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Time value of money

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• This simple formula provides a link between often quoted number and our theory – P S = kE/r E rearrange as P S /E = k/r E \$250/\$100 = .5/.20 = 2.5 Our model has a P/E ratio of 2.5. That’s low relative to the market (NASDAQ is 30 odd) but this is a company with No earnings growth! Drawbacks to this Simple formula are many and varied No growth in future earnings or dividends. No uncertainty in the future dividends. No consideration tax treatment of dividends versus capital gains.
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• 1. Investment Tools – Time Value of Money
• 2.
• Concepts Covered in This Section
• Future value
• Present value
• Perpetuities
• Annuities
• Uneven Cash Flows
• Rates of return
Time Value of Money
• 3.
• Time lines show timing of cash flows.
• Tick marks at ends of periods.
• Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
• 90% of getting a Time Value problem correct is setting up the timeline correctly!!!
Interest Rate Cash Flows CF 0 CF 1 CF 3 CF 2 0 1 2 3 i%
• 4. What’s the FV of an initial \$100 after 3 years if i = 10%?
• Finding FVs (moving to the right on a time line) is called compounding.
• Compounding involves earning interest on interest for investments of more than one period.
FV = ? Future Values 100 0 1 2 3 10%
• 5. Single Sum - Future & Present Value
• Assume that you can invest PV at interest rate i to receive future sum, FV
• Similar reasoning leads to Present Value of a Future sum today.
1 2 3 0 FV 1 = (1+i)PV FV 3 = (1+i) 3 PV PV FV 2 = (1+i) 2 PV 1 2 3 0 PV = FV 1 /(1+i) FV 1 PV = FV 2 /(1+i) 2 FV 2 PV = FV 3 /(1+i) 3 FV 3
• 6. PV Calculation for \$100 received in 3 years if interest rate is 10% Single Sum – FV & PV Formulas FV n = PV(1 + i ) n for given PV \$100   = 0.7513 = \$75.13. 1.10 PV = \$100 1       3
• 7. Question on PV of a given FV
• Ex 1. An investor wants to have \$1 million when she retires in 20 years. If she can earn a 10 percent annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:
• A. \$100,000.
• B. \$117,459.
• C. \$148,644.
• D. \$161,506.
• This is a single payment to be turned into a set future value FV=\$1,000,000 in N=20 years time invested at r=10% interest rate.
• PV =[ 1/(1+r) ] N FV
• PV = [ 1/(1.10) ] 20 \$1,000,000
• PV 10 = [0.14864](\$1,000,000)
• PV 10 = \$148,644
• 8. Perpetuities Perpetuity is a series of constant payments, A, each period forever. Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment. PV perpetuity =  [A /(1+i) t ] = A  [ 1/(1+i) t ] = A/i 1 2 3 4 5 6 7 A 0 A A A A A A PV 1 = A/(1+r) PV 2 = A/(1+r) 2 PV 3 = A/(1+r) 3 PV 4 = A/(1+r) 4 etc. etc.
• 9.
• Regular or ordinary annuity is a finite set of sequential cash flows, all with the same value A , which has a first cash flow that occurs one period from now.
• An annuity due is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately .
Annuities
• 10. Time line for an ordinary annuity of \$100 for 3 years. \$100 \$100 \$100 i% Ordinary Annuity Timeline 0 1 2 3
• 11. Difference between an ordinary annuity and an annuity due ? Ordinary Annuity vs. Annuity Due PMT PMT 0 1 2 3 i% PMT Annuity Due PV FV Ordinary Annuity PMT PMT PMT 0 1 2 3 i%
• 12. Annuity Formula and Perpetuities Intuition : Formula for a N-period annuity of A is: PV of a Perpetuity of A today minus PV of a Perpetuity of A in period N 2 4 6 8 10 12 14 1. Perpetuity of A per period in Period 0 -- PV 1 = A/i A 0 A A A A A A A A A A A A A 2 4 6 8 10 12 14 2. Perpetuity of A per period in Period 8 -- PV 8 = [1/(1+i)] 8 x (A/i) 0 A A A A A A 2 4 6 8 10 12 14 3. Annuity of A for 8 periods -- PV = PV 1 – PV 8 = (A/i) x { 1 – [1/(1+i)] 8 } A 0 A A A A A A A
• 13. Annuities & Perpetuities Again
• Rather than memorize the annuity formula, it is easier to calculate it as the difference between two perpetuities with the same payment.
• PV of an N-period annuity of \$A per period is:
• PV N =
• (A/i) x { 1 – [1/(1+i)] N }
• Calculating the PV of an annuity has 3 steps:
• Calculate (A/i)
• PV of a Perpetuity with payments of \$A per period.
• Calculate [1/(1+i)] N
• Discount factor associated with end of the annuity.
• Calculate PV N =
• (A/i) x { 1 - [1/(1 + i)] N }
• I think this is easier under pressure than memorizing the formula.
• 14. Question on FV of Annuity Due
• Ex 2.An individual deposits \$10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9 percent interest compounded annually. The amount of money in the account at the end of 10 years will be closest to:
• A. \$109,000.
• B. \$143,200.
• C. \$151,900.
• D. \$165,600.
• This is an annuity due of A=\$10,000 for N=10 years at i=9% interest rate.
• Annuity due must be adjusted by (1+i) to reflect payment is made at beginning rather than end of period.
• Also must adjust PV formula by (1+i) N for FV of annuity.
• PV N = (1+i) N (1+i) [ ( A/i) { 1 – [1/(1+i)] N } ]
• PV 10 = (1.09) 11 (\$10K/.09) {1 – [1/1.09] 10 }
• PV 10 = (2.58)(\$111,111){1 – [0.42]}
• PV 10 = \$165,601
• 15. Time line for uneven CFs: \$100 at end of Year 1 (t = 1), \$200 at t=2, and\$300 at the end of Year 3. \$100 \$300 \$200 Uneven Cash Flows 0 1 2 3 i%
• 16. Question on Uneven Cash Flows
• Ex 3.An investment promises to pay \$100 one year from today, \$200 two years from today, and \$300 three years from today. If the required rate of return is 14 percent, compounded annually, the value of this investment today is closest to:
• A. \$404.
• B. \$444.
• C. \$462.
• D. \$516.
• This is a set of unequal cash flows. You could do it as a sum of annuities but it is easier to calculate it directly in this case.
• Interest rate is i =14%.
• PV =  [ 1/(1+i) ] t FV t
• PV = \$100/(1.14) + \$200/(1.14) 2 + \$300/(1.14) 3
• PV = \$87.72 + \$153.89 + \$202.49
• PV = \$444.10
• 17. Uneven Cash Flows Intuition : PV of uneven cash flows is equal to the sum of the PV’s of regular cash flows that sum to the uneven cash flows. 2 4 6 8 10 12 14 1. Uneven cash Flows over 10 periods – PV = PV 10 + PV 4 5 0 \$100 \$100 \$100 \$100 \$100 \$500 \$500 \$500 \$100 \$500 2. Annuity of \$100 per period for 10 periods -- PV 10 = { 1 - [1/(1+i)] 10 } x (A/i) 2 4 6 8 10 12 14 0 \$100 \$100 \$100 \$100 \$100 \$100 \$100 \$100 \$100 \$100 3. Annuity of \$400 per period for 4 periods from period 5 -- PV 4 5 = [1/(1+i)] 5 x [ (A/i) x { 1 – [1/(1+i)] 4 } ] 2 4 6 8 10 12 14 0 \$400 \$400 \$400 \$400
• 18. Comparison of Compounding Periods Annually: FV 3 = \$100(1.10) 3 = \$133.10. Semiannually: FV 6 = \$100(1.05) 6 = \$134.01. 0 1 2 3 10% 100 133.10 0 1 2 3 5% 4 5 6 134.01 1 2 3 0 100
• 19. Questions on Time Value
• Develop an approach to problems on Time Value.
• Draw the Time line for the cash flows.
• Put in the cash flows from the problem.
• Identify if single payment, annuity, annuity due, or perpetuity.
• If uneven cash flows can you break it into sums of annuities?
• Identify what is to be calculated – PV, FV, N or i ?
• Write out the appropriate formula, put in values for the variables, and calculate.
• Best Study Tip: Do the problems, and then do some more and then do some more!! Practice using your calculator!!
• 20. Possible Time Value Questions
• Present Value Formula
• Given FV N , i, N – solve for PV N
• Given PV N , i, N – solve for FV N
• Given PV N , FV N , N – solve for i
• Given PV N , FV N , i – solve for N
• Perpetuity Formula
• Given A, i – solve for PV per
• Given PV per , i – solve for A
• Given PV per , A – solve for i
• Annuity Formula
• Given A, i, N – solve for PV
• Given A, i, N – solve for FV
• Given PV, i, N – solve for A
• 21. Bonds and Their Valuation
• Key features of bonds
• Bond valuation
• Measuring yield
• Assessing risk
• 22. Key Features of a Bond
• Par value : Face amount; paid at maturity. Assume \$1,000.
• Coupon interest rate : Stated interest rate. Multiply by par value to get dollars of interest. Often fixed but can float with market rate.
• Maturity : Years until bond must be repaid. Declines.
• Issue date : Date when bond was issued.
• Default risk : Risk that issuer will not make interest or principal payments.
• 23. Valuing a 5-Period Bond Time = 0 1 2 3 4 5 6 7
• Discounted Cash Flow Approach
• Current Bond Price = Present value of all future Cash Flows (Interest & Principal) at required return, k B .
Bond Price, P B t Coupon Interest, CP Face Value, FV
• 24. The Right Discount Factor
• The discount rate (k i ) is the opportunity cost of capital , i.e., the rate that could be earned on alternative investments of equal risk.
• k i = k* + IP + DRP + MRP + LP
• k* = Real rate of interest
• IP = Inflation risk premium
• DRP = Default risk premium
• LP = Liquidity risk premium
• 25. What’s the value of a 10-year, 10% coupon bond if k d = 10%? V B = ? Bond Valuation Example \$100 \$100 \$100 + \$1,000 0 1 2 10 10% ... = \$90.91 + . . . + \$38.55 + \$385.54 = \$1,000. V k k B d d  \$100 \$1 , (1 000 (1 1 10 10 . . . + \$100 (1 + k d + + + + ) ) )
• 26. Stocks and Their Valuation
• Features of common stock
• Determining common stock values
• 27.
• Represents ownership.
• Ownership implies control.
• Stockholders elect directors.
• Directors hire management.
• Management’s goal: Maximize stock price.
Features of Common Stock
• 28. Valuing Common Stock Time = 0 1 2 3 4 5 6 7 Uncertain Dividends, D t+i
• Dividend Discount Model
• Current Stock Price = Present value of all future Expected Cash Flows (Dividends) at required return, k S .
Stock Price, P S t
• 29.
• Constant Growth stock
• One whose dividends are expected to grow forever at a constant rate, g.
• Can link this to earnings by assuming that firm pays out a fixed percentage of earnings as dividends
• i.e. D t = k x E t where k equals payout ratio
Stock Value = PV of Dividends