time value of money and discounted cash flows
DESCRIPTION
Time Value of Money and Discounted Cash Flows. Compounding: Finding a future value for a current cash flow. Discounting: finding a present value for a future cash flow. Annuity:. Annuity: a sequence of equal cash flows, which pays at the end of each equal period. - PowerPoint PPT PresentationTRANSCRIPT
Time Value of Money and Discounted Cash Flows
Compounding: Finding a future value for a current cash flow
1
22
FV $1,000 (1 10%) $1,100
FV $1,100 (1 10%) $1,000 (1 10%) $1,210
(1 )nn iFV PV
Discounting: finding a present value for a future cash flow
PV=? FV=$1,000
0 1
0
1000909.09
(1 10%)PV
ni1FVPV
Annuity: Annuity: a sequence of equal cash flows, which pays
at the end of each equal period. Annuity Due: a sequence of equal cash flows, which
pays at the beginning of each equal period.
0 1 2 3 4
0 1 2 3 4
$0 $1,000 $1,000 $1,000 $1,000
$1,000 $1,000 $1,000 $1,000 $0
普通年金
到期年金
Finding future value for an annuity
0 1 2 3 4
$0 $1,000 $1,000 $1,000 $1,000FV=?
4(1 ) (1 10%)1000 4641
1 110%
niFVIFA PMT
i
Finding present value for an annuity
0 1 2 3 4
$0PV=?
$1,000 $1,000 $1,000 $1,000
41 (1 ) 1 (1 10%)1000 3169.9
10%
niPVIFA PMT
i
i1i
1i1i1FVIFAFVIFAD
i1ii11
i1PVIFAPVIFAD
n
in,in,
n
in,in,
Present and future values for annuity due
Perpetuity: an annuity with infinity periods, so the present value for a perpetuity becomes
iPMT
i
-ni)+(1-1PMT=PV
How to use Excel to compute FV FV(rate,nper,pmt,pv,type) Rate discount rate, if it is an annual rate, then put in
10%. If it is monthly rate, then state as 10%/12, or 0.83%.
Nper number of period. Pmt amount of annuity payment. pv amount of present value. Type 0 or 1. 1 for annuity due, zero (or omission for
annuity)
How to use Excel to compute PV
PV(rate,nper,pmt,fv,type) Rate discount rate, if it is an annual rate , then put in
10%. If it is monthly rate, then state as 10%/12, or 0.83%.
Nper number of period. Pmt amount of annuity payment. Fv amount of future value. Type 0 or 1. 1 for annuity due, zero (or omission for
annuity)
How to use Excel to compute rate and time period
RATE(nper,pmt,pv,fv,type) NPER(rate,pmt,pv,fv,type)
Note: When calculating rate and number of period, you need to specify the correct directions of the cash inflows (+) and cash outflows (-). At least one of the cash flows need to have a negative value.
PV of annuity and amortization of a mortgage loan
If someone borrow $300,000 from a bank for five years. The term is to pay an equal money at the end of each year, and interest rate is 10%.
$300,000 PMT *1- (1 10%)
10%
PMT = $300,000 / 3.7908 = $79,139
-5
year
A.Beg.
Balance
B.Paymen
t
C.Interest
（ A* i）
D.Principle
（ B-C）
E.End Bal.
（ A-D）
1 $300,000
$79,139
$30,000 $49,139 $250,861
2 $250,861
$79,139
$25,086 $54,053 $196,808
3 $196,808
$79,139
$19,681 $59,458 $137,349*
4 $137,349
$79,139
$13,735 $65,404 $71,945
5 $71,945 $79,139
$7,194* $71,945 $0
PV of annuity and amortization of a mortgage loan
Effective Interest RateInterest
Payment
Compound Number
in a year
Nominal
Rate
Period
Rate
Effective Annual Rate
Yearly 1 12% 12%
Semi-annually
2 12% 6%
Quarterly 4 12% 3%
Monthly 12 12% 1%
(1 +12%) -1 = 12%
(1 + 6%)2 1 12 36%.
(1 + 3%)4 1 12 55%.
(1 +1%)12 1 12 68%.
Effective Annual Rate (EAR) - Formula
1m
i+1=EAR
m
Continuous compounding:
1 , when mm
iie
m
Bond Valuation
0 1 2 3 n-1 n
0 C C C C Cpar
.....
n
n
=1tt i+1
Par
i+1
C=V
One firm issues a bond with 10-year, $10,000 par, paying 8% annual coupon, if the required rate for the bond is 10%, what will be the price you are willing to
pay ？
$800 $10,
$800*.
$10,$8, .
1 10%
000
1 10%
1 1 10%
010
000
1 10%7711
1
10
10
10
10
tt
PV(rate,nper,pmt,fv,type) = PV(10%,10, 800, 1000,0) = $8771.1
Yield to Maturity: the rate of return will be obtained if bond holders decide to hold the bond until maturity
One firm issues a bond with 10-year, $10,000 par, paying 10% annual coupon, if the current bond price is $9,500, then the yield to maturity is?
RATE(nper,pmt,pv,fv,type) = RATE(10,1000, -9500, 10000, 0) = 10.84%
Holding period return: the average rate of return that an bond investor has from purchasing the bond to selling the bond.
Four years ago, you purchase a 10-year, $10,000 par, 8% annual coupon for $9,000. Today after received the fourth coupon, you sell the bond for $11,089. What is your holding period return?
RATE(4,800,-9000,11089,0) =15.71%
Preferred Stock Valuation
Preferred stock pays fixed dividend, so the value of stock similar to present value of perpetuity.
pf
pfpf r
DP
A preferred stock with par of $100 paying 12%preferred dividend, if the required rate of return for the stock is 15%, then its price is ？ Preferred dividend per year $100*12%=$12
pfpf
pf
D $12P $80
k 0.15
Cont. Holding period return for a preferred stock
If an investor purchases the preferred stock from last page. Four years later, after the investor received the fourth dividend, he sell the stock. At the time of selling, the stock’s required rate of return raised to 20% due to the increased competition for the company issued the stock, what will be the investor’s holding period return?
Holding period return for preferred stock
The selling price for the stock after four years
So the rate of return can be calculated from
RATE(4,12,-80,60,0) =9.58%
pfpf
pf
D $12P $60
k 0.20
Common stock valuation
We can not predict all dividends for a stock over its entire life, so we have to make assumptions toward dividend behavior: Assuming Firm pay fixed dividend. Dividend increases in a constant growth rate Dividend first increases in high growth rates, then
increases in a constant growth rate.
P
D
1 + i
D
1 + i
D
1 + i
D
1 + i0
1 2 3 4 2 3 4
....
Public utility stocks often pay fixed dividend
ABC Utility usually pays $3 dividend per share each year. If the required rate of return for ABC is 14%, what will be its reasonable price?
s
DP =$3/0.14=$21.43
k
Large and established firms usually pay constant growth dividends.
PD
1+ i
D
1+ i
D
1+ iD
g
i
g
1+ i
g
1+ i0
1 2 30
2 3
2
2
3
3
1
1
1 1.... ....
PD
1+ i
D
1+ i
D
1+ i.... D
1 g
1 i
1 g
1+ i
g
1+ i....1
2 32
43 0
2 3
2 3
14
k
D P P
P
D
P
P P
P
D
P
P g P
P
D
Pgs
1 1 0
0
1
0
1 0
0
1
0
0 0
0
1
0
1
1 0 (1 )P P g
PD
k g01
s
P
D
k gtt+1
s
Dividend growth, payout ratio and return on equity
g = retention * ROE
Assuming ROE=20%, payout=50%, beginning equity = 100
A.
Beg. EQ
B. Earnings
(A*ROE)
C.
Dividend
(B*50%)
D. Ret. Earnings
(B-C)
E.
End EQ
(A+D)
100 20 10 10 110
110 22 11 11 121
121 24.2 12.1 12.1 133.1
133.1 26.6 13.3 13.3 146.4
Dividend growth, payout ratio and return on equity
Firm A has EPS of $1, assuming that market required rate of return toward its stock is 15%. Observe the stock prices on different ROEs (10% 、 15% 、 20%) and different payout ratios:
ROE=10% ROE=15% ROE=20%
40% payout
50% payout
100% payout
1
0
D =$1*0.4=0.4
g=10%(1-.4)=6%
0.4P 4.44
0.15 0.06
1
0
D =$1*0.4=0.4
g=15%(1-.4)=9%
0.4P 6.67
0.15 0.09
1
0
D =$1*0.4=0.4
g=20%(1-.4)=12%
0.4P 13.33
0.15 0.12
1
0
D =$1*0.5=0.5
g=10%(1-.5)=5%
0.5P 5.00
0.15 0.05
1
0
D =$1*0.5=0.5
g=15%(1-.5)=7.5%
0.5P 6.67
0.15 0.075
1
0
D =$1*0.5=0.5
g=20%(1-.5)=10%
0.5P 10.00
0.15 0.10
1
0
D =$1*100%=1
g=10%(1-1)=0%
1P 6.67
0.15 0
1
0
D =$1*100%=1
g=10%(1-1)=0%
1P 6.67
0.15 0
1
0
D =$1*100%=1
g=10%(1-1)=0%
1P 6.67
0.15 0
Dividend growth, payout ratio and return on equity
When ks > ROE, the higher the payout, the higher stock price.
When ks < ROE, the lower the payout, the higher stock price.
When ks = ROE, payout is irrelevant to stock price.
When payout ratio is 100%, thus no retention, the growth rate becomes zero, so the ROE is irrelevant to stock price.
Dividend first increases in high growth rates, then increases in a constant growth rate.
An up and rising firm this year issues $2 dividend (D0). The analyst expects the dividend will increase in 30% for two years, and 20% for another year before the dividend grows perpetually in 10%. If the required rate of return for the stock is 15%, how much will you pay for the stock?
$ D0
=2
$ D2
=2.6(1+0.3)
=3.38
$ D3
=3.38(1+0.2)
=4.06
30% 30% 20% 10% 10%
$ D4
=4.06(1+0.1)
=4.46
$ D1
=2(1+0.3)
=2.6
PD
k g34
s
4 46
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.
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