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.
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
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.
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
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%
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
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
MRP = Maturity 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 + + + + ) ) )
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