Finance Lecture2

MBAM 614 Finance The Time Value of Money I • MBAM614 Class 2 - 1 Summary of Last Class 1. Goal of Financial Manager is Increase Shareholder Wealth 2. Principal Financial Decisions are: Capital Budgeting (Business Strategy); Capital Structure (Financial Strategy); and Working Capital Management 3. FCF from Assets = NCF to Bondholders + NCF to Shareholders (based on TA = TL + E) 4. Compare Across Time or Firms With Common Size Statements 5. Ratios Analysis Looks at: Levels, Trends, Comparison • MBAM614 Class 2 - 2 1 Agenda 1. Future Value and Compounding 2. Present Value and Discounting 3. Basic Present Value Equation 4. Future Value of Multiple Cash Flows 5. Present Value of Multiple Cash Flows • MBAM614 Class 2 - 3 What Are Future Cash Flows Worth Today? We Would Rather Have $100 Today than $100 One Year From Now However, $100 One Year From Now is Still Valuable How Much is the Certainty of $100 in One Year Worth Today? What is the “Time Value” of $100 in One Year? Answer is the Key to Valuing Investments • MBAM614 Class 2 - 4 2 How Much is $100 Invested Worth in One Year? Depends on the Current Interest Rate, r if r = 6%, then t=0 (today) t=1 (one year later) $100 principal interest $100 + 6 $106 ( = $100 * 0.06 or $100 * 6%) • MBAM614 Class 2 - 5 How Much is $100 Invested Worth in One Year? Future Value in One Year, FV1, Has Two Parts original investment, principal, or Present Value interest FV1 = $100 + $6 = $100 * ( 1 + 0.06 ) • MBAM614 Class 2 - 6 3 How Much is $100 Invested Worth in Two Years? Assuming r = 6% and That We Re-invest Interest t=0 (today) t=1 (one year later) t=2 (two years later) $100 principal interest Value after one year $100 + 6 $106 principal interest $106.00 + 6.36 $112.36 FV2 = $100 * (1 + 0.06) * (1 + 0.06) = $112.36 • MBAM614 Class 2 - 7 Compound Interest Earning Interest on Interest is Called “Compounding” or “Compound Interest” FV2 = PV * (1 + r) * (1 + r) = PV * (1 + r)2 Future Value of an Investment, PV, After t Years Using Compound Interest is: FVt = PV * (1 + r)t We Say We Have Compounded the Present Value Forward to Find its Future Value • MBAM614 Class 2 - 8 4 Future Value Interest Factor Term (1 + r)t is Called “Future Value Interest Factor” FVIF(r,t) = (1 + r)t FVt = PV * FVIF(r,t) • MBAM614 Class 2 - 9 Example: Investing $1,200 For 5 Years You Can Buy a GIC That Pays 8% for 5 Years for $1200 Today. What is the GIC Worth at Maturity if it Pays Simple Interest? If it Pays Compound Interest? What is the Appropriate FVIF? Simple Interest [ FVt = PV * (1 + t*r) ] : FV5 = $1200 * (1 + 5 * 0.08) = $1200 * (1.40) = $1680 Compound Interest [ FVt = PV * (1 + r)t ] : FV5 = $1200 * (1 + 0.08)5 = $1200 * (1.4693) = $1763.16 Future Value Interest Factor [ FVIF(r,t) = (1 + r)t ] : FVIF(8%,5) = (1 + 0.08)5 = 1.4693 • MBAM614 Class 2 - 10 5 Example: How Much Interest is Earned? Simple Interest (Interest on Principal Only): Interest Earned = FV5 - $1200 = $1680 - $1200 = $480 Compound Interest (Includes Interest on Interest): Interest Earned = FV5 - $1200 = $1763.16 - $1200 = $563.16 How Much Interest on Interest Will Be Paid if Interest is Compounded? (Interest on Principal and Interest) - (Interest on Principal Only) = $563.16 - $480 = $83.16 • MBAM614 Class 2 - 11 Example: How Much Must We Invest? If We Want to Have $100 in Our Bank Account Three Years From Today, How Much Must We Invest at 5% Today? FVt = PV * (1 + r)t We Know: – Length of the Investment, t = 3 – Future Value, FV3 = $100 – Compound Interest Rate, r = 5% – Need to Solve for PV • MBAM614 Class 2 - 12 6 Example: How Much Must We Invest? FVt = PV * (1 + r)t $100 = PV * (1 + 0.05)3 $100 = PV (1.05)3 PV = $100 = $86.38 (1.1576) Must Invest $86.38 Today to Have $100 in Three Years $100 in Three Years is the Same as $86.38 Today! • MBAM614 Class 2 - 13 Basic Present Value Equation 1 (1 + r)t 1 FVIF(r,t) Basic Present Value Equation Present Value Interest Factor or, the Discount Factor PV = FVt * 1 (1 + r)t = r is Usually Called the Discount Rate Method is Called Discounting (Future) Cash Flows or Discounted Cash Flow (DCF) Analysis • MBAM614 Class 2 - 14 7 Example: Certificates of Deposit (CD’s) CD’s Come in Two Flavors. One Type Specifies an Annual Compound Interest Rate and the Value of the CD at Maturity. How Much Would You Expect to Pay For a CD That Will Be Worth $1500 in 4 Years if the Interest Rate is 73 %? Really Looking for the Present Value of $1500 Four Years From Now Discounted at 73 %: 1 (1 + r)t 1 (1 + 0.0725)4 $1500 (1.3231) PV = FVt * = $1500 * = = $1133.71 We Would Expect to Pay $1,133.71 For This CD • MBAM614 Class 2 - 15 Example: Buying a Used Car Given Any Three of the Variables in the Basic PV Equation, We Can Solve for the Fourth: You Have Your Eye on a 1997 Toyota Supra That Costs $12,000. The Owner Has Graciously Offered to Wait for Payment Until After You Finish University. If You Wait to Pay, You Must Pay Him $17,000 in Three Years. What Interest Rate is This? $12,000 = $17,000 * 1 / (1 + r)3 (1 + r)3 = $17,000/$12,000 (1 + r) = 1.1231 or r = 12.31% • MBAM614 Class 2 - 16 8 FV of Multiple Cash Flows Instead of Buying the Supra Now, You Would Like to Save $3000 in Each of the Next 4 Years. If You Start Now and Save Another $3000 at the Beginning of Each Year, How Much Will You Have to Spend on a Car at the End of Year 4 Assuming You Earn 8% on Your Money? Compound Forward One Period at a Time Compound Each Cash Flow Forward Separately Methods are Equivalent • MBAM614 Class 2 - 17 Compound One Period At A Time t=0 1 2 3 4 $ 0.00 $3240.00 $6739.20 $10518.33 $14599.80 3000.00 × 1.08 3000.00 × 1.08 3000.00 × 1.08 3000.00 × 1.08 $6240.00 $9739.20 $13518.33 $3000.00 • MBAM614 Class 2 - 18 9 Compound Separately t=0 $3000.00 1 $3000.00 2 $3000.00 × 1.083 3 $3000.00 × 1.082 × 1.08 4 $ 3240.00 $ 3499.20 $ 3779.14 $ 4081.46 $14599.80 × 1.084 • MBAM614 Class 2 - 19 PV of Multiple Cash Flows Congratulations! Reader’s Digest Has Just Selected You as The Grand Prize Winner in Their Annual Contest! You Have Won $200,000 (paid in 5 annual installments starting immediately, or you may take $160,000 now)! If the Discount Rate is 14%, Which Would You Choose? Discount Each Cash Flow Separately Discount Backward One Period at a Time Methods are Equivalent • MBAM614 Class 2 - 20 10 Discount Separately t=0 $40,000 $35,088 $30,779 $26,999 $23,683 1 $40,000 × 1/1.14 2 $40,000 × 1/1.142 3 $40,000 4 $40,000 × 1/1.143 × 1/1.144 $156,549 Choose the $160,000 Now! • MBAM614 Class 2 - 21 Discount One Period At A Time t=0 1 2 3 4 $116,549 $ 92,866 $ 65,867 40,000 ÷ 1.14 40,000 ÷ 1.14 40,000 ÷ 1.14 $156,549 $132,866 $105,867 $35,088 40,000 ÷ 1.14 $75,088 $40,000 Choose the $160,000 Now! • MBAM614 Class 2 - 22 11 Example: Selling A House Your Parents Would Like to Sell Their House But They Haven’t Had Much Luck So Far. Last Week, Someone Offered to Rent the House for Three Years at $600 per Month (paid at the end of the year) and Then Purchase the House for $175,000. Today, They Received an Offer of $150,000. They Were Asking $160,000 and They’re Not Too Happy With the Offer so They Have Asked Your Advice. You Know They Can Earn 10% on Their Money. What do You Recommend? Two Equivalent Methods: – Calculate the PV of Both Alternatives and Compare – Calculate the FV of Both Alternatives and Compare • MBAM614 Class 2 - 23 Example: PV of Renting The House t=0 $ $ $ 0 6,545 5,950 1 $7,200 × 1/1.10 2 $7,200 × 1/1.102 3 $7,200 + 175,000 $136,890 × 1/1.103 $149,385 • MBAM614 Class 2 - 24 12 Present Value Tables Instead of Calculating the PVIF (or FVIF), You Can Use a Present Value Table (or FVIF Table, see Appendix A.1 and A.3) Present Value of $1 to be Received After t Periods Period 1 2 3 4 5 6 7 8 1% 0.9901 0.9803 0.9706 0.9610 0.9515 2% 0.9804 0.9612 0.9423 0.9238 0.9057 3% 0.9709 0.9426 0.9151 0.8885 0.8626 4% 0.9615 0.9246 0.8890 0.8548 0.8219 5% 0.9524 0.9070 0.8638 0.8227 0.7835 6% 0.9434 0.8900 0.8396 0.7921 0.7473 To Find PVIF(4%,3) 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 • MBAM614 Class 2 - 25 Interpolation With Present Value Tables Present Value Tables Can Be Used to Estimate PVIF’s. This is Called “Interpolation.” To Find PVIF(3.5%,6) Present Value of $1 to be Received After t Periods 1% 0.9901 0.9803 0.9706 0.9610 0.9515 2% 0.9804 0.9612 0.9423 0.9238 0.9057 3% 0.9709 0.9426 0.9151 0.8885 0.8626 4% 0.9615 0.9246 0.8890 0.8548 0.8219 5% 0.9524 0.9070 0.8638 0.8227 0.7835 6% 0.9434 0.8900 0.8396 0.7921 0.7473 Period 1 3.5% is Halfway 2 Between 3% and 4% 3 so PVIF is Probably 4 Halfway Between 5 0.8375 and 0.7903 or ≈ 0.8139 (actual value is 0.8135) 6 7 8 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 • MBAM614 Class 2 - 26 13 • MBAM614 Class 2 - 27 Key Points $1 Tomorrow is Worth Less Than $1 Today Basic PV Equation: PV = FVt * 1 (1 + r)t If We Know Any 3 Variables, Can Solve For the Last Summing the PV of Individual CF’s is Equivalent to Discounting Back One Period at a Time • MBAM614 Class 2 - 28 14 MBAM 614 Finance The Time Value of Money Part II • MBAM614 Class 2 - 29 Summary of Last Section $1 Tomorrow is Worth Less Than $1 Today Basic PV Equation: PV = FVt * 1 (1 + r)t If We Know Any 3 Variables, Can Solve For the Last Summing the PV of Individual CF’s is Equivalent to Discounting Back One Period at a Time • MBAM614 Class 2 - 30 15 Agenda 1. Present and Future Value of an Annuity 2. Present Value of a Perpetuity 3. Net Present Value 4. EAR, APR and Interesting Rates • MBAM614 Class 2 - 31 Annuities A stream of multiple, identical cash flows occurring at the end of each period for some fixed number of periods is called an “Ordinary Annuity” (or Annuity) Example: A 4 year annuity paying $500 annually. t=0 1 2 3 4 $500 $500 $500 $500 • MBAM614 Class 2 - 32 16 Present Value of an Annuity Could PV each cash flow individually Use Annuity Present Value (APV) Equation: 1 APV = $C * 1 (1 + r)t r Called the Present Value Interest Factor for Annuities or PVIFA(r,t) APV = $C * 1 - PVIF(r,t) r • MBAM614 Class 2 - 33 Example: How Much Can You Borrow? If you make 36 monthly payments of $100 at 1.5% per month, how big a loan can you get? (This is equivalent to finding the APV of a $100 per period, 36 period annuity at 1.5% per period.) t=0 1 2 35 36 $100 $100 1 - ... 1 $100 $100 PVIFA(1.5%,36) = (1+0.015)36 0.015 = 27.66 so APV = $C * PVIFA(r,t) = $100 * 27.66 = $2,766 • MBAM614 Class 2 - 34 17 Example: Finding Loan Payments If you borrowed $12,000 for that Supra today and repaid the loan in 5 equal annual installments with 12% interest annually, what would your loan payments be? This is a 5 period (t = 5), 12% per period (r = 12%) annuity that has a present value of $12,000 (APV = $12,000). We need C. 1 PVIFA(12%,5) = 1 (1+0.12)5 0.12 = 3.6048 and $C = APV / PVIFA(r,t) = $12,000 / 3.6048 = $3,328.92 • MBAM614 Class 2 - 35 PVIFA Tables (Table A.2) Present Value of an Annuity of $1 per Period for t Periods To Find PVIFA(12%,5) = 3.6048 Periods 1 2 3 4 5 6 7 8 6% 0.9434 1.8334 2.6730 3.4651 4.2124 7% 0.9346 1.8080 2.6243 3.3872 4.1002 8% 0.9259 1.7833 2.5771 3.3121 3.9927 9% 0.9174 1.7591 2.5313 3.2397 3.8897 10% 0.9091 1.7355 2.4869 3.1699 3.7908 12% 0.8929 1.6901 2.4018 3.0373 3.6048 4.9173 4.7665 4.6229 4.4859 4.3553 4.1114 5.5824 5.3893 5.2064 5.0330 4.8684 4.5638 6.2098 5.9713 5.7466 5.5349 5.3349 4.9676 • MBAM614 Class 2 - 36 18 Example: Loan Length How long would it take to repay your $12,000 loan if you paid $3,000 at the end of every year and the annual interest rate was 10%? Here C = $3,000, r = 10%, and the APV = $12,000. We need t. Since APV = $C * PVIFA(r,t) then PVIFA(r,t) = APV / $C PVIFA(10%,t) = $12,000 / $3,000 = 4.0 Look up PVIFA(10%,t) = 4.0 in PVIFA Tables Solve for t directly • MBAM614 Class 2 - 37 Example: Loan Length Using PVIFA Tables Present Value of an Annuity of $1 per Period for t Periods To Find PVIFA(10%,t) = 4.0 Periods 1 2 3 4 5 about a of the way 6% 0.9434 1.8334 2.6730 3.4651 4.2124 7% 0.9346 1.8080 2.6243 3.3872 4.1002 8% 0.9259 1.7833 2.5771 3.3121 3.9927 9% 0.9174 1.7591 2.5313 3.2397 3.8897 10% 0.9091 1.7355 2.4869 3.1699 3.7908 12% 0.8929 1.6901 2.4018 3.0373 3.6048 between 5 and 6 t . 5a 6 7 8 4.9173 4.7665 4.6229 4.4859 4.3553 4.1114 5.5824 5.3893 5.2064 5.0330 4.8684 4.5638 6.2098 5.9713 5.7466 5.5349 5.3349 4.9676 • MBAM614 Class 2 - 38 19 Example: Exact Loan Length Solution 1 PVIFA(10%,t) = 1 (1+0.10)t 0.10 (1+0.10)t = = 4.0 or 1 - 0.4 = 1 (1+0.10)t so 1 (1 - 0.4) = 1.6667 ln[ (1+0.10)t ] = ln[ 1.6667 ] or t * ln[ 1.10 ] = ln[ 1.6667 ] t = ln[ 1.6667 ] / ln[ 1.10 ] = 0.5109 / 0.0953 = 5.36 years • MBAM614 Class 2 - 39 Example: 0% Financing! Slashin’ Sammy’s Stereo frequently advertises 0% financing for 12 months on new purchases - you simply make 12 equal monthly payments. If you enquire, you will find that they offer a 10% discount if you opt to pay cash instead. Is this really 0% financing? You could buy a $1000 stereo for $900 today (APV = $900) or for $1000/12 = $83.33 per month (C = $83.33) for 12 months (t = 12). $900 = $83.33 * PVIFA(r,12) or PVIFA(r,12) = $900/$83.33 = 10.80 r = 1.66% per month (or 19.90% per year!) To solve, you must use a Financial Calculator, Tables, or a lot of patience (guess after guess,…)! • MBAM614 Class 2 - 40 20 Example: To Rebate or Not to Rebate Keyes Mazda has a sale on Mazda Miatas at $18,999 (all-in). To sweeten the deal, you can choose 5% APR (5%/12 per month) financing for 36 months or a $1,100 rebate. If your bank is offering you a 10% (10%/12 per month) loan for the same 36 months, should you take the rebate? Your payments under the two options would be: Bank: APV = $18,999 - $1,100 = $17,899; t = 36 months; r = 0.833% $17,899 = C * PVIFA(10%/12,36) or C = $17,899/30.9912 = $577.55 Mazda: APV = $18,999; t = 36 months; r = 5%/12 = 0.417% $18,999 = C * PVIFA(5%/12,36) or C = $18,999/33.3657 = $569.42 • MBAM614 Class 2 - 41 Perpetuity A stream of identical cash flows occurring at the end of each period forever is called a “Perpetuity” Perpetuity Present Value: C r PPV = • MBAM614 Class 2 - 42 21 Example: Perpetual Preferred Shares One very common type of Perpetuity is a perpetual preferred share. Typically, these shares pay a fixed quarterly dividend (the annual amount is expressed as a percentage of par value) as long as the issuing firm is in business. What would a 7% perpetual preferred share with $50 par value be worth if today’s interest rate was 10% per year (or 2.5% per quarter)? C, the dividend, is ($50 * 7%)/4 = $0.875; r = 2.5% so PPV = $0.875 / 2.5% = $0.875 / 0.025 = $35.00 Note that the dividend rate (7%/4) is unrelated to the interest rate (2.5%) and that the “par value” is unrelated to the actual value (except that it determines the actual cash flows) • MBAM614 Class 2 - 43 Growing Perpetuity A stream of cash flows occurring at the end of each period that grows by a constant amount each period forever is called a “Perpetuity” Present Value of a Growing Perpetuity: When r > g, GPPV = C r-g • MBAM614 Class 2 - 44 22 Example: Valuing the Assets of a Firm Amgen, Inc. is a global biotechnology company that discovers, develops, manufactures and markets human therapeutics based on advances in cellular and molecular biology. For the year ending Dec 31 1998, Amgen had free cash flows to assets of $367.8 MM and experienced sales growth of 13.2%. Assuming that everyone expected FCFs to grow at the same rate the previous year’s sales and that Amgen expected to earn a 20% return on its assets, what were Amgen’s assets worth at year end 1998? Value of Assets = PV(Expected Future Free Cash Flows) = PV(Growing Perpetuity) Expected FCF1999 = FCF1998 * (1 + g) = $367.8 * 1.132 = $416.3 MM Value of Assets = $416.3 / (20% - 13.2%) = $6,123 MM • MBAM614 Class 2 - 45 Net Present Value The Net Present Value of an investment is the present value of its benefits less (net of) the present value of its costs. NPV = PV(Cash Inflows) – PV(Cash Outflows) When NPV > 0, benefits exceed costs (good investment) In the previous example, Amgen’s assets produced cash inflows worth $6,123 MM at the end of 1998. If you could buy those assets for $6,000 MM, what was the NPV of the investment? Was it a good investment? Why? NPV = $6,123 - $6,000 = $123 MM which is > 0, so yes! • MBAM614 Class 2 - 46 23 Interesting Rates The Stated or Quoted Interest Rate is the rate over some period before considering compounding effects. The Annual Percentage Rate (APR) is the rate per period times the number of periods per year. The APR is a Quoted or Stated Rate for one year. Note that the APR does not account for compounding The Effective Annual Interest Rate (EAR) is the rate, on an annual basis, that reflects the total interest paid annually including compounding. EAR = (1 + APR/m)m - 1 - m is the number of compounding periods per year - APR/m is the quoted rate per compounding period When comparing rates, ALWAYS convert to EAR’s • MBAM614 Class 2 - 47 Example: Quarterly Compounding How much would $1000 invested at 8% per year compounded quarterly grow to after one year? The 8% is a Quoted Rate but it is not the Quoted Rate for one compounding period. Since one compounding period is 1/4 of a year, the Quoted Rate for one compounding period is 8%/4 or 2% In one year, your $1000 would earn interest of 2% compounded 4 times. FV = $1000 * (1+0.02) * (1+0.02) * (1+0.02) * (1+0.02) = $1,082.43 EAR = (1 + 0.02)4 -1 = 0.0824 or 8.24% • MBAM614 Class 2 - 48 24 Example: Choosing a Loan You would really like to get that stereo from Sammy’s but you don’t have $900. You’ve talked to your bank, a savings & loan, and a financing company. This is what they offered: – Bank: 12.25% compounded semi-annually – Savings & Loan: 11.5% compounded daily – Finance company: 13% simple interest – In addition, Visa has a special promotion offering 12% compounded monthly Assuming you will not repay your loan for exactly one year, which should you choose? • MBAM614 Class 2 - 49 Example: Choosing a Loan All rates are expressed as an annual rate but they are not directly comparable. You really want to know how much interest you will be required to pay under each. Find and compare the EAR’s. Recall, Bank: S&L: Finance Co.: Visa: • MBAM614 EAR = (1 + Quoted Rate/m)m - 1 EAR = (1 + 12.25%/2)2 - 1 = (1.06125)2 - 1 = 12.63% EAR = (1 + 11.5%/365)365 - 1 = 12.19% EAR = 13% EAR = (1 + 12%/12)12 - 1 = 12.68% Class 2 - 50 25 Key Points 1 1. 1 (1 + r)t r = $C * PVIFA(r,t) C r C r-g APV = $C * 2. PPV = GPPV = 3. 4. NPV = PV(Cash Inflows) – PV(Cash Outflows) EAR = (1 + QR/m)m - 1 Class 2 - 51 5. • MBAM614 26