of Deferred Annuity Today: January 1, 2010 Beginning: December 31, 2012 Annuity will pay \$12,500 a year At end of each year for 2 years? \$12,500 \$12,500 Rate of return, 12% Calculate PV 1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/ \$12,500 \$12,500 PV of Deferred Annuity: #1? \$12,500 \$12,500 1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 Two Step Process 1. Calculate PV of annuity as of beginning of annuity period 2. Discount single value to its present value at time zero 1/1/10 12/31/10 12/31/11 12/31/12 12/31/ PV of Deferred Annuity: #1? \$12,500 \$12,500 1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 PV of Deferred Annuity: #2 PV annuity for period with no payments = \$12,500 \$12,500 1/1/10 12/31/10 12/31/11 12/31/12 12/31/13 PV single amount n = 2, i = 12% FV = \$21,126 PV factor = PV = \$16,841 PV ordinary annuity n = 2, i = 12% Annuity = \$12,500 PV factor = PV = \$21, PV annuity for entire period = \$12,500 ( ) = \$16, Learning Objectives Application of time value of money Notes receivable / Notes payable Bonds Effective interest amortization Expected cash flow Monetary Assets, Liabilities Monetary assets Cash and claims to receive cash Amount fixed or determinable Monetary liabilities Obligations to pay cash Amount fixed or determinable

20 Monetary Assets, Liabilities Time frame important Cash exchanged one year or less at face value Use simple interest (if interest rate stated) Cost of using PV > benefit Monetary Assets, Liabilities Time frame important Cash exchanged more than one year Use compound interest at present value of future cash flows Receive utility bill, \$500; pay in 30 days Made sale on account, \$1,000; collect in 60 days Loan \$15,000 to vendor, 8% interest, due in 90 days Note With Rate Purchase equipment Sign note, face value, \$1,000 rate, 4.5% Market rate, 4.5% Due in two years Pay \$1,092 in two years (FV) Jan 1 Equipment 1, Note payable 1, Note With Rate Dec 31 expense (1, %) payable 45 Dec 31 expense (1, %) payable 47 Dec 31 Note payable 1, payable 92 Cash 1, Unreasonable Stated Rate Exchanging cash for non-cash asset Time frame greater than one year Discount future amount at market rate Stated rate on note 15%, market rate for borrower 6% Differing Rates 1 Purchase inventory on January 2, 2011 FMV inventory unknown Seller accepts note Face value, \$100,000 Stated interest rate, 2% Term, 4 years (due 12/31/2014) Buyer s interest rate from bank, 10%

21 Differing Rates 1 Differing Rates 1 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Calculation of i = 2% (stated rate), n=4 PV FV\$1 = FV \$100, = FV \$108,243 = FV 121 Calculation of i = 10% (market rate), n=4 FV PV\$1 = PV \$108, = PV \$73,931 = PV 122 Differing Rates 1 1/2/11 Inventory 73,931 Discount on note payable 34,312 Note payable 108,243 Differing Rates 1 Recognize interest expense for period 1/2/11 Inventory 73,931 Discount on note payable 34,312 Note payable 108,243 Dec 31 expense (73,931 10%) 7, Discount on note payable 7,393 Dec 31 Int exp ((73, ,393) 10%) 8, Discount on note payable 8, Differing Rates 2 Purchase inventory on January 2, 2011 FMV inventory unknown Seller accepts note Face value, \$100,000 Stated interest rate, 8% Term, 4 years (due 12/31/2014) Buyer s interest rate from bank, 5% Differing Rates 2 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Calculation of i = 8% (stated rate), n=4 PV FV\$1 = FV \$100, = FV 125 \$136,049 = FV

22 Differing Rates 2 Differing Rates 2 1/2/11 12/31/11 12/31/12 12/31/13 12/31/14 Calculation of i = 5% (market rate), n=4 FV PV\$1 = PV \$136, = PV \$111,928 = PV 127 1/2/11 Inventory 111,928 Discount on note payable 24,121 Note payable 136,049 Dec 31 expense (111,928 5%) 5, Discount on note payable 5,596 Dec 31 Int exp ((111, ,596) 5%) 5, Discount on note payable 5, Learning Objectives Bonds issued at discounts, premiums Effective interest amortization Need Two Billion Dollars Intel needs cash to build new factory Large debt broken into small pieces \$2,000,000, Sell 2,000,000 \$1,000 bonds 130 Bonds Receive cash when issued Promise to pay Face value on maturity date (future value) semiannually (ordinary annuity) Issue price of bond is PV of future value + PV of ordinary annuity 131 Issued At Par General Electric issued bonds Face value of \$50 million Mature in five years Coupon interest rate of 9% Issued par, market rate = coupon rate Cash 50,000,000 Bonds payable 50,000,

23 Calculate Annuity Coupon rate used to compute annuity (periodic interest payments) payment = Face value Coupon rate Time Payments Semi-annual interest payments I = P R T I = \$50,000, /12 I = \$2,250,000 only loan 133 expense 2,250,000 Cash 2,250, Payment At Maturity Make last interest payment expense 2,250,000 Cash 2,250,000 Pay principal (face value) in full Bonds payable 50,000,000 Two Rates Rate printed on bond called Coupon rate Stated rate Contract rate Market interest rate called Effective-rate Yield-to-maturity Cash 50,000, Two Rates Coupon interest rate Determines semi-annual payment Market interest rate Determines bond market price (PV) Effective interest expense Market rate > coupon rate, bond issued at discount Coupon rate > market rate, bond issued at premium 137 Issued At Discount A \$1,000 bond issued at a discount Market rate > coupon rate Bond sells for less than face value For example Quoted at 88 3/8 Sells for 88 3/8% of face value Bought or sold for \$

24 Bond issued at discount Face value \$5,000 Term 3 years Coupon interest rate 7% Market interest rate 10% Compounded semi-annually Market interest rate per period 5% Number of periods 6 Bond Principal payment = Face value Coupon rate Time \$175 = \$5, /2 Issue price \$4,618 Discount \$ Payment 1 Payment 2 Payment 3 Single amt 140 Bond Issue Price Of Bond \$3,730 Discount at market rate, 10%, semi-annually \$5,000 of the Face (a single amount) + of the Payments (an annuity) = value of Bond (Issue Price of the Bond) \$888 \$175 \$175 \$175 \$4,618 Payment 1 Payment 2 Payment 3 Single amt 141 Use market rate of interest to calculate present value 142 Issue Price Of Bond \$ 3,730 of the Face of the Annuity = \$ 4,618 of the Bonds Bond discount effective-interest amortization schedule (A) Beginning A*MR*1/2= (B) Effective (C) Annuity Payment B C= (D) Discount Amortized F(up) D= (F) Discount Remaining A+D= (G) Ending , , , , , , , , , , , , ,000 Also called issue price of bonds, or market value of bonds 143 Effective interest expense Beg Bal Market Rate Time \$4,618 10% 1/2 = \$231 payment Face value Coupon rate time \$5,000 7% 1/2 = \$

25 Zero Coupon Bond Zero Coupon Bond Principal of the Face (a single payment) + of the Payments (an annuity) = Issue Price of the Bond Payment 1 Payment 2 Payment 3 Single amt Issued At Premium A \$1,000 bond issued at a premium Market rate < coupon rate Bond sells for more than face value For example Quoted at 110 ¼ Sells for % of face value Bought or sold for \$1, Bond issued at premium Face value \$6,000 Term 3 years Coupon interest rate 12% Market interest rate 8% Compounded semi-annually Market interest rate per period 6% Number of periods 6 payment = Face value Coupon rate Time \$360 = \$6, /2 Issue price \$6,627 Discount \$ Of Bond \$ 4,740 of the Face + 1,887 of the Annuity = \$ 6,627 of the Bonds \$6,627 is greater than face amount of \$6,000, bonds are issued at premium of \$ Bond premium effective-interest amortization schedule (A) Beginning A*MR*1/2= (B) Effective (C) Annuity Payment C B= (D) Premium Amortized F(up) D= (F) Premium Remaining A D= (G) Ending , , , , , , , , , , , , ,000 Effective interest expense Beg Bal Market Rate Time \$6,627 8% 1/2 = \$265 payment Face value Coupon rate time \$6,000 12% 1/2 = \$

26 Learning Objectives Expected cash flow Amount Timing Uncertainty Cash Flow Issues Expected Cash Flows Concepts Statement No. 7 requires expected cash flow approach that uses a range of cash flows and incorporates the probabilities of those cash flows FASB states a company should discount expected cash flows by the risk-free rate of return Expected Cash Flows Pure Rate (2% to 4%) No possibility of default No expectation of inflation Expected Inflation Rate (0% or more) Credit Risk Rate ( 0% or more) Risk-Free Rate of Return Pure rate + Expected inflation rate = Risk Free Rate Expected Cash Flows Expected Cash Flows cash flow uncertain Estimate amount using expected value Discount to PV using risk-free rate Expected value paid at end of 5 years Assume risk free rate of 5% Calculate present value Amount Probability Expected Amount Probability Expected \$100,000 10% \$10,000 \$100,000 10% \$10,000 \$200,000 60% \$120,000 \$200,000 60% \$120,000 \$300,000 30% \$90,000 \$300,000 30% \$90,000 Expected value \$220, Expected value \$220,

### CHAPTER 6. Accounting and the Time Value of Money. 2. Use of tables. 13, 14 8 1. a. Unknown future amount. 7, 19 1, 5, 13 2, 3, 4, 7

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