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
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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.
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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
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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%
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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
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 + + + + ) ) )
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Notas del editor
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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