1. MEASURING WEALTH: TIME VALUE OF MONEY CHAPTER 3
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2. A. future value of a single amount
FVn = PV (1+ K)n , FV = PV (1+K/q)qn
q = the number of times interest is compounded
(1+ K)n , (1+K/q)qn , FVIF1 k,n,nq
Find the future value after 20 years of $100 deposited in an account paying 10% rate of interest , compounded annually
Solution: FV20 = $100 ( )20 =
$100 x = $672.75
If any three of the four variables (FV, PV, k, n) are known, the fourth can be calculated.
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3. B. Present value of a single amount PV = FVn [1/(1+k)n ] ,
1/(1+k)n = PVIF1k10%,n
Find the present value of $20,000 to be received 5 years from today, if the opportunity cost of capital is 10 percent
Solution: PV5 = $20,000 [ 1/(1+.10 )5 ] = $20,000 x = $12,418
Relationship between Present value, rate of retune, and time Figure 3-1, P70)
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4. C. Annuity Problems-
Future value of ordinary annuity: payments are the same and the first payment occurs at the end of the year
FVAn = PMT X FVIFA1n,k% ,
FVAn = PMT X [(1+k)n -1]/k
Find the future value at the end of year 3 of 3 equal payment of $1,000 each at 10 percent rate of return. The first payment occurs one year from today.
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5. Solution : FVA3y = $1,000 X FVIFA1 3y,.10% = $1,000 X 3.310 = $ 3,310
If any three of the four variables ( PMT, k, n, FVA) are known the fourth variable can be calculated.
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6. D. Annuity Due (annuity in advance)
AD has payments at the beginning of each
period, and the future value is found one
period after the last payment.
Year
OAN $ $ $1000
AND $ $ $1000
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7. Find the future value of $1,000 , 20 payment annuity due at 8 percent
FVAD3 = PMT X FVIFA120y,8% x (1+.08) ,
= $1,000 X X 1.08 =$ 9,423.
E. Present value of an annuity
E1. Ordinary annuity :
PVOAN = PMT x PVIFA1 n,k%
E2: Annuity due:
PVAND = PMT X PVIFA1 n.k% x (1+k)
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8. F. Complex Cash Flow Problems:
A complex cash flow is one where the cash flow amounts are not equal and consistent but instead either miss periods or change in amount.
Problem. Find the present value of a stream of cash flows such that cash flow will be $1,000 at the end of each year for years 4 through 20.
= $1,000(PVIFA1 20y,10% – PVIFA1 3y,10% )
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9. Problem: Find the present value of the following cash flow stream at 10 percent cost of capital
Year zero = ,000, Year 1 ($50,000),
Year 2 through 10 $25,000, Year 10 $80,000.
Solution: NPV =
[($100,000) x 1 + ($50,000) x .9091
+ $25,000 ( )
+ $80,000 x ] = $16,273.
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10. Multiple inflows and outflows net present value example (P.81)
A rental property can be purchased for $40,000. The property will generate cash flows of $8,000 at t he end of each year for 7 years, before equipment replacement. A new $6,000 furnace will be needed at the end of 5 years, and the building will be worth an estimated $50,000 at the end of 7 years.
If the opportunity cost of capital is 10%, what is
the project’s NPV?.
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11. G. Effective Interest Rate : K = (1+ K` / q)q -1 3-6,
K` = nominal annual rate of return rate,
q = the number of times interest is compounded
Find the future value of $100 after 3 years at 10 percent rate of interest, if the interest is compounded semiannually
Solution: FV3 = $100 (1+ .10/2)3x2 = $134 ,
k = (1+.10/2)2 -1 =10.25%
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12. Quarterly compounding example (P.83-84)
Find the present value of $10,000
to be received in years at an
nominal required return of 16% a year,
compounded quarterly.
PV = $10,000 [1/(1+.16/4)10x4]
= $10,000 x = $2,083
What is the effective rate of interest?
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13. H. Alternate Payment Patterns
Fractional time period. Payments occur at some
date within a period.
Find the future value of $100 deposit in a savings account that has semiannual compounding and pays 5 percent each 6-months period and the money is left for 3.5 years.
Solution: FV3.5y = $100 (1.05)7 = $ or
Alternatively the problem can be solved by
finding effective rate of interest.
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14. I. Payments Spread Evenly across the Year:
If payment series are spread over the year, with a payment received at the end of each period of 1/q of a year:
Then effective rate per payment period is:
Kq = (1 + k)1/q – 1
Problem: if you receive $1,000 a year for 10
years, divided into equal daily payments,
With an effective annual interest rate of
10%. Find the present value of the payments.
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15. To solve for PVA1n,k use equation 3-4b, page 77
Solution:
K365 = ( )1/365 – 1 =
PVA 3650 days =
($1,000/365) PVA 13650days, %
= 6,446.07
To solve for PVA1n,k use equation 3-4b,
page 77
PVA1n,k = [1 -1/(1+k)n ] /k,
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16. Example (page 86) number in million $
Year ets.
Year-end
Mid-year
Daily* per day for 10
Years, *30,000,000/365 =$82.19
If payments received at year-end, the present
value = PVA 10y,10% = $30m PVIFA110y,10%
= $30m x = $184,338,000
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17. b. If payments are received at mid-year, present value = PVA 10y =
= $30m PVA1 10y,10% x (1+.10).5 = $193,335,325
c. If payments are spread evenly over the year on a daily basis.
Kq = ( 1+.10)1/ = %
PV = ($30m/365) PVA13650, %
= 193,382,189
Difference between daily cash flow and mid-year cash flow is only .02%!
Difference between daily cash flow and year-end cash flow is 4.68%!
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18. Using midyear cash flow to approximate
daily cash flow- example on page 87
Y ½ y y y y y3
$1m/ $1m/ $1m/ $1m/ $1m/ $1m/3
FVA3y
The compound cost of Boeing’s investment at the end of years 3 is
$1m/3 x FVIFA13y,10% x (1+.10)1/2 = $115,718,576
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19. J. Present value of a No-Growth Perpetuity PVp = PMT/k
K. Present value of a Constant- Growth
Perpetuity, PVgp = PMT1 / k-g
L. Contentious Compounding
A situation in which interest is added
continuously rather than at discreet points in
Time. FVn = PV (ekxn), PV = FVn/ (ekxn),
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20. The effective interest rate with Continuous compounding:
K = ek` – 1, if k`= 10% then k = e =10.52
K/ = ln (1+k)
An investment of $1,000 at a 10% nominal
interest rate with continuous compounding
will grow in 5 years to:
FV5 = $1,000 e.10x5 = $1,000 x
= $1,648.70
Table on page shows the amount to which
$1,000 will grow with various compounding schemes
if invested at 10% a year for 5 year.
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21. Future value of an annuity FVAn = PMT [ek` n -1 ]/ k`
Find the future value of an annuity of $10,000 a
year f or years, with payments spread
continuously over the year. The money I s
expected to earn an effective annual rate of
10.517percent.
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22. The future value of the annuity FVA20 =
FVA20 = $10,000 [e.10x20 -1] /.10
= $10,000[ ]/.10 = $638,910
With continuous compounding,
K` = ln ( ) = 10 percent
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23. Present value of an annuity PVAn = PMT [1-1/ek`n]/k`
Example: ( page 98)
Searcy hat co. is considering a Building
expansion that will provide cash flows of
$100,000 a year, arriving continuously
over the years, for 20 years. At the end of the
20- years period, the investment can be sold
For $200,000. The effective required return
is percent a year.
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24. the present value of the benefits is:
PVAn = PMT [1-1/ek`n]/k`
K` = ln( ) = 10 percent,
e.10x20 =7.3891
the present value of the benefits is:
PV = $100,000[1 -1/ ] /.10 +
$200,000/e.10x20
= $100,000 [1 -1/7.3891]/.10 +
$200,000/ = $891,732
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