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Chapter 5

Chapter 5. Discounted Cash Flow Valuation. Overview. Important Definitions Finding Future Value of an Ordinary Annuity Finding Future Value of Uneven Cash Flows Finding Present Value of an Ordinary Annuity Finding Present Value of Uneven Cash Flows

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Chapter 5

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  1. Chapter 5 Discounted Cash Flow Valuation

  2. Overview • Important Definitions • Finding Future Value of an Ordinary Annuity • Finding Future Value of Uneven Cash Flows • Finding Present Value of an Ordinary Annuity • Finding Present Value of Uneven Cash Flows • Valuing Level Cash Flows: Annuities and Perpetuities • Comparing Rates: The Effect of Compounding Periods • Loan Types and Loan Amortization

  3. Important Definitions • Annuity: A level stream of cash flows for a fixed period of time. • Ordinary Annuity: Cash flows take place at the end of each period. • Annuity Due: Cash flows takes place at the beginning of each period. • Perpetuity: A level stream of cash flows forever. • Note: For now computations will be shown using formulations and calculator entries. I recommend focusing on calculator entries because it is much more efficient in getting answers. Formulations are provided for demonstration purposes. You should also review Excel file to see how excel functions can be used.

  4. Finding Future Value of an Ordinary Annuity

  5. Finding Future Value of Uneven Cash Flows • Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? • FV can be computed for each cash flow at a common future period (in this case 5 years from today) • FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

  6. Finding Future Value of Uneven Cash Flows – Continued • Calculator solution of the previous problem would involve determining FV of each payment in five years from today and then adding them • Total FV = 136.05 + 349.92 = 485.97

  7. Finding Present Value of an Ordinary Annuity

  8. Finding Present Value of Uneven Cash Flows • Find the PV of each cash flow and add them • Year 1 CF: 200 / (1.12)1 = 178.57 • Year 2 CF: 400 / (1.12)2 = 318.88 • Year 3 CF: 600 / (1.12)3 = 427.07 • Year 4 CF: 800 / (1.12)4 = 508.41 • Total PV = 178.57 + 318.88 + 427.07 + 508.41 • Total PV = 1,432.93

  9. Finding Present Value of Uneven Cash Flows – Continued

  10. Finding Present Value of Uneven Cash Flows – TI BA II PLUS • When using CF make sure to clear previous entries • CF  2ND CLR WORK

  11. Example • You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? • This cash flow set has two parts • PV of annuity payments 39 years from today • PV of a single payment in 39 years • See the timeline below

  12. Example – Continued • PV of annuity 39 years from today • PV of single payment in 39 years

  13. Example – Continued – TI BA II PLUS • PV of annuity 39 years from today • PV of single payment in 39 years

  14. Example – Continued – TI BA II PLUS

  15. Finding Future or Present Value of an Annuity Due • Annuity Due cash flows take place at the beginning of each period – this means that there is a payment now • The immediate payment requires that we account its impact on FV and PV correctly • The basic adjustments • Treat the annuity as if it were an ordinary annuity and determine FV or PV • Then multiply the answer by (1 + r) • FV or PV of Annuity Due = (FV or PV of Ordinary Annuity) x (1 + r) • If you are computing PMT then the adjustment is slightly different • PMT of Annuity Due = (PMT of Ordinary Annuity) / (1 + r) • This is because immediate payment with annuity due reduces amount borrowed leading to lower overall payment • I suggest using calculator settings for PMT, N, and I/Y computations

  16. Finding Future or Present Value of an Annuity Due – Continued • Consider the above annuity due. • This is a five year annuity due. • When you compute FV it is computed at the end of year 5. • Assume 10% rate.

  17. Finding Future or Present Value of an Annuity Due – TI BA II PLUS • To compute FV and PV of an annuity due you need to change MODE to BGN • 2ND BGN • 2ND SET • 2ND QUIT • Once complete you will see BGN on the display • To return to END follow the same steps • END mode is not displayed

  18. Example • Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

  19. Non Annual Interest Earnings (Compounding of Interest Rate) and Payments • Remember that FVt = PV0 x (1 + r)t • When dealing with non annual interest and payment you should make two adjustment • Interest rate should be periodic • Number of periods should reflect the total number of periods given number of years and frequency of interest earnings in a year • Formulations should be adjusted as follows • If “m” is the number of compounding in a year then where you see “r” divide it by “m” and multiply “t” by “m” • This assumes that “t” is the number of years • FVt = PV0 x (1 + r/m)t X m • Excel adjustment are same as formulations that can be made in a function dialog box or data cells

  20. Non Annual Interest Earnings (Compounding of Interest Rate) and Payments – Continued • FVt = PV0 x (1 + r/m)t X m • Entering the above equation into TI BA II PLUS • Method 1: • Keep P/Y = 1 (so is C/Y = 1) • N = t X m (Number of periods) • I/Y = r/m (Periodic interest rate) • Rest of the information is entered as before • Method 2: • Change P/Y = m (so is C/Y = m) • N = t X m (Number of periods) • I/Y = r (Annual interest rate) • Rest of the information is entered as before • Note that P/Y has to be checked for consistency every time

  21. ExampleBuying a Car • Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly. If you take a 4-year loan, what is your monthly payment?

  22. ExampleBuying a Car – TI BA II PLUS • Note the adjustment 2ND P/Y 12 ENTER 2ND QUIT

  23. ExampleCredit Card • Note: Finding interest rate and number of periods in annuity formulations are relatively time consuming. By this point you should be more comfortable with your calculator. From now on, solutions will be shown based on calculator entries. • You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month (18% per year compounded monthly). How long will you need to pay off the $1,000.

  24. Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the annual interest rate compounded monthly? ExampleBorrowing Money from Unlikely Resources

  25. Present Value of a Perpetuity • Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity • You can think of a perpetuity as an annuity that goes on

  26. ExamplePerpetuity • What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? • PV = $10,000 / 0.08 = $125,000

  27. Effective Annual Rate (EAR) vs. Annual Percentage Rate (APR) • Effective Annual Rate (EAR) • This is the actual rate paid (or received) after accounting for compounding that occurs during the year • If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. • Sometimes it is also called Annual Percentage Yield (APY) or Effective Rate • Annual Percentage Rate (APR) • This is the annual rate that is quoted by law • By definition APR = periodic rate times the number of periods per year • Consequently, to get the periodic rate we rearrange the APR equation: • Periodic rate = APR / number of periods per year • Sometimes it is also called Stated Rate, Quoted Rate or Nominal Rate • You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the periodic rate

  28. Rate Conversions • APR to EAR • EAR to APR • Calculator convention • NOM = APR • EFF = EAR • C/Y = m

  29. ExampleRate Conversion • You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? • Compare EAR and choose the larger one since you are investing. If you were borrowing then you would choose the lower EAR alternative • 5.25% compounded daily • 2nd ICONV • NOM 5.25 ENTER (EFF) • C/Y 365 ENTER (EFF) • CPT 5.39 • 5.30% compounded semiannually • 2nd ICONV • NOM 5.30 ENTER (EFF) • C/Y 2 ENTER (EFF) • CPT 5.37 • Choose 5.25% compounded daily

  30. ExampleRate Conversion – Continued • Let’s verify the choice. Suppose you invest $100,000 in each account. How much will you have in each account after 5 years? • Daily Compounded Account: • Semiannually Compounded Account: • You have more money in the first account.

  31. Amortized Loan with Fixed Payment • Each payment covers the interest expense plus reduces principal • Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.

  32. Amortization Table • Computations: • Interest Paid = Beginning Balance X Interest Rate • Principal Paid = Total Payment – Interest Paid • Ending Balance = Beginning Balance – Principal Paid

  33. Amortization Table – Continued • If you want to determine the outstanding balance of the loan after 2 years: • 2nd AMORT • 2 ENTER 2 ENTER • This will allow you to see loan information at that point in time. • You can change P1 and P2 to get the data for the specified payment range

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