INTRODUCTORY FINANCE BCOR 2200
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Chapter 4 Introduction to Time Value of Money Chapter Outline 41 Future Value and Compounding 42 Present Value and Discounting 43 More on PV and FV Key Concepts and Skills Be able to compute the future value of an investment made today Be able to compute the present value of cash to be received at some future date Be able to compute the return on an investment Be able to compute the number of periods required for an investment to grow to certain amount given a growth rate Learn the Rule of 72 s Here is the Idea Get 100 today or Get 100 in one year Which is better Obviously getting the 100 today is better Why If you want to buy something today you can If you want to buy something in one year You can lend the 100 today for one year And have more than 100 in one year So if I won t get the money for one year I need to get more than 100 How much more Talk about that in a soon What we will do Chapter 4 How much is a payment worth today PV if we get it sometime in the future FV Given the FV Calculate the PV What is a payment worth in the future FV if we make the payment today PV Given the PV Calculate the FV Chapter 5 What if there are a whole bunch of payments The Notation FV PV1rt OR PV FV1rt FV Future Value PV Present Value r The interest rate t the number of periods years We can solve for each of these variables If we have the other 3 And talk about what the variables mean 41 Future Value and Compounding Future Value What will a payment made today be worth later Save 100 for 1 year at 10 interest What will we have in 1 year t1 r10 PV100 FV In 1 yearthe FV PV1 r1 100111 110 Save 100 for 2 years at 10 interest Leave 110 in bank for a second year 11011 121 1001111 100112 General Notation 1001 rt 1 rt is sometimes called the Future value Interest Factor Table 42 r 7 v I i 0 7 if 5 Number of l e Bales w Future value interest periods 5 10 15 20 factors f i i 77 quot quot39i J39quotquot 1 10500 11000 11500 12000 2 11025 12100 13225 14400 3 11576 13310 15209 117280 4 142155 14641 17490 20736 5 1127183 16105 20114 248133 80 100 in 5 years is worth 10016105 16105 This is equal to 1001105 16105 But nobody uses tables anymore Use your calculator Simple Interest vs Compound Interest 100 in Five Years at 10 per year Interest Factor 1 rt 115 161051 Future Value 100161051 16105 10 Interest on the 100 in each year is 10 Over five years it is 50 The extra 6105 50 1105 is interest on interest aka COMPOUND INTEREST Using the Calculator Financial Calculators have a Time Value of Money TVM function N number of periods YR interest per period year PMT periodic payments will get to this later PV Present Value FV Future Value N OPV PMT FVN i1 r r With PMT O W pV 1rN Using the Calculator Note the Negative in front of the PV The Idea is you are paying the PV so it is an outflow Be conscious of this when using the TVM function Make sure your calculator is set to 1 period per year Payments Occur at the end of the period N5 YR1O PV1OO FV16105 OR N5 YR1O PV1OO FV16105 42 Present Value and Discounting What is the PV of 1 000 in one year discounted at 7 How much put away now so I have 1000 in one year What is 1000 paid in one year worth today PV 1 0001 07 93458 Or TVM Function on calculator FV 1000 r 7 N 1 PV 93458 1000 in two years discounted to today at 7 PV 1 0001 072 87344 Or TVM Function on calculator FV 1000 r 7 N 2 PV 87344 See Page 99 and Appendix D for Calculator instructions Clicker Question What is the PV of 10000 paid in 5 years at 12 What is the FV in 8 years of 30000 paid today at 9 Note Even though the calculator Will show negative numbers as F V and PV outputs we Will report positive numbers PV 16000 and FV 49200 PV 10000 and FV 49200 PV 4000 and FV 49200 PV 4000 and FV 59777 PV 5674 and FV 59777 F1901 Clicker Answer PV of10000 paid in 5 years at 12 N 5 YR 12 FV 10000 PV 5674 FV in 8 years of 30000 paid today at 9 N 8 YR 9 PV 30000 FV 59777 PV 5674 and FV 59777 The answer is E 43 More on PV and FV Some examples Example Your company can pay 800 for an asset it believes it can sell for 1200 in 5 yrs Similarinvestments pay 10 What does similar mean Another investment with the same risk a stock or bond issued by another company pays 10 So is paying 800 for something that can be sold in 5 yrs for 1200 a good idea PV FV1rt 1 2001 15 74511 Or N 5 FV 1200 r 10 PV74511 FV PV1rt 800115 128841 Or N 5 PV 800 r1O FV 128841 Bad Idea Either You should pay less than 800 pay 74511 to get 1200 You should pay 800 to get more than 1200 get 128841 Clicker Question An investment costs 20000 and you expect to hold it for 10 years Investments with similar risk earn 12 per year If the projected sale price is 75000 is this a good investment idea A YES B NO C MAYBE Clicker Answer If the projected sale price exceeds the calculated FV then it is a good idea FV PV1rt 200001121O 20000310585 62117 Or using the Calculator TVM function N 10 YR 12 PV 20000 FV 62117 75000 gt 62117 so invest Here s the idea Over 10 years the invest which costs 20k and pays 75k has a return greater than 12 If it paid only 62117 the return would be 12 Since it pays more 75k it must have a higher return than the required return Determine the Discount Rate Solve for r aka IIY R PV1 rt FV 1 rt FVPV 1 r FVPVlt1tgt r FVPV1t 1 o What rate is need to increase 200 to 400 in 10 years r 400200110 1 0071 718 o What rate is need to increase 200 to 400 in 8 years r 40020018 1 00905 905 Calculator N 8 PV 200 FV 400 YR not found N 8 PV 200 FV 400 YR 905 N 8 PV 200 FV 400 YR 905 Clicker Question What rate is needed to increase 20000 to 80000 in 10 years Rate not Found 718 1487 0 2000 3000 Clicker Answer r FVIPV1 0 1 8020i1quot 1 4lt1 1 39gt 1 01487 1487 Or N 10 FV20 FV 80 IYR 1487 The answer is C Determine the Number of Periods Solve for t or N PV1 rt FV 1 rt FVPV In1 rt InFVPV tIn1 r InFVPV t lnFVPVln1 r But we ll just use the machine How many years are needed to increase 200 to 400 at 718 Calculator PV 200 FV 400 lYR 718 N 99964 z 10 Calculator PV 200 FV 400 lYR 718 N 99964 z 10 Note PV and FV or PV and FV both work How many years are needed to increase 200 to 400 at 905 Calculator PV 200 FV 400 YR 905 N 80007 z 8 See page 110 for Excel tips for calculating FV PV r amp N in Excel 20 Clicker Question How many years are needed to get 1000000 if you invest 22095 at 10 per year 2026 22 2226 4O Not Found Clicker Answer t lnFVPVIn1 r In1000000220950In117 40 Or PV 22095 FV 1000000 IIYR 10 N 40 The answer is D Rule of 72 s If FVPV 2 your money doubles then rt z 72 Example Start with 200 and get 400 then FVPV 2 200 to 400 in 10 years at 718 10718 718 z 72 200 to 400 in 8 years at 905 8905 724 z 72 If the value of your stock doubles in 5 years what is the approximate annualized compounded return rt z 72 r5 z 72 r z 725 144 23 Clicker Question You own a house that you believe has doubled in value over the last 20 years Using the Rule of 7239s estimate the approximate annual return on the house 360 525 1000 A B C 720 D E 2000 Clicker Answer rt z 72 r20 z 72 r20 7220 36 That answer is A Check this answers using the calculator s TVM function N 20 PV 1 FV 2 lYR 353 z 36 Bonus Question Assume you will earn 12 How long to quadruple in price Quadruple is double twice rt z 72 72r zt 7212 6 years So double in 6 double twice in approximately 12 years Check this r 12 PV 1 FV 4 N 1223 z 12 25 Recap FV PV1rt Solve for any of the four variables 1 FV 2 PV 3 r also called lYR 4 talso called N Use the TVM function on the calculator to solve for the one variable not given Be sure to understand the economic meaning of the values 26 Table 44 Page 112 IIL Symbel e Pi Present value whet l uture eeeh flewe are werth ledey Flt Future value whet eeeh llewe ere warlh In the luture r Interest rate rate of return er dieeeunt rate per penned ty pipelly39 but not elweye one year t 7 Number of periods typically but net always the number of years G 39 Ceeh emeunl Future value at C invested at rpereenl per pellet fer tperieds Fm C24l1 rl39 The term 1 I rl le called the future value teeter Present value at Ste be received in tperiede at r percent per peried Pi Gill r39 The term hill I rifle celled the presenrvelue fetter The heeie present value equetien giving the relationship between present and future value is Pl 1 FWli r39 27 Chapter 3 Working with Financial Statements Chapter Outline 31 Standardized Financial Statements 32 Ratio Analysis 33 The Du Pont Identity 34 Internal and Sustainable Growth 35 Using Financial Statement Information Key Concepts and Skills Know how to standardize financial statements for comparison purposes Know how to compute and interpret important financial ratios Know the determinants of a firm s profitability and growth Understand the problems and pitfalls in financial statement analysis 31 Standardized Financial Statements The basic idea 0 Divide everything on the page by the biggest number on the page Balance Sheet The biggest number is Total Assets or Liabilities plus Equity Income Statement The biggest number is Sales Standardized Balance Sheet Table 31 my man canmxmnu man 1 nl amquot an mm mm mm s 1 mm 2097 ms Mwa mm aims isquot mm mama s mum mm mm ussds w mm and mumam w assms mum and Owners Equity 2mm cum m4quot mm m m mm ma mm habumasann mm em Standardized Balance Sheet Table 32 Standardized Bis mmocx cnnmnmm Cummansm mm sn I ummm 3 2001 m zon 2mm 2508 Enema mm Gwen ages cm n m saws NEW Ham m mm mm assm lelmus nu Dwnnrs Emmy mm mums CMMD mm mmsmpms mm unm nas mm W ab mesnw My my Standardized Balance Sheet This is Table 31 laying over Table 32 Pnumocx canvaxmnu cnmnm srnu sum a nmnm L um um mm in mm shun a m mmlun n 2001 mam mm 2m 7 ma Chem hum Cumquot ism mm mm isseb w um and cumww 19mm umnuu m Ownw 5qu mm mm Anmums mm Nmes mm a w mm mm a x mums enva Commquot mm mm swab mumn mm mm anamamhuzsann My aw Standardized Income Statement This is Table 33 laying over Table 34 PHUFFIDEK EDHFDHA HUH lu39l39l H lnsuma Statement aiammt 395 In millims Baez3 533 1 1m L3 Gas var gems said 1 44 ea 2 E eprermrnzquotl 2TB 11 Earmnge befare ir39al rexel a i EELer 395 E91 23939 Inleresl paid 141 1 Taxable Incarna E EEilll 935 Times 31 15 H I Met Inecume 363 TE 7 E39M391ElrSI39IESS 1231 52a in amen m rammed earnings 91 115 Some other stuff we will need later for the quotRatio Analysis EBTNnt ExpTax Exp363141187691 EBITDA NI Int Exp Tax Exp Dep Exp 691 276 967 32 Ratio Analysis Instead of values show as fractions of other values ratios Ratio Categories 1 Short Term Solvency Firm s ability to pay current bills 2 Long Term Solvency Firm s ability to meet LT debt obligations 3 Asset Management aka Turnover Ratios Efficiency measures 4 Profitability Ratios 5 Market Value Ratios Some things to think about as we look at ratios 1 Definition of the Ratio How is it computed Is it always the same It will be for us Sometime the ending B S value is used Sometimes the Average value 2 What is the unit of Measure Dollars years Dollars of assets 3 What are HIGH values What are LOW values For the company over time for the industry for the sector for all companies 10 Category 1 ShortTerm Solvency First Some Notation Current Assets CA Current Liabilities CL Total Assets TA A Total Liabilities TL L Total Debt TD Debt D Total Equity TE E 31 Current Ratio CACL 708540 131 32 Quick Ratio CA nvCL 708 44254O 053 Inventory is the least liquid current asset 33 Cash Ratio CashCL 98540 018 Cash is the most liquid current asset Clicker Question CurrentAssets 2007 2008 Current Liabilities 2007 2008 I TA Did the firm s shortterm solvency improve or deteriorate between the end of 2007 and then end of 2008 A ShortTerm Solvency Improved B ShortTerm Solvency deteriorated C The evidence is mixed Clicker Answer quotties 2007 2008 Quick Ratio CL 7os3944254o 053 1 Each ShortTerm Solvency ratio is higher at the end of 2008 The answer is A Category 2 LongTerm Solvency Balance Sheet 34 Total Debt Ratio Debt to Assets DA 9973588 028 Equity to Assets EA 25913588 072 35 Debt to Equity DE 9972591 038 028078 039 z 038 36 Equity Multiplier EM AE 35882591 138 1EA 1o72 138 1DE1038138 EM AE D EE EE DE 1 DE Note that all these are also quotFinancial Leverage Measures In general the more levered the less likely a firm is to repay its debt Category 2 LongTerm Solvency Continued Income Statement 37 Times Interest Earned TIE EBITIn Exp 691141 49 times 38 Cash Coverage EBITDAIn Exp 967141 69 times Clicker Question Current Assets 2007 2008 Liab and Equity 2007 2008 I 09 LT Ass ts 3990 31950 i Did the firm s Longterm solvency improve or deteriorate between the end of 2007 and then end of 2008 A LongTerm Solvency Improved B LongTerm Solvency Deteriorated Clicker Answer 2007 gmia 2008 g aa Current Assets f saae Liab and Equity quot I u LT solvency deteriorated Leverage increased The firm borrowed 200 and bought new assets Category 3 Asset Management or Efficiency 39 Inventory Turnover COGSInventory 1344422 32 times This is the value of inventory sold during the year COGS divided by the amount of inventory on hand at the end of the year 2008 So during the year the firm sold 32 times amount of inventory on hand at year s end 310 Days Sales in Inventory 365lnventory Turnover 36532 114 days Since the firm sold the current amount of inventory 32 times over the last year the current inventory will be sold in 132 years or 114 days 311 Receivables Turnover SalesAR 2311188 123 times Really should be quotCredit Salesquot not total Sales We don t have that Would be number of times in the year credit was extended and then collected Often Average AR is used as opposed to Ending AR Cat 3 Asset Management or Efficiency Continued 312 Days Sales in Receivables 365Receivables Turnover 365123 30 days Credit was extended and collected 123 times over the last year or 1123 years or 30 days 313 Asset Turnover SalesAssets 23113588 064 For each dollar of assets the firm generated 064 in sales Capital Intensity AssetsSales 35882311 156 It takes 156 in Assets to generate 1 in Sales Clicker Question Account Calculate 64 I T Inv soldInv on hand 5 DSI Days per year of times Inv turned in per year 399 O 1 AT Amount soldAmount quotemployedquot to generate sales 73 04 Clicker Answer 10000 7 5000 T Inv soldInv on hand Calculate 1 Inventory Turnover IT 2 Days Sales in Inventory DSI 3 Asset Turnover AT IT COGSInv 5000S 1000 5 DSI Days per year of times Inv turned in per year DSI 365IT 3655 73 AT Amount soldAmount quotemployedquot to generate sales AT SalesAssets 1000025000 04 The Answers is E Category 4 Profitability 314 Profit Margin PM NISales 3632311 157 Income Statement quotBottom Line divided quotTop Line Accounting Profit per Dollar of Sales Think of PM as a measure of Efficiency not Profitability Measures the expenses needed to generate sales Sales Expenses NI 315 Return on Asset ROA NIAssets 3633588 10 This is the accounting profit per unit of Assets Compare with Asset Turnover Ratio AT SalesAssets 23113588 064 Sales are 64 of Assets and Profits are 10 of Assets PM ROAAT 010064 157 Think of ROA as a measure of Efficiency not Profitability Think of assets as some number of trucks How much profit is generated from these trucks The more you generate the more efficient the business It is a function of sales more is better and expenses less is better 316 Return on Equity ROE NIEquity 3632591 14 This measures the accounting profit per unit of Equity ROE ROA x EM NIAssets x AssetsEquity 50 Profit ROE is a function of Efficiency RCA and Leverage EM 50 increase profit by increasing efficiency or increasing leverage Remember Efficiency has two components Increasing Sales and Decreasing Expenses Increase Efficiency by Increasing Sales or Decreasing expenses Category 5 Market Value Measures But First of Shares Outstanding is 33 m and Price is 88 per share 317 Earnings per Share EPS NIShares 36333 11Share So each owners share earned 11 318 PriceEarnings Ratio PE or PE PriceEPS 88S11 8 times So pay 8 for 1 of earnings PE and EPS is can be compared across different stocks Why pay more for a dollar of earnings 319 MarkettoBook Price per ShareBook Val of Equity per Share Book Value of Equity per Share 259133 7852 MarkettoBook 887852 112 aka PricetoBook and is the inverse of BooktoMarket BM 23 Clicker Question 460 m 1504 m Crocks CROX Whole Foods WFMI Given the market data above which stock does the market expect to have higher growth A CROX B WFMI C There are Mixed Signals 24 Clicker Answer Whole Foods WFMI 1504 m CROX 415S110 377 46082 561 415561 074 WFMI 1831105 1744 1504140 1074 18311074 170 WFMI has a greater PE and MarkettoBook So quotthe marketquot is paying more for a dollar of WFMI s earnings and more for a dollar of its equity The answer is B 25 Recap Table 35 page 63 Shortterm solvency or liquidity ratios Current ratio Curremrasfc39lelts Current liabilities Quick ratio Current assets Inventory Current liabilities Cash ratio Longterm solvency or financial leverage ratios Total assets 7 Total equity Total assets Debtequity ratio Total debtTotal equity Equity multiplier Total assetsTotal equity Total debt ratio Times interest earned ratio EBIT Interest Cash coverage ratio W Interest V Asset utilization or turnover ratios Cost of goods sold Inventory turnover Inventory Days sales in inventory amp Inventory turnover Sales Receivables turnover Accounts receivable Cash Current liabilities IV 365 days Days sales In receivables Receivables turnover Sales Total asset turnover Total assets Capital intensity m Sales Profitability ratios Profit margin Net Income Sales Net income Return on assets ROA i Total assets Fleturn on equity ROE W Total equity ROE Net income X Sales x Assets Sales Assets Equity Market value ratios Price per share Earnings per share Markermbook ratio Market value Eer share Book value per share Priceearnings ratio Remember Think of RCA and m as Efficiency Ratios not Profitability Ratios 33 DuPont Analysis A method of calculating the contribution of different parts to overall profitability Also called Profitability decomposition Profitability is measured by ROE NIE 1 Decompose profitability into broad measures of Efficiency RCA and leverage EM Efficiency 9 RCA NIA Leverage 9 EM AE ROEROAXEM9NIENIAXAE Profit equals Efficiency times Leverage 2 Decompose Efficiency into Sales generated from Assets AT and expenses needed to generate the sales PM Sales Generated from Assets Asset Turnover 9 AT SalesA Earnings kept from each dollar of sales 9PM NISales ROAATXPM 9 NIASalesAx NISales Efficiency is a function of sales and exp 33 DuPont Analysis continued 3 Decompose AT SalesA into different types of Sales Manufactured Products Servicing Consulting AT Product SalesA ServicingA ConsultingA What breakdown categories are appropriate Depends on the company and its business 4 Decompose PM NISales into different expenses COGSSales SGampASaes Int ExpSales Tax ExpSales Dep ExpSales PM NISales Sales ExpensesSaes 23 33 DuPont Analysis continued ROE NlE Profit EM AE Lew mgo I ROA NlA Efficiency AT SalesA Revenues PM NlSales Expenses Equip SalesA ServicingA ConsultingA COGSSales SGampASales Int ExpSales Tax ExpSales Dep ExpSales Clicker Question Account Firm 1 Firm 2 Assets 10ooo 7 20000 Equity 5000 55000 Nil 1aner Sales 5000 10000 For both firms calculate 1 ROE NlE 2 RCA NlA 3 EM AE 4 AT SalesA 5 PM NlSales Firm 2 is more profitable greater ROE than Firm 1 because A B C D It is more efficient in generating sales from its assets It is more efficient in limiting its expenses It is more efficient in limiting its tax liability It is more levered Clicker Answer Firm 1 Account M i a Q j ssooo 33 li i Firm 1 For both firms calculate 1 ROE NIE 2 RCA NIA 3 EM AE 4 AT SalesA 5 PM NISales Firm 2 ROE greater for Firm 2 because EM is bigger EM measures leverage The answer is D 34 Internal and Sustainable Growth By definition for a firm to Grow Assets must grow 1 External Growth Sell stocks or bonds talk about this later 2 Internal Growth Retain Earnings talk about this now Internally Financed Growth is a function of 1 The Earnings aka NI as a percentage of Assets 2 The Earnings Retained by the business as a percentage of NI So How much did you make and how much did you keep First some definitions 321 Dividend Payout Ratio DivNI 121363 33 13 Payout 13 of Accounting Profits 322 Retention Ratio aka Plowback Ratio RENl 242363 23 Also equaltolDivNl Often denoted as quotbquot 32 34 Internal and Sustainable Growth Let b RENl Plowback Ratio ROA NlA Accounting profits per unit of assets Internal Growth ROA x b1 ROA x b For the example company Prufrock ROA 3633588 1012 and b 242262 06667 How many digits after the decimal point It depends Internal Growth 01012 x 066671 01012 x 06667 00724 724 If the company plows back 23 of NI which increases assets and RCA is 1012 then the firm grows at 724 without external financing Note that growth can be improved if ROA is improved How can ROA be improved Increase Sales or Decrease Expenses Which parts of sales or expenses are best suited for improvement How do you breakdown sales and expenses into different categories 33 34 Internal and Sustainable Growth continued Note that retaining earnings increases Retained Earnings the Equity account on the 85 But this does not increase Debt liabilities So overtime the DE ratio decreases So to maintain the same DE ratio the firm must sell some debt This leads to the Sustainable Growth Rate Sustainable Growth ROE x b1 ROE x b ROE 1401 Sustainable Growth 01401 x 066671 01401 x 06667 01030 1030 Internal Growth Rate 724 Sustainable Growth 1030 Sustainable growth implies borrowing to maintain the same DE ration Borrowing means increasing leverage 34L 34 Internal and Sustainable Growth continued So Growth is determined by four things 1 Sales generated from Assets in place Assets Use Efficiency AT SalesAssets 2 NI aka Earnings kept from those sales Operating Efficiency PM NlSales Sales ExpensesSales 3 Portion of NI Retained Plowback Ratio b RENl 4 Financing Policy How much more is borrowed relative to earnings retained Leverage Equity Multiplier EM AE 35 Clicker Question RG5in 7 in Calculate the Internal and Sustainable Growth rate for the firm Internal 10 Sustainable 20 Internal 8 Sustainable 18 Internal 7 Sustainable 14 Internal 5 Sustainable 11 Internal 2 Sustainable 6 W909 Clicker Answer ASA 1219 Internal Growth ROA x b1 ROA x b 01051 0105 005 Sustainable Growth ROE x b1 ROE x b 02051 0205 011 Answer is D 34 Recap Table 37 Page 70 internal growth rate Internal growth rate m 1 ROA X b where ROA Return on assets Net incomeTotal assets 2 Plowback retention ratio Addition to retained earningsNet income 1 Dividend payout ratio The internal growth rate is the maximum growth rate that can be achieved with no external financing of any kind Sustainable growth rate Sustainable growth rate ROE X b 1 ROE X b where ROE Return on equity Net incomefl39otal equity b Plowback retention ratio Addition to retained earningsNet income 1 Dividend payout ratio The sustainable growth rate is the maximum growth rate that can be achieved with no external equity financing while maintaining a constant debtequity ratio 35 using Financial Statement Information Compare to same firm over time Compare to firms within industry SIC codes North American industry Classification System NAICS or quotNakesquot See Table 38 page 73 But use your own common sense and knowledge about the company or industry See box on Page 78 Chapter 5 Discounted Cash Flow Valuation Valuing Multigle CFs Chapter Outline 1 FV and PV of Multiple Cash Flows Using the calculator 2 Valuing Annuities and Perpetuities 3 Comparing Rates of Different Compounding Penods Comparing apples to apples given the different way rates are quoted annual semiannual monthly 4 Loan Types and Loan Amortization Definitions of different finance contracts 51 Multiple CF s To find the FV of multiple CF s Calculate the FV of each individual CF Then ADD the individual CFs FVs together Example Receive 100 att 0 andt 1 Calc value the FV att 2 The first 100 increases twice The second 100 increases once 1001082 100108 22464 51 Multiple CF s Figure 51 A The time line D 1 2 I I I Time I 1 years Cash aws 100 100 B Calculating the future vaIue O 1 2 I I I 5 Time I 1 1 years Cash 110 100 100 103 439EU B W1 08 Future vallues 208 22464 Example 51 Page 117 You currently have 7000 in an account at t 0 You will deposit 4000 at the end of each of the next 3 years att 1t2andt3 How much will you have at time 3 at 8 7000 at t 0 with 3 years of interest 9 70001083 8818 4000 at t 1 with 2 years of interest 9 40001082 4666 4000 at t 2 with 1 year of interest 9 40001081 4320 4000 at t 3 4000 4000 0 1 2 3 I l 7090 8818 40iiii 4666 4an 4331 4000 21804 Example continued Same Example but now How much will you have at time 4 at 8 7000 att 0 with 4 years of interest 9 70001084 9523 4000 att 1 with 3 years of interest 9 40001083 5039 4000 att 2 with 2 years of interest 9 40001082 4666 4000 att 3 with 1 year of interest 9 40001081 4320 El 1 2 3 4 Williiii 9 523 4000 5 13 El 4000 4 555 4000 4 320 23543 Calculations FVatt3 FVatt4 70001083 8818 70001084 9523 40001082 4888 40001083 5039 40001081 4320 40001082 4888 4000108O 4000 40001081 4320 21804 23548 Clicker Question You currently have 600 in an account You will deposit 1000 at time 1and at time 2 How much will you have at time 2 if your account earns 10 1 600 2600 2826 3600 3826 F1909 Clicker Answer The 600 in the account now at time 0 will earn 10 for 2 years 6001 0102 726 The first 1000 deposited at time 1 will earn 10 for 1 year 10001 010 1100 The second 1000 deposited at time 2 will earn no interest 1000 Sum 726 1100 1000 2826 The Answer is c Another FV Example 2000 at the end of each year for 5 years Calculate FV at time 5 at 10 0 I 2 3 4 5 W W W W W W Time W W W W W W Wyem 52000 52000 52000 52000 52000 0 I 2 3 4 S W W W I W W TWme W W I W W yvars 2000 2000 2000 52000 5 200000 XWW 220000 1 3 242000 266200 X J y 2 925 20 Now Calculate the PV of Multiple CFs You need 1000 at t 1 and 2000 at t 2 How much do you need to invest today if you earn 9 Or what is the PV of these cash flows at 9 O 1 2 39 1 1 91743 4 1000 168336 1 2000 260079 1000109 20001092 260079 Think about the PV this way Invest 2601 at 9 Show that you can withdraw 1000 at t 1 and 2000 at t 2 260079109 283486 at t 1 283486 1000 183486 withdraw 1000 at t 1 1 834861 092 2000 at t 2 So if you invest 2601 at 9 you can withdraw 1000 at time 1 and 2000 at time 2 Clicker Question How much would you have to invest now to be able to withdraw 500 in one year and 800 in two years if your investment earned 10 F1901 1000 1116 1200 1226 1300 Clicker Answer In order to withdraw 500 in one year and then 800 in two years you must invest the sum of the PV of these withdrawals PV of 500 in one year 50011 455 PV of 800 in two years 800112 661 Sum 455 661 1116 The answer is B 52 Annuities The word Annuity has two definitions Economic Definition 1 All CFs are the same 2 CFs occur at regular intervals Annually Semiannually Quarterly Monthly 3 All CFs are discounted at the same rate The Financial Product 1 Pay an insurance company or a bank a lump sum today 2 Receive CFs at regular intervals for a fixed period or until you die 3 Sometimes you pay now or make regular payments starting now and then receive payments when you retire at 65 Same pattern of Cash Flow rules for Loans you pay Purchased Annuities you are paid 15 Formula for PV of an Annuity PVA PVA C1 Hay r PVA C1 11 rtr Same thing but typed Text Book s Notation 11 rt Present Value Factor PVF PVA C1 PVFr Other Notation 1 11 rtr Present Value Annuity Factor PVAF PVA CPVAF 16 PV of 1000 per for 5 years 6 ltIII I1ISKI 0 1 2 3 4 5 I I I I I I Time Present value I I I I I years calculated by 1000 1000 1000 1000 1000 discounting each I 3006 J cash flow separately 94340 quot 89000 83962 I063 I39 A 79209 4 1105 39 5 N728 1 05 421237 Total present value 7 Of We ll use the TVM function N 5 YR 6 PMT 1000 PV 4212 You don t have to enter anything for FV since it is zero 17 Clicker Question An investment pays 15000 at the end of each of the next four years Assume a discount rate of 8 Calculate the present value of the investment Or you can say calculate the PV of this annuity 15000 20000 29682 49682 60000 F1909 Clicker Answer Use the TVM Function to calculate the PV the annual payments of 15k per year for 4 years N 4 PMT 15000 IYR 8 FV 0 PV 49682 The answer is D Consumer Loans Consumer loans have monthly payments Credit card loans Mortgages Rates are quoted at the monthly rate times 12 Called the Annual Percentage Rate APR OrAPRMonthly 12 APRMonthly really means 1 per month 18 APRMonthly really means 15 per month 20 Example The bank quotes a rate of 12 APR on a 5 year car loan You can afford to pay 500 per month How much can you borrow t 5 x 12 60 r1212 1 C 500 PVA o1 11 rtr 5001 11 00160oo1 5001 11 00160oo1 500449550 2247752 21 Using the TVM Calculator Function Same Example Calculate the loan size for a 5 year 12 APR loan with 500 monthly payments Two ways to do this Set to 1 payments per year and enter 1 N 60 PMT 500 IN 1 FV 0 PV 2247752 Set to 12 payments per year and enter 1 N 60 PMT 500 IN 12 FV 0 PV 2247752 recommend you keep your calculator set to 1 payment per year and adjust your inputs 22 Finding Loan Payments You want to borrow 100000 to buy a house Calculate the monthly payments on a 30 year 9 APR loan r 912 075 t 30 x 12 360 PV C1 11 rtr PV CPVAF C PV1 11 rtr C PVPVAF C 1000001 11 00075360ooo75 C 1000001242819 80462 Can also use this notation PV CPVAF C PVPVAF Using TVM Function Set to 1 payment per year N 360 lYR 075 PV 100000 PMT 80462 23 Clicker Question You put 8000 on your credit card The card has a stated rate or 18 APRMonthly Calculate the monthly payments assuming the credit card uses a 30 year payback period 30 121 333 3284 8120 m00wgt 24 Clicker Answer N 30 x12 360 PV 8000 YR 1812 15 PMT 121 The answer is B Bonus Question Now assume that you actually read the fine print and find that a single late payment will cause the creditcard rate to increases to 30 APRMonthly Calculate the new monthly payment N 360 PV 8000 New YR 3012 25 PMT 200 25 Finding Annuity Payments Before we did PV of an annuity You want to buy an annuity from an insurance company You will pay 100000 today for equal monthly payments for the next 30 years The Annuity is offered at 9 Calculate the payments Same Calculation PV 100000 r 912 075 t 30 x 12 360 Using TVM Function N 360 lYR 075 PV 100000 PMT 80462 26 Finding the Number of Periods Put 2000 on your credit card and choose to pay 40 per month How many months to payoff at 18 APRMonthly PV o1 11 rtr PVC 1 11 rtr rPVC 1 11 rt 1 rt 11 rPVC t n1 r n11 rPVC t n11 rPVCn1 r n11 0015100040n1015 9311 What about using the TVM function 27 Finding of Periods using TVM Function 18 annual 1812 15 per month YR 15 PV 2000 PMT 40 N 9311 Y R15 PV 2000 PMT 40 N 9311 Potential Errors Using the TVM function Set at 1 PYR Both PV and PMT positive YR 15 PV 2000 PMT 40 N 3759 wrong Set at 12 PYR YR 15 PV 2000 PMT 40 N 5166 wrong YR 18 PV 2000 PMT 40 N 9311 28 Clicker Question You charge 5000 on your credit card The card has a stated rate or 24 APRMonthly Calculate the number of months needed to payoff the debt assuming monthly payments of 150 24 36 56 6O 94 F1909 29 Clicker Answer PV 5000 YR 2412 2 PMT 150 N 5548 So you will payoff the loan in 56 months The Answer is C 30 Future Value of Annuities FVA C1 lt 1r FVA CFVAF Example Make 5000 annual contributions to retirement fund for 40 years Earn 10 per year t40 r10 C5000 FVA 500011040 41010 500044259 2212963 44259 is called the Future Value Annuity Factor FVAF Calculator TVM Function N 40 lYR 10 PMT 5000 FV 2212963 31 Clicker Question To save for your child s education you invest 1000 a year for 18 years in a college savings account The account earns 10 per year What will be the value of the account in 18 years What if you don t start saving until your kid is 5 save for 13 years WPPF J 18000 over 18 years and 13000 over 13 years 36000 over 18 years and 26000 over 13 years 36900 over 18 years and 26300 over 13 years 45599 over 18 years and 24523 over 13 years 45599 over 18 years and 30250 over 13 years 32 Clicker Answer 1000 per year over 18 years at 10 N 18 YR 10 PMT 1000 FV 45599 1000 per year over 13 years at 10 N 13 YR 10 PMT 1000 FV 24523 The answer is 45599 over 18 years and 24523 over 13 years D So roughly half as much 2452345599 54 if you don t start until year 5 How much do you have to save over 18 years to have 100k N 18 YR 10 FV 100000 PMT 2193 Note that 1 O000045599 2193 33 Annuities Due An Annuity Due means the payments are made at the beginning of each period not at the end of each period So the First payment is made immediately not at the end of the first period The figure below shows the payment timing of a four year 1 00 Annuity and an Annuity Due 0 1 2 3 4 i Annuity 100 100 100 100 Annuity Due 100 100 100 100 34 Compare an Annuity to an Annuity Due Is the PV of a 4 yr Annuity Due greater than a regular 4 yr annuity Would you rather get the Annuity of the Annuity Due The 4 yr Annuity Due is the same as a 3 yr regular Annuity plus an extra 100 now at time zero 0 1 2 3 3 yr Annuity 100 100 100 4 yr Annuity Due 100 100 100 100 PV3yrAnnuity N3 lY10 PMT1OO PV24869 PV 4 yr Annuity Due 24869 100 34869 PV4yr regular Annuity N4 lY1O PMT1OO PV31699 35 Clicker Question You will pay 1000 per month to rent apartment for a year The lease requires monthly payments at the beginning of each month Assume a 12 APRMonthly discount rate Calculate the NET BENEFIT in present value terms to the landlord of receiving the rent payments at the beginning of each month as opposed to the end of each month A 1343 B 1000 C 743 D 500 E 113 36 Clicker Answer PV of a 12 month 1000 Annuity discounted at 12 APR N12 Y12121 PMT1000 PV11255 PV of a 12 month 1000 Annuity Due discounted at 12 APR N 11 Y1212 1 PMT 1000 PV 10368 PV of Annuity Due 10368 1000 11368 Net Benefit 11368 11255 113 The Answer is E 37 Perpetuities Level stream of cash flows forever PV Perpetuity C1r C1 r2 C1 r3 C11r11 r2 11 r3 o The term in brackets is a convergent sequence for any value of r between 0 and 1 Since each subsequent term is divided by 1 r taken higher and higher power the ratio becomes a smaller number approaching zero Eventually you are just adding zeros Economically payments made 100 years out divided by 1rtaken to the 100th power aren t really adding anything For0 lt rlt 1 1 r391 1 r392 1 r393 1r PV Perpetuity C1r Cr The PV of 500 per year forever discounted at 8 50008 6250 38 Review of Formulas and Symbols quot a ha 3 Pv a 9 m Allin FV Fu ure value whal cash llews are wonh ln lhe lulure al lime l lnleresl vale rale cl relurn or discounl vale per perlorHyplcally bm nol always one year 1 Number el penuds lyplcally bm nol always lne number oi years 0 Cash amounl Furure value 0 l inveelerl per period var periods al rpercenl per period Fl ex 1 A rllr Klr Asenes el ldenhcal cash ows ls called an annuily and lne lerrn in A r r lllr called me ennmly lulure value Iaclan lll Presanl value 0 0 per period lcr lperioes al rpereenr per pence v a cxll 11 r39lr The lerrn I 7 ml l rls called he armtu presenl valu Iaclpr lV Flasen vlue ol a perpamily of 0 per paliod FV a c and perpewily calculations r r Aperpelully has in same cash ow Every year lurever Page 137 53 Comparing Rates APR vs EAR Banks and other financial institutions must quote loan and deposit rates as Annual Percentage Rate APR An APR is the periodic rate times the number of periods 5 every six months is 10 APR SemiAnnual 1 every month is 12 APR Monthly 2 every quarter is 8 APR Annual An EAR Effective Annual Rate is the annual rate that is equivalent to a given APR What annual rate is the same as 5 compounded twice a year 10 APR SemiAnnual 105105 11025 So the equivalent is 11025 1 01025 1025 So either pay me 10 APR S A or 1025 per year They are the same 40 APR and EAR Formulas APR Annual Percentage Rate APR Periodic Rate x of periods 5 every six months 10 APR SA 12 APR Monthly 1 every month EAR Effective Annual Rate Annual rate that is equivalent to the compounded APR 10 APRSemi Annual 5 compounded twice 105105 1 1025 EAR 12 APRMonthly 10112 1 1268 EAR Let m of periods Calculate the EAR if the quoted rate is 18 APR monthly APR 18 m 12 EAR 1 APRmm 1 1 0181212 1 101512 1 1956 You would be indifferent between payingearning 1956 per year and 18 APR monthly 15 per month 41 APR and EAR Formulas Convert EAR to APR APR m1 EAR1m 1 What semiannual APR is equivalent to 10 EAR EAR 10 m 2 APR 21112 1 976 Show this works 976 APR means 488 each 6 month period EAR 1 APRmm 1 104882 1 10 42 Terminology and Calculators EAR is also called Effective Rate EFF Effective Annual Yield EAY Annual Percentage Yield APY APR is also called Nominal NOM Using the function in your HP Calculator Calculate EAR for 18 APR monthly Function Buttons Disglay Set payments per year to 12 12 Yellow Shift PY 12 Input 18 APR 18 Yellow Shift NOM 18 Calculate EAR Yellow Shift EFF 195618 43 Clicker Question You want to invest for one year A bank is offering 310 APRQuarterly on a three month CD and 325 on a oneyear CD Assume you can rollover the threemonth CD at the same 310 rate three more times So assume rates will not change until you have rolled over the threemonth CD three more times Which investment alternative is better A The threemonth CD 4 times B The oneyear CD once C There is not enough information 44 Clicker Answer Compare 310 APRQuarterly to 325 Annual So calculate the EAR of 310 APRQuarterly m 4 APR 310 EAR 1 APRmm 1 1 0031044 1 1 007754 1 00314 314 Since 310 APRQuarterly is the same as 314 Annual 325 Annual is better than 310 APRQuarterly The Answer is B 45 54 Loan Types and Amortization There are a number of different ways a loan can be repaid 1 Ways in which interest can be paid 2 Ways in which the loan amount called the principal can be repaid There are common loan structures for different types of loans Loans made to people called consumer or retail loans Loans make to business by banks Called institutional or business loans Loans make to business by the market The company sells Commercial Paper to the market The company sells Bonds to the market long term loans The way the loan is repaid is called the Amortization Schedule 46 54 Loan Types and Amortization Amortization refers to how the loan is repaid 1 Pure Discount Loans Only one payment Payment Includes both the principal loan amount and interest Amount loaned is the Present Value of the one payment The payment is the Future Value of the loan amount EC payment is also called the Face Value of the loan also 47 1 Pure Discount Loans Continued Common loan structure for a Bank CD Calculate the payment for a 100 2 year loan at 5 annual rate FV PV1 rt 1001052 11025 100 is the loan amount and 1025 is the interest Common loan structure for a TBill Calculate the amount loaned for a 10000 FV 1 year loan at 6 PV FV1 rt 10000106 943396 10000 943396 56604 is the interest Common loan structure ZeroCoupon corporate bond a zero Calculate the amount loaned for a 1 000 FV fiveyear loan at 10 PV FV1 rt 1000115 62092 1000 62092 37908 is the interest 48 2 InterestOnly Loans or Bullet Maturity Loans Periodic interest payments are made Quoted as a percentage of the Face Value The loan amount is repaid at the end This is the common structure for corporate bonds A 1000 fouryear loan is structured to make 8 interestonly ANNUAL payments Payments 1 000008 80 made at the end of each year 0 1 2 3 4 I I I I I I I I I 80 80 80 80 1000 1080 49 3 Amortized Loans or SelfAmortizing Loans Periodic payments include both interest and principal One possible type of amortizing loan A small business borrows 5000 for 5 years at 9 o The contract requires borrower to repay 1 000 each year Amortization Table Beginning Interest Principal Total Ending Year Balance at 9 Payment Payment Balance 1 5000 450 1000 1450 4000 2 4000 360 1000 1360 3000 3 3000 270 1000 1270 2000 4 2000 180 1000 1180 1000 5 1000 90 1000 1090 0 Totals 1350 5000 6350 Notice the total payment decreases as the balance decreases 50 Another type of amortizing loan The contract calls for equal payments each period Solve forthe equal annuity payments PVA C1 11 rtr PVA CPVAF C PVAPVAF Same loan 5 yr 9 5000 loan but with equal payments C 50001 11095009 128546 N 5 W 9 PV 5000 PMT 128546 Amortization Table Beginning Interest Principal Ending Year Balance Payment Owed Paid Balance 1 5000 1285 450 835 4165 2 4165 1285 375 911 3254 3 3254 1 285 293 993 2261 4 2261 1285 204 1082 1179 5 1179 1285 106 1179 0 Totals 6427 1427 5000 Note Since the loan is repaid slower the total interest paid is higher than the loan described on the previous slide 51 Recap Loan Types 1 Pure Discount Also called Zero Coupon Single payment consisting of both interest and principal at maturity Very common for shortterm debt and some corporate bonds 2 Interest Only Also called Bullet Maturity or Coupon Bond Periodic interest payments with entire principal repaid at maturity Very common for longterm debt Often at maturity new bonds are sold and the proceeds are used to repay the maturing bonds 3 Amortizing or SelfAmortizing Common for consumer loans Common for small business loans Each payment consists of interest and principal Balance of loan decreases over time 52 Chapter 6 Interest Rates and Bond Valuation Chapter 6 Outline 1Bonds and Bond Valuation 2More on Bond Features 3Bond Ratings 4Some Different Types of Bonds 5Bond Markets 6nfation and Interest Rates 7Determinants of Bond Yields Key Concepts and Skills Know the important bond features and bond types Understand bond how to calculate bond prices and why prices fluctuate Understand bond ratings Understand the impact of inflation on interest rates Understand the Term Structure of Interest Rates and the determinants of bond yields Understand the determinants of bond yields 61 Bonds and Bond Valuation Firs some terminology A 30 year 1000 bond pays 100 per year annual payments and repays the 1000 in thirty years The market requires a 10 return on loans to this company COUPON 100 COUPON RATE CouponFace 1001000 10 COUPON PERIOD Annual FACE VALUE or PAR VALUE 1000 TIME or TERM TO MATURITY 30 years YIELD TO MATURITY YTM required rate 10 A Bond is Defined or Identified by 1 Issuer A Corporation corps GE GM A Municipality Munis Boulder New York State E470 Public Highway Authority The Federal Government Govies Issued to finance the deficit and debt 2 Maturity Usually a date but in this class we ll use a time to maturity A ten year bond is a 2019 but we ll call it a ten year 3 Structure Coupon Zero Amortizing A Bond is Defined or Identified by 4 Coupon Rate if applicable 7 SemiAnnual 10 Annual 5 Bond Features Callable Convertible we ll talk about this later 6 Rating Rating agencies SampP Moodys Fitch issue public rating BBB or higher are Investment Grade BB or lower are NonInvestment Grade junk Example GE 5125 of 2028 five and an eighth of 28 GE corp bond Matures in 2028 Pays a 5125 SA coupon AAA rated Calculate the price of the bond 30 year 1000 Face or Par Value 10 annual coupon and 10 required rate so YTM 10 Two Components 1 Thirty payments of 100 discounted at 10 PV 10011100112100113 100113O 94269 N 30 IIYR 10 PMT 100 PV 94269 2 One payment of 1 000 in thirty years PV 1000113O 5731 N 30 IIYR 10 FV 1000 PV 5731 Or both at the same time Total Price 94269 5731 1000 N 30 IIYR 10 PMT 100 FV 1000 PV 1000 Equal to the Face Value or Par Value Calculate the price of the bond Same bond with a different YTMs Why might the bonds YTM change The issuer s credit rating has changed All rates have changed 30 year 1000 Face or Par Value 10 annual coupon 8 required rate YTM 8 N 30 IIYR 8 PMT 100 FV 1000 PV 1225 Price 1225 Great than the Face Value or Par Value Premium 30 year 1000 Face or Par Value 10 annual coupon 12 required rate YTM 12 N 30 IIYR 12 PMT 100 FV 1000 PV 839 Price 839 Less than the Face Value or Par Value Discount Clicker Question A 20 year 1000 facevalue bond pays an 7 annual coupon and has a discount rate of 9 Calculate the bonds price Is it priced at par a discount or a premium A 217 Discount B 234 Discount C 817 Discount D 1000 Par E 1196 Premium Clicker Answer 20 year 1000 facevalue bond pays an 7 annual coupon and has a discount rate of 9 PMT 1000X00770 N20 YR9 PMT 7O FV1000 PV817 The price is 817 The answer is C Important thing about a bond s price If YTM Coupon Rate 9 Price Par N 30 IIYR 10 PMT 100 FV 1000 PV 1000 If YTM gt Coupon Rate 9 Price lt Par Priced at a discount N30 IYR12 PMT100 FV1000 PV839 If YTM lt Coupon Rate 9 Price gt Par Priced at a premium Why Bonds are Issued at Par By tradition a bond is issued at par This means the YTM equals the Coupon Rate A company that wants to borrow money goes to the market to determine what rate of return lenders bond buyers want to earn Call this is the Required Return Or the Reguired Yield Or the Yield to Maturitv YTM The company then sets the coupon rate of the bond equal to the YTM and the bond is issued at par Changes in Time Back the 1 000 30 year 10 coupon 10 YTM bond N 30 lYR 10 PMT 100 FV 1000 PV 1000 Now one year has passed Recalculate the Bonds value N 29 lYR 10 PMT 100 FV 1000 PV 1000 So still 1000 since Coupon Rate YTM Examine two components 1 Twentynine 100 payments discounted at 10 No FV N 29 lYR 10 PMT 100 PV 93696 2 One 1 000 payment in twentynine years No PMTs N 29 lYR 10 FV 1000 PV 6304 29 yrs Total Value 93696 6304 1 000 30 yrs Total Value 94269 5731 1 000 One fewer annuity payment but 1000 FV received one year sooner Changes in Rate Back to 30 years to a maturity but required rate is now 11 Why N 30 IIYR 11 PMT 100 FV 1000 PV 91306 So less than 1 000 since denominator r increased Examine two components 1 Thirty 100 payments discounted at 11 N 30 IIYR 11 PMT 100 PV 86938 2 One 1000 payment in thirty years N 30 IIYR 11 FV 1000 PV 4368 11 YTM Total Value 86938 4368 91306 10 YTM Total Value 94269 5731 1 000 Why is price lower Price Changes as the Interest Rate Changes Call this Interest Rate Risk It is the Volatilitz ofthe bond s value caused by changes in rates The Price function PV C1 r1 C1 r2 C1 r3 C FV1 rt C Coupon Fixed FV Face Value Fixed t Time to Maturity slow and can see that one coming r YTM or the discount rate So what will affect the price of the bond by tomorrow Changes in Rates So bond holders incur interest rate risk Determinants of Interest Rate Risk What causes Interest Rate Risk Or what increases Interest Rate Risk We will se there are three factors that increase Interest Rate Risk 1 Interest rate risk Increases as Time to Maturity INCREASES Longer Maturity higher risk 2 Interest rate risk Increases as the Cougon Rate DECREASES Lower coupon higher risk 3 Interest rate risk Increases as the Starting YTM DECREASES Lower YTM higher risk 16 Time to Maturity The longer the term to maturity ceteris paribus the greater the Interest Rate Risk The change in price for the 30 year bond is greater than for the 15 year or 5 year bonds Coup Rate t r PV chg 10 5 10 1000 10 5 9 1039 389 10 15 10 1000 10 15 9 1081 806 10 30 10 1000 10 30 9 1103 1027 Starting YTM The Lower the Starting YTM The higher the interest rate risk The change in price is Greater when the YTM starts at 5 than when the YTM starts at 10 or 15 Coup Rate t r PV chg 10 30 5 1769 10 30 4 2038 1520 10 30 10 1000 10 30 9 1103 1027 10 30 15 672 10 30 14 720 717 Figure 62 from the text Table and Chart of the PriceYield Relationship Red is a 30 Year Bond Blue is a 1 Year Bond 3 3quot noun 6 39 value 5 i J Interest rate risk and time to maturity 239000 W 1i76862 30year bond 1500 310475 Wear bond 13900 E E quot 3quot 591667 500 u 50211 L I I Interest 5 10 15 20 r 3 Wquot Value of a Band with a 10 Percent Coupon Rate for Different Inlerest Rates and Maiiurities Time to Maturity Interest Fiaie 1 Year 30 Years 59quot 104762 1 75862 10 100000 1 00000 15 95652 67170 20 91567 5021 1 Fig 62 Compare the slope of the RED LINE at 5 and 20 o It is much steeper at 5 greater slope Steeper means greater change in Price Rise for given change in rates Run Interest Rate Risk is measured by the slope Bond l value8 in 3 7L 7 Interest rate risk and time to maturity 239000 7 smssse 30year bond 1500 S1 41672 1year bond LOGO Z 397 u a 91667 i 500 1 50211 i i l l Interest 7 rate 33 20 Coupon Rate The lower the coupon rate the greaterthe Interest Rate Risk The change in price is Greater for the 5 coupon bond than the 75 coupon or 10 coupon Coup Rate t r PV chg 5 30 10 529 5 30 9 589 1143 75 30 10 764 75 30 9 846 1067 10 30 10 1000 10 3O 9 1103 1027 21 Recap Interest rate risk Increases as the Coupon Rate DECREASES Lower coupon higher risk Interest rate risk Increases as the Starting YTM DECREASES Lower YTM higher risk Interest rate risk Increases Time to Maturity INCREASES Longer Maturity higher risk 22 Clicker Question Consider Bonds 1 and 2 Bond 1 10 year bond 10 annual coupon 10 YTM Bond 220 year bond 10 annual coupon 10 YTM Consider Bonds 3 and 4 Bond 3 10 year bond 5 annual coupon 5 YTM Bond 4 10 year bond 10 annual coupon 5 YTM Which of the following is true about interest rate risk Bond 1 gt Bond 2 and Bond 3 gt Bond 4 Bond 2 gt Bond 1 and Bond 3 gt Bond 4 Bond 1 gt Bond 2 and Bond 4 gt Bond 3 Bond 2 gt Bond 1 and Bond 4 gt Bond 3 powgt 23 Clicker Answer Bond 1 10 year bond 10 annual coupon 10 YTM Bond 2 20 year bond 10 annual coupon 10 YTM Longer Maturity 9 Greater Interest Rate Risk 9 2 gt 1 Bond 3 10 year bond 5 annual coupon 5 YTM Bond 4 10 year bond 10 annual coupon 5 YTM Lower Coupon 9 Greater Interest Rate Risk 9 3 gt 4 Which of the following is true about interest rate risk Dowgt Bond 1 gt Bond 2 and Bond 3 gt Bond 4 Bond 2 gt Bond 1 and Bond 3 gt Bond 4 Bond 1 gt Bond 2 and Bond 4 gt Bond 3 Bond 2 gt Bond 1 and Bond 4 gt Bond 3 24 SemiAnnual Coupon Bonds There are special rules for bonds that pay SemiAnnual coupons Example 21 year 5125 S A bond YTM 588 Calculate the price Coupon Rate 5125 means 0051252 0025625 NOT 003 or 0026 or 00256 or Use FULL PRECISSIONll Coupon payment 0025625 x 1000 25625 YTM 588 means 005882 00294 NOT 003 or 0029 or 3 or 29 entered into the calculator Number of SemiAnnual Periods 21 x 2 42 N 42 lYR 294 PMT 25625 FV 1000 PV Greater or less than 1000 PV 90962 25 Clicker Question A 15 year 1000 facevalue bond pays an 825 SEMIANNUAL coupon and has a YTM of 10 Calculate the bonds price Hint Both the Coupon Rate and the YTM are quoted at twice the periodic rate Always A 850 B 865 C 950 D 1 000 E 1 200 26 Clicker Answer A 15 year 1000 facevalue bond pays an 825 SEMIANNUAL coupon and has a YTM of 10 Calculate the bonds price Hint Both the Coupon Rate and the YTM are quoted at twice the periodic rate Always N 15 x 2 30 r 102 5 PMT 0082521000 4125 FV 1000 PV 865 so the answer is B 27 62 More on Corporate Bond Features Differences between Debt and Equity Debt Not an ownership interest Creditors do not have voting rights Interest is considered a cost of doing business and is taxdeductible Creditors have legal recourse if interest or principal payments are missed Excess debt can lead to financial distress and bankruptcy Equity Ownership interest Common stockholders vote to elect the board of directors and on other issues Dividends are not considered a cost of doing business and are not tax deductible Dividends are not a liability of the firm until declared Stockholders have no legal recourse if dividends are not declared An allequity firm cannot go bankrupt 28 A Bond is Formally called a Bond Indenture It is a contract between the issuer and the lender Contract Terms Includes The Basic Terms of the Bonds Interest rate Payment Structure Repayment Schedule or Amortization Total Amount of Bonds Issued Assets used as security Debentures unsecured Collateralized secured by financial securities Mortgage bonds secured by real property land or buildings Notes unsecured debt with original maturity less than 10 years Seniority Which bonds gets paid 1st Senior debt Get paid first Subordinate debt Get paid second 9 More Termsi More Terms of a Bond Contract Sinking Fund Provisions Money set aside to repay the bond is called a sinking provision How much money is set aside and when Call Provisions Price and time when the company can buy back or call the bond if it wants Conversion Provisions Price and time when the bond holders can convert the bond to shares of stock if they want Details of Restrictive Covenants Must maintain a DE ratio below a certain level May not sell certain assets the company can t sell its trucks May not issue new debt seniorto this issue Must limit dividends payments to stockholders Why 30 63 Bond Ratings Bonds that are sold to the public are rated by one of the major creditrating agencies SampP Moody s Fitch The agencies are private companies that are paid by the issuer to produce ratings Many investment funds mutual funds insurance companies endowments will not buy unrated debt Unrated or bonds rated BB or below is called non investment grade or junk Some low quality bond issuers will buy bond insurance The issuer pays 6 to the bond owners and 5 to the insurance company The net of 11 is less than the issuer would have paid without the insurance If not insured the issuer probably would not have been able to sell bonds at all 31 64 Types of bonds Bonds are Defined by 1 The type of issuer Government Municipality Corporation 2 How is the interest paid Is there a coupon Annual SemiAnnual Is the coupon rate fixed or does it float Do you buy it at a discount today and receive all the interest along with principal at the end ZeroCoupon 3 How is the principal paid All at the end Bullet Maturity A little each period SelfAmortizing 32 Government Bonds Called Treasury Securities Treasum Site TBills Pure discount bonds Original maturity of one year or less TNotes Coupon debt Original maturity between one and ten years TBonds Coupon debt Original maturity greater than ten years TIPS Treasury Inflation Protection Securities Principal Amount face value adjusts with CPI TIPS Rates amp Terms TIPS in Depth 33 Municipal Bonds Issued by state and local governments and Authorities Schools Districts It is how they borrow money to build bridges schools airports buy open space Interest earned by the bond owners is taxexempt at the federal level Example A corporate bond pays 8 and a Muni bond pays 5 o If your marginal tax bracket is 30 which bond to you prefer You keep 1 030 of the corporate coupon 1030008 0056 56 so the 8 corporate bond is better If your marginal tax bracket is 40 which bond to you prefer 1040008 0048 48 so the 5 muni bond is better At what tax bracket are you indifferent between the bonds 1 T008 005 T 1 005008 0375 375 34 More on Munis Either General Obligation or Revenue bonds GO bonds are backed by the full faith and credit of the issuer All a city s or state s taxes and revenue sources are committed to paying the coupons and principal Revenue bonds Revs are backed by revenue from a specific project E470 Tos DIA Landing and gate fees 35 Floating Rate Bonds Floaters Coupons set at an Index rate plus a spread Index rates Tbill rate Tbonds rate LIBOR If the coupon is Tbills 200 bps And the Tbi rate is 475 Then the Floater pays 675 Price of a Floater is usually close to Par A company s default risk premium is 200 bps above Tbills So coupon rate is set to Tbills 200 Then the Coupon Rate Discount Rate aka YTM f Coupon Rate YTM 9 Price Par As TVM changes Tbill rate changes and coupon rate changes But coupon rate and discount rate are still equal 9 Price Par Unless default risk changes Say default risk increase to 300 bps Then Coupon Rate Tbill 200 lt YTM TBill 300 9 Price lt Par 36 LIBOR London InterBank Offered Rate The average rate charged by London banks for offshore dollar loans Offshore dollars are called eurodollars So LIBOR is also called the eurodollar rate Eurodollars are not the same as euros Why London bank rates and not US bank rates London banks are less regulated US banks charge the Fed Funds rate plus LIBOR is determined more by natural supply and demand But is also affected by the Fed Funds rate 37 65 Bond Markets Dealers at securities firms stand ready to buy and sell certain bonds They are linked electronically and by phone This is called an OvertheCounter market OTC Distinguish this from a big room full of traders called an exchange There are an extremely large number of bond issues Multiple bonds from each company Generally illiquid low volume in any single bond issue Most recent treasury at each maturity is an exception Only one Federal government The most recent treasury at each maturity is called the onthe run treasury Read about bond quotes on pages 183 to 187 38 66 Interest Rates and Inflation Inflation decreases purchasing power Nominal return is amount paid Real Return factors out inflation Measures increase in purchasing power Let Nominal Return 10 Inflation 3 100110 110 only buys 1 101 03 1 680 more stuff Not 10 more stuff Look before ex ante at expected inflation h Calculate the required nominal return Look aftenNards ex post at measured inflation i Calculate the realized real return 39 ExAnte or before you hold the bond Use EXPECTED INFLATION h Example You will hold a bond for one year The Required real return is 5 Expected Inflation is 3 Calculate the required nominal return on a bond R Nominal Return r Required Real Return 5 h expected inflation 3 R 1 r1 h 1 105103 1 815 40 Clicker Question You require a real return of 10 on a bond You expect inflation to be 5 Calculate the nominal return on the bond A 50 B 55 C 105 D 150 E 155 Clicker Answer r required return 10 h expected inflation 5 R Nominal Return 1r1h1 110105 1 155 The Answer is E What is the ESTIMATED Nominal Return 10 5 15 42 ExPost or after you have held the bond Use REALIZED or MEASURED INFLAITON i Example The nominal return was 815 Measured Inflation was 2 Calculate the realized real return R Nominal Return 815 r Required Real Return i measured inflation 2 r 1 R1 i 1 10815102 1 603 gt 5 43 Clicker Question You have earned 12 nominal on a bond Over the period measured inflation was 4 Calculate the realized REAL return on the bond A 400 B 769 C 800 D 869 E 1200 Clicker Answer R 12 nominal on a bond i 4 r1R1i1 1 121 04 1 769 The answer is B What is the ESTIMATED realized real return 1248 45 67 The Determinants of Bond Yields YTM The Term Structure of Interest Rates Relationship between nominal rates and the term to maturity Usually refers to yields of ontherun US Treasury bonds No default risk premium No or very little liquidity premium USTreasurvqov yield curve site Graphical depiction is called The Yield Curve US Treasury Yield Curve 450 222009 1 mo 3 mo 6 mo 1 yr 2 yr 3 yr 5 yr 7 yr 10 yr 20 yr 30 yr Does the yield curve slope up or down Upward usually Short rates are lower than long rates Downward or Inverted rare Short rates are higher than Long rates See the 20yr to 30yr rates in the chart on the last slide Three components of government rates 1 Required Real rate 2 Interest Rate Risk Premium Longer Maturity more interest rate risk See slide 13 3 Expected Inflation Premium Usuallyincreasing in maturity But what if it isn t People might expect Deflation Deflation is lower prices over time 80 required nominal rates decrease overtime maybe 47 UpwardSloping Yield Curve Shown with constant real rate Not necessarily the case Increasing interest rate risk premium Increasing inflation premium Interest A Upwardsloping term structure rate Nominal interest rate Interest rate risk premium Inflation premium I Real rate I Time to maturity 48 Some Notes on Terminology YTM vs Pure Discount Rates The YTM for a 20 year Tbond is an average rate for all 40 bond payments 20 semiannual coupons Each payment has it s own Pure Discount rate Averaged together sort of they are the YTM for the whole bond The Yield Curve usually refers to YTMs averaged rates The Term Structure usually refers to Pure Discount rates YTM is sometimes called the Promised yield You only earn the YTM if The bond is held to maturity Each coupon is reinvested at the same YTM rate Think of YTM as A rough indication of what the bond pays A way to price the bond Pricing convention The price equals the PV of the CFs discounted by the YTM 49
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