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**Unformatted text preview: **Future Value
The value of a lump sum or stream of cash payments at a future point in time: FVn = PV(1+r)^n interest rate (up in IR up in FV) Number of periods (yrs) compounding interval Future Value depends on: 1 The Power of Compound Interest
41 Future Value of One Dollar ($) 36 20% 31 26 21 16 11 15% 6 1 1 3 5 7 9 11 13 15 17 19 21 10% 5% 0% 23 25 2 Periods Future Value of $200 (4 Years, 7% Interest )
FV4 = 262.16 FV3 = 245.01 FV2 = 228.98 FV1 = 214 PV = $200
0 1 2 3 4 End of Year Compounding: the process of earning interest in each successive year
3 Compounding Intervals
m compounding periods FVn = PV (1+r/m)^m*n The more frequent the compounding period, the larger the FV!
4 Compounding More Frequently Than Annually
FV at end of 2 years of $125,000 deposited at 5% interest Annually = $137,812.50 For semiannual compounding, m equals 2: FV2=125,000(1+.05/2)^2*2 = $137,976.61 For quarterly compounding, m equals 4: FV =125,000(1+.05/4)^4*2 =$138,060.76
5 The Stated Rate versus the Effective Rate
Stated rate: the contractual annual rate charged by lender or promised by borrower Effective rate: the annual rate actually paid or earned EAR = ((1+r/m)^m) 1
6 M=# of compounds Effective Rates: Always greater Than or Equal to Stated Rates For annual compounding, effective = stated EAR = (1+.05/1)^1 1=5% For semiannual compounding EAR = (1+.05/2)^2 1=5.06% For quarterly compounding EAR = (1+.05/4)^4 1=5.09%
7 Future Value of Ordinary Annuity (End of 5 Years, 5.5% Interest Rate)
$1238.82 $1174.24 $1113.03 $1055.00 $1000.00_ ________ TTL = 5581.09 $1,000 0 1 $1,000 2 $1,000 3 $1,000 4 $1,000 5 End of Year FV=PMT*((1+r)^n 1)/r) = $5581.09
How is annuity due different ? 8 Future Value of Annuity Due (End of 5 Years, 5.5% Interest Rate)
$1,306.96 $1,238.82 $1,174.24 $1,113.02 $1,055.00
$1,000 0 $1,000 1 $1,000 2 $1,000 3 $1,000 4 5 n=6 End of Year FV =PMT(((1+r)^n 1)/r)*(1+r) = $5,888.04
Annuity due: payments occur at the beginning of each period 9 Present Value
Today's value of a lump sum or stream of cash payments received at a future point in time: FVn = PV = ( + r ) 1 n PV =
10 FVn (1 + r )
n The Power of High Discount Rates
Present Value of One Dollar ($)
1.00 0% 0.75 0.5 5% 0.25 10% 15% 20% 0 2 4 6 8 10 12 14 16 18 20 22 24 Periods 11 Present Value of $200 (4 Years, 7% Interest )
Discounting
0 1 FV1 = $200 PV = $186.92 PV = $174.69 PV = $163.26 PV = $152.58 2 FV2 = $200 3 FV3 = $200 4 FV4 = $200 End of Year 12 discounting: the process of converting a future cash flow into a present value Present Value of Ordinary annuity (5 Years, 5.5% Interest Rate)
0 1 $1,000 2 $1,000 3 $1,000 4 $1,000 5 $1,000 End of Year $947.87 $898.45 $851.61 $807.22 $765.13 PV=(PMT/r)*[1(1/(1+r)^n)] = $4,270.28
13 Present Value of Annuity Due (5 Years, 5.5% Interest Rate)
0 $1,000 1 $1,000 2 $1,000 3 $1,000 4 $1,000 5 End of Year $947.87 $898.45 $851.61 $807.22 PV = (PMT/r)*[1(1/((1+r)^n))]*(1+r)= $4,505.15
14 Present Value of Perpetuity ($1,000 Payment, 7% Interest Rate)
Stream of ________ annual cash flows that lasts _________ PV = PMT
t =1 1 t (1 + r ) PV = PMT*(1/r) =$1000*(1/.07) =$14,285.71
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