Discounted Cash Flow Valuation MBAC 6060 Chapter 4 Discounted Cash Flow Valuation

Chapter Outline: 4.1 One-Period Case 4.2 Multi-Period Case 4.3 Compounding Periods 4.4 Simplifications 4.5 Loan Amortization 4.6 What Is a Firm Worth?

Key Concepts and Skills: Compute the future value and/or present value of a single cash flow or series of cash flows Compute the return on an investment Use a spreadsheet to solve time value problems Understand Annuities and Perpetuities 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 1 year instead: You can lend the $100 today for one year And have more than $100 in one year So if I don’t get the money for one year, I need to get more than $100 How much more? Talk about that in a soon!

FV = C0 x (1+r)t PV = Ct /(1+r)t 4.1 and 4.2 One Cash Flow One- and Multi-Period Cases: FV = C0 x (1+r)t PV = Ct /(1+r)t FV = Future Value PV = Present Value Ct = Cash Flow at time t r = The interest rate We will solve for each of these variables If we have the other 3. And talk about what the variables mean

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? t = 1 r = 10% PV = $100 FV = ? In 1 year the FV = PV( 1 + r)1 = $100(1.1)1 = $110 Save $100 for 2 years at 10% interest Leave $110 in bank for a second year: $110(1.1) = $121 $100(1.1)(1.1) = $100(1.1)2 General Notation $100(1 + r)t  (1 + r)t is sometimes called the Future Value Interest Factor

Table A.3 (page 966 in the book) Actually shows at TABLE of FVIFs Use the table to look up FVIF for 10% in 5 years: 1.6105 So $100 in 5 years is worth $100(1.6105) = $161.05 But nobody uses tables anymore! My grandfather used tables! Use your spreadsheet (or calculator)! This is equal to $100(1.10)5 = $100(1.6105) = $161.05 Using Excel: =FV(rate, nper, pmt, [pv], [type]) =FV(.10,5,0,100) = -161.05 Why is the Excel answer negative? 

This is the formula that is in Excel: 0 = PV + FV/(1+r)t Solve for FV: FV = -PV(1 + r)t = -100(1.1)5 = -161.05 This formulation allows for “signing: cash flows: Positive is an inflow Negative is an outflow The assumption is If PV is positive (get money now) Then FV must be negative (pay money later) But by convention, we report positive values

Simple Interest vs. Compound Interest $100 in Five Years at 10% per year: Interest Factor: (1 + r)t = 1.15 = 1.61051 Future Value: $100(1.61051) = $161.05 10% Interest on the $100 in each year is $10 Over five years it is $50 The extra $61.05 - $50 = $11.05 is interest on interest Also called COMPOUND INTEREST

Present Value Present Value: What will a payment made later be worth today? Receive $5,000 in 12 years discounted by 6% interest. Calculate the Present Value: PV = FV/( 1 + r)t = $5,000/(1.06)12 = $2,484.85 Using Excel: =PV(rate, nper, pmt, [fv], [type]) =PV(.06,12,0,5000) = -2484.85 $2,484.85 < $5,000 $2,484.85 is the PV of $5,000 (at 6% over 12 years) $5,000 is the FV of $2,484.85 (at 6% over 12 years)

Review Question: What is the PV of $10,000 if you receive the money in 5 years and it is discounted at 12% per year? What is the FV in 8 years of $30,000 paid today if it earns 9% per year? Note: Even though Excel will show negative numbers as FV and PV outputs, we still report positive values. PV = $16,000 and FV = $49,200 PV = $10,000 and FV = $49,200 PV = $4,000 and FV = $49,200 PV = $4,000 and FV = $59,777 PV = $5,674 and FV = $59,777

Review Answer: The answer is E. PV of $10,000 paid in 5 years at 12%: =PV(rate, nper, pmt, [fv], [type]) =PV(.12,5,0,10000) = -5,674 FV in 8 years of $30,000 paid today at 9%: =FV(rate, nper, pmt, [pv], [type]) =FV(.09,8,0,30000) = -59,777 The answer is E.

Interpreting PV and FV Example: Example: Your company can pay $800 for an asset it believes it can sell for $1,200 in 5 yrs. Similar investments pay 10% What does similar mean? Another investment with the same risk (a stock or bond issued by another similar company) pays 10% So is paying $800 for something that can be sold in 5 yrs for $1,200 a good idea? PV = FV/(1+r)t = $1,200/(1.1)5 = $745.11 In Excel: =PV(.10,5,0,1200) = -745.11 FV = PV(1+r)t = $800(1.1)5 = $1,288.41 In Excel: =FV(.10,5,0,800) = -1288.41 It is a Bad Idea! You need to earn 10% so either: You should pay less than $800 (pay $745.11) to get $1,200 You should pay $800 to get more than $1,200 (get $1,288.41)

Review Question: An investment costs $20,000 and you expect to hold it for 10 years. Investments with similar risk earn 12% per year. If the projected sale price is $75,000, is this a good investment idea? YES. NO. MAYBE.

Review Answer: =FV(rate, nper, pmt, [pv], [type]) If the projected sale price exceeds the calculated FV, then it is a good idea: FV = PV(1+r)t = $20,000(1.12)10 = $20,000(3.10585) = $62,117 Or using Excel: =FV(rate, nper, pmt, [pv], [type]) =FV(.12,10,0,20000) = -62,117 $75,000 > $62,117 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 $62,117, 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 PV(1 + r)t = FV (1 + r)t = FV/PV 1 + r = (FV/PV)(1/t) r = (FV/PV)(1/t) – 1 What rate is need to increase $200 to $400 in 10 years? r = ($400/$200)(1/10) – 1 = 0.071 = 7.18% What rate is need to increase $200 to $400 in 8 years? r = ($400/$200)(1/8) – 1 = 0.0905 = 9.05% In Excel: =RATE(nper, pmt, pv, [fv], [type]) =RATE(8,0,200,400) = #NUM =RATE(8,0,-200,400) = 9.05% =RATE(8,0,200,-400) = 9.05%

Review Question: What rate is needed to increase $20,000 to $80,000 in 10 years? Rate not Found 7.18% 14.87% 20.00% 30.00%

Review Answer: r = (FV/PV)(1/t) – 1 = (80/20)(1/10) – 1 = (80/20)(1/10) – 1 = (4)(1/10) - 1 = 0.1487 = 14.87% Or =RATE(nper, pmt, pv, [fv], [type]) =RATE(10,0,-20,80) = 0.1487 = 14.87% The answer is C. Note: =RATE(10,0,-1,4) = 0.1487 = 14.87% =RATE(10,0,1,-4) = 0.1487 = 14.87%

See page 99 for tips to calculating FV, PV, RATE & NPER in Excel Determine the Number of Periods: Solve for t (or N) PV(1 + r)t = FV (1 + r)t = FV/PV ln(1 + r)t = ln(FV/PV) t[ln(1 + r)] = ln(FV/PV) t = [ln(FV/PV)]/[ln(1 + r)] But we’ll just use the machine: How many years are needed to increase $200 to $400 at 7.18% =NPER(rate, pmt, pv, [fv], [type]) =NPER(.0718,0,200,400) = #NUM! =NPER(.0718,0,-200,400) = 10.00 =NPER(.0718,0,200,-400) = 10.00 (–PV and +FV or +PV and –FV both work) See page 99 for tips to calculating FV, PV, RATE & NPER in Excel

Review Question: How many years are needed to get $1,000,000 if you invest $22,095 and earn 10% per year? 20.26 22 22.26 40 Not Found

Review Answer: t = [ln(FV/PV)]/[ln(1 + r)] = ln(1,000,000/22,095)/ln(1.1) = 40 Or =NPER(rate, pmt, pv, [fv], [type]) =NPER(.10,0,-22095,1000000) = 40 =NPER(.10,0,22095,-1000000) = 40 The answer is D.

Rule of 72’s If FV/PV = 2 (your money doubles) then (r)(t) ≈ 72 Example: Start with $200 and get $400 then FV/PV = 2 $200 to $400 in 10 years at 7.18% (10)(7.18) = 71.8 ≈ 72 $200 to $400 in 8 years at 9.05% (8)(9.05) = 72.4 ≈ 72 If the value of your stock doubles in 5 years, what is the approximate annualized compounded return? (r)(t) ≈ 72  (r)(5) ≈ 72  (r) ≈ 72/5 = 14.4%

Review Question: You own a house that you believe has doubled in value over the last 20 years. Using the Rule of 72’s, estimate the approximate annual return on the house. 3.60% 5.25% 7.20% 10.00% 20.00%

Review Answer: (r)(t) ≈ 72 (r)(20) ≈ 72  (r)(20) ≈ 72/20 = 3.6 That answer is A Check this answers using the calculator’s TVM function: N= 20 PV = -1 FV = 2 I/YR = 3.53 ≈ 3.6 Bonus Question: Assume you will earn 12%? How long to quadruple in price? Quadruple is double twice: (r)(t) ≈ 72  72/r ≈ t  72/12 = 6 years So double in 6, double twice in approximately 12 years Check this: r = 12 PV = -1 FV = 4 N = 12.23 ≈ 12

Recap: FV = PV(1+r)t FV PV r (also called RATE) Solve for any of the four variables FV PV r (also called RATE) t (also called N or NPER) Use the Excel functions to solve for the one variable not given Be sure to understand the economic meaning of the values

The FV of Multiple CFs For any one CF: FV = PV(1+r)t For multiple CFs, the FV is the sum of each FV Example: Receive $100 at t = 0 and t = 1. Calc the FV at t = 2 if the rate is 8% The first $100 increases twice. The second $100 increases once. FV = $100(1.08)2 + $100(1.08) = $224.64

Another Example: You currently have $7,000 in an account (at t = 0) You will deposit $4,000 at the end of each of the next 3 years (at t = 1, t = 2 and t = 3) How much will you have at time 3 at 8%? $7,000 at t = 0 with 3 years of interest  $7,000(1.08)3 = $8,818 $4,000 at t = 1 with 2 years of interest  $4,000(1.08)2 = $4,666 $4,000 at t = 2 with 1 year of interest  $4,000(1.08)1 = $4,320 $4,000 at t = 3  $4,000 = $4,000

Example continued Same Example, but now… How much will you have at time 4 at 8%? $7,000 at t = 0 with 4 years of interest  $7,000(1.08)4 = $9,523 $4,000 at t = 1 with 3 years of interest  $4,000(1.08)3 = $5,039 $4,000 at t = 2 with 2 years of interest  $4,000(1.08)2 = $4,666 $4,000 at t = 3 with 1 year of interest  $4,000(1.08)1 = $4,320

Calculations: FV at t = 3: FV at t = 4: $7,000(1.08)3 = $8,818 $4,000(1.08)2 = $4,666 $4,000(1.08)1 = $4,320 $4,000(1.08)0 = $4,000 $21,804 FV at t = 4: $7,000(1.08)4 = $9,523 $4,000(1.08)3 = $5,039 $4,000(1.08)2 = $4,666 $4,000(1.08)1 = $4,320 $23,548

Now Calculate the PV of Multiple CFs: You need $1,000 at t = 1 and $2,000 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%? $1,000/(1.09) + $2,000/(1.09)2 = $2,600.79

Think about the PV this way: Invest $2,601 at 9%. Show that you can withdraw $1,000 at t = 1 and withdraw $2,000 at t = 2: $2,600.79(1.09) = $2,834.86 (at t = 1) $2,834.86 - $1,000 = $1,834.86 (withdraw $1,000 at t = 1) $1,834.86(1.09)2 = $2,000 (available at t = 2) So if you invest $2,601 at 9%, you can withdraw $1,000 at time 1 and $2,000 at time 2 The PV of $1,000 at time 1 and $2,000 at time 2 is $2,601

Review Question: If your investment earns 10%, how much do you need to invest now to be able to withdraw $500 in one year and $800 in two years? $1,000 $1,116 $1,200 $1,226 $1,300

Review Answer: PV of $500 in one year = $500/(1.1) = $455 In order to withdraw $500 in one year and then $800 in two years, you must invest the sum of the PVs of these withdrawals. PV of $500 in one year = $500/(1.1) = $455 PV of $800 in two years = $800/(1.1)2 = $661 Sum = $455 + $661 = $1,116 The answer is B.

4.3 Compounding Periods We will cover this after section 4.5

4.4 Multiple CFs Some Terms: Annuity Perpetuity Growing Perpetuity A stream of constant cash flows that lasts for a fixed number of periods Perpetuity A constant stream of cash flows that lasts forever Growing Perpetuity A stream of cash flows that grows at a constant rate forever Growing Annuity A stream of cash flows that grows at a constant rate for a fixed number of periods

Word Annuity has two definitions Economic Definition: All CFs are the same CFs occur at regular intervals (Annually, Semi-annually, Quarterly, Monthly…) All CFs are discounted at the same rate The Financial Product: Pay an insurance company or a bank a lump sum today Receive CFs at regular intervals for a fixed period or until you die 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) A Purchased Annuity (you are paid)

Formula for PV of an Annuity (PVA) 𝑃𝑉𝐴= 𝐶 𝑟 1− 1 1+𝑟 𝑡 PVA = (C/r)[1 - 1/(1 + r)t] (Same thing but typed) Text Book’s Notation: 1/(1 + r)t = Present Value Factor (PVF) PVA = C{[1 - PVF]/r} Other Notation: {[1 - 1/(1 + r)t]/r} = Present Value Annuity Factor (PVAF) PVA = C{PVAF}

We’ll use Excel’s PV function: PV of $1,000 per for 5 years @ 6%: We’ll use Excel’s PV function: =PV(rate, nper, pmt, [fv], [type]) =PV(.06,5,1000) = -4,212.36

Future Value of an Annuity You will receive $50 per year for next 10 years When you get the money, you will deposit it in a bank and earn 7% Calculate the FV of a 10 yr, $50, 7% annuity? Formula: FVA = C{[(1 + r)t -1]/r} = C{FVAF} Excel: =FV(rate, nper, pmt, [pv], [type]) =FV(0.07,10,50) = -690.82 Note: 10 x $50 = $500 < $690.82 $690.82 - $500 = $190.82 is interest and interest-on-interest

PV of a Growing Annuity A growing stream of cash flows with a fixed maturity: 1 C 2 C×(1+g) 3 C ×(1+g)2 T C×(1+g)T-1

Growing Annuity Example A defined-benefit retirement plan pays $20,000 per year for 40 years Payments increase by 3% each year. Calculate the PV at retirement if the discount rate is 10%? 1 $20 2 $20×(1.03) 40 $20×(1.03)39

Growing Annuity Example What if you will not retire for five more years? Calculate today’s value of the retirement plan Recall: A defined-benefit retirement plan pays $20,000 per year for 40 years. Payments increase by 3% each year and the discount rate is 10% At retirement the plan is worth: PV = C/(r-g){1 – [(1 + g)/(1 + r)]T} = 20,000/(.1 – .03){1 – [1.03/1.1)]40} = $265,121.57 Five years earlier it is worth: 265,121.57/(1.1)5 = 164,619.64

PV Annuity vs. Growing Annuity There is no good way (I think) to calculate the PV of a growing annuity in Excel 𝑃𝑉Annuity= 𝐶 𝑟 1− 1 1+𝑟 𝑡 𝑃𝑉 Growing Annuity= 𝐶 𝑟−𝑔 1− 1+𝑔 1+𝑟 𝑡

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 $100 Annuity and an Annuity Due: In Excel, use the “Type” indicator =PV(rate, nper, pmt, [fv], [type]) Type = 1 for payments at the end of the period Example 

Compare an Annuity Due to an Annuity Is the PV of a 4 yr Annuity Due greater than or less than the PV a regular 4 yr annuity? Would you rather be paid the Annuity Due or the Annuity? Assume a 10% discount rate The 4 yr Annuity Due is the same as a 3 yr (regular) Annuity plus an extra $100 now (at time zero): 3 yr Annuity: =PV(.10,3,100) = 249 PV 3 yr Annuity + $100 = 349 In Excel =PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100,,1) = 349 =PV(rate, nper, pmt, [fv], [type]) = PV(.10,4,100) = 317

Review Question: You will pay $1,000 per month to rent apartment for a year The lease requires monthly payments at the beginning of each month. Assume a 12% APR-Monthly discount rate. Note: 12% APR means 1% per month 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. $1,343 $1,000 $743 $500 $113

Review Answer: PV of a 12 month, $1,000 (Regular) Annuity discounted at 12% APR. 12% APR means 1% per month =PV(.01,12,1000) = 11,255 PV of a 12 month, $1,000 Annuity Due discounted at 12% APR: =PV(.01,12,1000,,1) = 13,368 Net Benefit = $11,368 – 11,255 = $113 The Answer is E

PV of a Perpetuity A perpetuity is a level stream of CFs that lasts forever The PV of a Perpetuity equals: The denominator in each successive term in the brackets has a larger exponent, so the value approaches zero This is known as a “convergent sequence” The value to which it converges is: 𝑃𝑉= 𝐶 1+𝑟 + 𝐶 1+𝑟 2 + 𝐶 1+𝑟 3 +⋯ 𝑃𝑉=𝐶 1 1+𝑟 + 1 1+𝑟 2 + 1 1+𝑟 3 +⋯ 𝑃𝑉=𝐶 1 𝑟 = 𝐶 𝑟

Perpetuity Example A company’s preferred stock will pay $5 annual dividend forever Preferred stock of similar risk has a 6% return Calculate the price of the preferred stock PV = C/r = $5/.06 = $83.33

PV of a Growing Perpetuity CFs grow at a constant rate (g) forever The PV of a Growing Perpetuity equals: If g < r (and it pretty much has to be), then 𝑃𝑉= 𝐶 0 1+𝑔 1 1+𝑟 1 + 𝐶 0 1+𝑔 2 1+𝑟 2 + 𝐶 0 1+𝑔 3 1+𝑟 3 +⋯ 𝑃𝑉= 𝐶 0 1+𝑔 1 1+𝑟 1 + 1+𝑔 2 1+𝑟 2 + 1+𝑔 3 1+𝑟 3 +⋯ 𝑃𝑉= 𝐶 0 1+𝑔 1+𝑟 1 + 1+𝑔 1+𝑟 2 + 1+𝑔 1+𝑟 3 +⋯ 1+𝑔 1+𝑟 <1

PV of a Growing Perpetuity If g < r, then each fraction is less than one So this is also a convergent sequence If converges to: 𝑃𝑉= 𝐶 0 1+𝑔 1+𝑟 1 + 1+𝑔 1+𝑟 2 + 1+𝑔 1+𝑟 3 +⋯ 𝑃𝑉= 𝐶 0 1+𝑔 𝑟−𝑔 = 𝐶 0 1+𝑔 𝑟−𝑔 = 𝐶 1 𝑟−𝑔

Growing Perpetuity Example A company has a fixed finance and operating policy such that its common stock dividend will grow at a fixed rate of 4% forever. It is expected to pay a dividend of $2 in one year Similar stocks have a required return of 14% Calculate the price of the common stock PV = C1/(r – g) = $2/(0.14 – 0.04) = $20 Now assume increased efficiency will allow the company increase its growth rate to 6% forever But the change will not affect the next dividend Calculate the new price of the common stock PV = C1/(r – g) = $2/(0.14 – 0.05) = $25

4.5 Loan Amortization How will loan’s principal will be repaid? Pure Discount Loan Two CFs: Borrower receives money at beginning Borrows makes a single payment at end The single payment covers both interest and principal Zero-coupon Bonds, CP, T-Bills and CDss Interest-Only Loans Borrower makes periodic payments which are just interest Principal (and the final interest payment) paid at the end Most bonds (Gov, Corp, Muni) Amortizing (or Self-Amortizing) Loans Periodic payments include principal and interest Payments are calculated using Excel PMT function Loan amount is PV of the “annuity” payments Consumer loans

Self-Amortization Loans Example: Consider a 5 year, $3,000 self-amortizing at 10% annual interest. Calculate the annual payments: Use Excel PMT function: =PMT(rate, nper, pv, [fv], [type]) =PMT(.10,5,3000) = -791.39 The Amortization Schedule shows the portion of each fixed payment that is interest the portion that repays the loan

Amortization Schedule 5 year, $3,000, 10% annual, self-amortizing loan Annual Payments are $791.39 Beg Bal = Previous Period’s End Bal = 2,508.61 Interest = Beg Balance x Rate = 2,508.61 x 0.1 = 250.86 Principal = Payment – Interest = 791.39 – 250.86 = 540.53 End Bal = Beg Bal – Prin = 2,508.61 – 540.53 = 1,968.08 At the end of year 2, $1,968.08 not yet repaid Period Beg Bal PMT Interest Principal End Bal 1 $3,000.00 $791.39 $300.00 $491.39 $2,508.61 2 $250.86 $540.53 $1,968.08 3 $196.81 $594.58 $1,373.50 4 $137.35 $654.04 $719.46 5 $71.95 $719.44 $0.01

Amortization Schedule What is the ending balance at time 2? It is the amount that still has to be paid It is also the PV of remaining payments At time 2, there are 3 more $791.39 payments at 10% =PV(rate, nper, pmt, [fv], [type]) =PV(.1,3,791.39) = 1,968.08 Also the value of a “balloon payment” due at time 2 Period Beg Bal PMT Interest Principal End Bal 1 $3,000.00 $791.39 $300.00 $491.39 $2,508.61 2 $250.86 $540.53 $1,968.08 3 $196.81 $594.58 $1,373.50 4 $137.35 $654.04 $719.46 5 $71.95 $719.44 $0.01

4.3 Compounding Periods Annual Percentage Rate (APR) APR = Periodic Rate x # of Periods Periodic Rate = APR / # of Periods A credit charges 18% APR monthly The rate is actually 18/12 = 1.5% per month What annual rate is equivalent? Called the Effective Annual Rate (EAR) EAR = (1.015)(1.015)(1.015)…(1.015) – 1 = (1 + 0.18/12)12 – 1 = 19.56% So indifferent between paying 18% APR Monthly and paying 19.56% per year

4.3 Compounding Periods Example: An investment pays 12% APR Semi-Annual Calculate the FV of $1,000 invested for 3 years EAR = (1 + APR/m)m = (1 + 0.12/2)2 - 1 = 12.36 % FV = C(1 + EAR)T = $1,418.52 = $1,000(1.1236)3 = $1,418.52 Combine the calculations: FV = C(1 + APR/m)(m x T) = $1,000(1 + .12/2)(2 x 3) = $1,418.52

Three thousand, six hundred and eighty-six percent! 4.3 Compounding Periods Formulas: EAR = (1 + APR/m)m – 1 APR = m[1 + EAR)1/m – 1] Example: Payday Loans cost $75 for a $500 two-week loan Calculate the APR and EAR Assume there are exactly 26 two-week periods per year. Periodic rate = $75/$500 = 15% APR = Periodic Rate x # of Periods = 0.15 x 26 = 390% EAR = (1 + APR/m)m – 1 = (1.15)26 – 1 = 3686% Yes, Really. Three thousand, six hundred and eighty-six percent!

Compounding Periods in Excel EAR is called EFFECT APR is called NOMINAL Assume 8% mortgage (monthly rate). Calculate the EAR: EFFECT(nominal_rate, npery) EFFECT(.08,12) = 8.30% A one-year CD pays 5%. What is the equivalent quarterly rate? NOMINAL(effective_rate, npery) NOMINAL(.05,4) = 4.91% The periodic rate is 4.91%/4 = 1.23%

Continuous Compounding EAR = (1 + APR/m)m – 1 As m increases, the EAR for 10% increases As m approaches infinity  EAR = er – 1 Called “Continuous Compounding” 10% Continuously Compounded EAR = e.10 – 1 = 10.52% m EAR 1 10.00% 2 10.25% 4 10.38% 12 10.47%

4.6 A Company’s Value A company is worth the present value of its cash flows The problems are Determining the size of the cash flows Determining the timing of the cash flows Determining the correct discount rate for the cash flows But we will value stocks and bonds by discounting the cash flows paid to the owners