Chapter 5
Chapter 5
The Time Value of Money
TIME VALUE OF MONEY
DISCOUNTED CASH FLOW
A sum of money in hand today is worth more than the same sum promised with certainty in the future.
Think in terms of money in the bank
The value today of a sum promised in a year is the amount you'd have to put in the bank today to have that sum in a year.
Example: Future Value (FV) = $1,000
k = 5%
Then Present Value (PV) = $952.38 because
$952.38 x .05 = $47.62
and $952.38 + $47.62 = $1,000.00
THE FUTURE VALUE OF AN AMOUNT
FV1 = PV + kPV
FV1 = PV(1+k)
FV2 = FV1 + kFV1
FV2 = FV1(1+k)
Substitute for FV1
FV2 = PV(1+k)(1+k)
FV2 = PV(1+k) 2
In General,
FVn = PV(1+k) n
THE FUTURE VALUE OF AN AMOUNT
Define
Future Value Factor for k and n =
[FVFk,n] = (1+k)n
then FVn = PV [FVFk,n]
[FVFk,n] = (1+k)n is tabulated for common combinations of k and n in Appendix A-1
The Future Value Factor for k and n
FVFk,n = (1+k) n
k
n 1% 2% 3% 4% 5% 6% ...
1 1.0100 1.0200 1.0300 1.0400 1.0500 1.0600 ...
2 1.0201 1.0404 1.0609 1.0816 1.1025 1.1236 ...
3 1.0303 1.0612 1.0927 1.1249 1.1576 1.1910 ...
4 1.0406 1.0824 1.1255 1.1699 1.2155 1.2625 ...
5 1.0510 1.1041 1.1593 1.2167 1.2763 1.3382 ...
6 1.0615 1.1262 1.1941 1.2653 1.3401 1.4185 ...
7 . . . . . .
. . . . . . .
Example 5-1
How much will $850 be worth if deposited for three years at
5% interest?
Solution: FVn = PV [FVFk,n]
FV3 = $850 [FVF5,3]
Look up FVF5,3 = 1.1576
FV3 = $850 [1.1576]
= $983.96
Problem Solving Techniques
Equations all contain four variables
(In this case PV, FVn, k, and n)
Every problem will give three and ask for the fourth.
Example 5-2
Ed Johnson sold land to Harriet Smith for $25,000. Terms: $15,000 down, $5,000 a year for two years. What was the "real" purchase price if the interest rate available to Ed is 6%?
Solution: Price today is $15,000 plus PV of two $5,000 payments in future
FVn = PV [FVFk,n]
$5,000 = PV [FVF6,1]
$5,000 = PV [1.0600]
PV = $4,716.98.
and
$5,000 = PV [FVF6,2]
$5,000 = PV [1.1236]
PV = $4,449.98
$15,000.00 + $4,716.98 + $4,449.98 = $24,166.96
The terms of sale imply an equivalent discount of $833.04
even though the real estate records indicate a price of $25,000
The Opportunity Cost Rate
Example 5-2 (continued)
6% was available to the seller
nothing was actually invested at that rate
In a sense, seller lost income at that rate by giving
the deferred payment terms.
Suppose Harriet Smith borrows to pay for land at 10%.
Her opportunity cost rate is 10%
And the deferred payment terms are worth
a discount of $1,317.31 to her
Deferred terms are worth more to the recipient than to the donor!
The opportunity cost of a resource is the amount it could earn
in the next best use.
More on Problem Solving Technique
If unknown is k or n, can't solve equations algebraically
Solve for factor and use table
Example 5-3
What interest rate will
grow $850 into $983.96 in three years?
Solution:
PV = FVn[PVFk,n]
$850.00 = $983.96 [PVFk,3]
PVFk,3 = $850.00 / $983.96 = .8639
Find .8639 in Table A-2, along the row for three years and read 5% at top
Example 5-4
How long does it take money invested at 14% to double?
Solution:
FVn = PV [FVFk,n]
FVF14,n = FVn / PV = 2.0000
(Search for 2.0000 in Appendix A-1,
along the column for k = 14%
Table value is between 5 and 6 years)
COMPOUND INTEREST AND NON-ANNUAL COMPOUNDING
Compound Interest
Earning interest on previously earned interest
The Effective Annual Rate (EAR)
The rate of annually compounded interest equivalent to the nominal rate compounded more frequently
Compounding Final balance
Annual $112.00
Semiannual $112.36
Quarterly $112.55
Monthly $112.68
Table 5-2 Year End Balances at Various Compounding Periods
$100 Initial Deposit and knom = 12%
In general:
COMPOUNDING PERIODS AND THE TIME VALUE FORMULAS
Time periods must be compounding periods and the interest rate must be the rate for a single compounding period
Semiannually: k = knom / 2 n = years ( 2
Quarterly: k = knom / 4 n = years ( 4
Monthly: k = knom / 12 n = years ( 12
Example 5-7
Save up to buy a $15,000 car in 2 1/2 years.
Bank pays 12% compounded monthly.
How much must be deposited each month?
Solution:
k = knom/12 = 12%/12 = 1%
n = 2.5 yr ( 12 mo/yr = 30 months
FVAn = PMT [FVFAk,n]
$15,000 = PMT [FVFA1,30]
$15,000 = PMT [34.785]
PMT = $431.22
Generalizing:
PVA = PMT(1+k)-1 + PMT(1+k)-2 + . . . + PMT(1+k)-n
PVA = PMT
PVA = PMT [PVFAk,n]
Appendix A-4
AMORTIZED LOANS
Principal is paid off gradually during loan's life
Generally structured so that a constant payment
is made periodically, usually monthly
Each payment contains one month's interest and
an amount to reduce principal
Interest is charged on the month beginning loan balance
As loan's principal is reduced interest charges become smaller
Since monthly payments are constant successive payments contain larger proportions of principal repayment and smaller proportions of interest
Example 5-10
How much is the monthly payment on a $10,000 loan
to be repaid in monthly installments over four years
at 18% (compounded monthly)?
Solution:
k = knom/12 = 18%/12 = 1.5%
n = 4 yrs ( 12 mo/yr = 48 months
PVA = PMT [PVFAk,n]
$10,000 = PMT [PVFA1.5,48]
$10,000 = PMT [34.0426]
PMT = $293.75
Example 5-11
How much can you borrow at 12% compounded monthly over three years if you can make payments of $500 per month?
Solution:
k = knom/12 = 12%/12 = 1%
n = 3 yrs ( 12 mo/yr = 36 months
PVA = PMT [PVFAk,n]
PVA = $500 [PVFA1,36]
PVA = $500 [30.1075]
PVA = $15,053.75
A loan is always a PVA problem
Amount borrowed is always PVA
Loan payment is always PMT
LOAN AMORTIZATION SCHEDULES
Beginning Interest Principal Ending
Period Balance Payment @ 1% Reduction Balance
1 $15,053.75 $500.00 $150.54 $349.46 $14,704.29
2 $14,704.29 $500.00 $147.04 $352.96 $14,351.33
3 _________ $500.00 _______ _______ __________
4 _________ $500.00 _______ _______ __________
. . . . . .
. . . . . .
. . . . . .
MORTGAGE LOANS
Early payments are nearly all interest
Later Payments are nearly all principal
Example
A thirty year, $100,000 mortgage at 12% (compounded monthly)
has a monthly payment of $1,028.61
First month's interest is $1,000 (1% of $100,000)
Only $28.61 is applied to principal
The first payment is 97.2% interest
Reverses in last months
Tax Effect of Mortgage Payments
Mortgage interest is tax deductible
Effective first payment at 28%:
Payment $1,028.61
Tax Savings 280.00
Net $ 748.61
Payoff Timing
Halfway through a mortgage's life, it isn't half paid off:
Present value of the second half of the payment stream
The amount one could borrow
making 180 payments of $1,028.61
PVA = PMT [PVFAk,n]
= $1,028.61 [PVFA1,180]
= $1,028.61 [83.3217]
= $85,705.53
Total Interest Paid
Total payments = $1,028.61 ( 360 = $370,299.60
Less original loan = 100,000.00
Total Interest = $270,299.60
Tax Savings @ 28% 75,683.89
Net Interest Cost $194,615.71
Example 5-12
The Baxter Corporation began 10 years of quarterly $50,000 sinking fund deposits today at 8% compounded quarterly. What will the fund be worth
in 10 years?
Solution: k = 8%/4 = 2%
n = 10 yrs ( 4 qtrs/yr = 40 qtrs
FVAdn = PMT [FVFAk,n] (1+k)
FVAd40 = $50,000 [FVFA2,40] (1.02)
FVAd40 = $50,000 [60.4020] (1.02)
= $3,080,502.00
The Present Value of an Annuity Due
PVAd = PMT [PVFAk,n] (1+k)
CONTINUOUS COMPOUNDING
FVn = PV (ekn)
Where k = nominal rate in decimal form
n = years
e = 2.71828...
Example 5-15:
a. Future value of $5,000 at 6 1/2% compounded
continuously for 3 1/2 years
b. The Equivalent Annual Rate (EAR) of 12%
compounded continuously?
Solution:
a. FVn = PV (ekn)
FV3.5 = $5,000 (e(.065)(3.5))
= $5,000 (e.2275)
= $5,000 (1.255457)
FV3.5 = $6,277.29
b. Deposit $100 for one year:
FVn = PV (ekn)
FV1 = $100 (e(.12)(1))
= $100 (e.12)
= $100 (1.1275)
= $112.75
Initial deposit = $100,
Interest earned = $12.75,
EAR = $12.75 / $100 = 12.75%
First find the future value of the $75,000
FVn = PV [FVFk,n]
FV8 = $75,000 [FVF4,8]
= $75,000 [1.3686]
= $102,645
Then the savings annuity must provide
$500,000 - $102,645 = $397,355
FVAn = PMT [FVFAk,n]
$397,355 = PMT [FVFA1,24]
$397,355 = PMT [26.9735]
PMT = $14,731
Example 5-17
The Smith family plans to buy a $200,000 house using a traditional thirty year mortgage.
Banks allow roughly 25% of income to be applied to mortgage payments.
The Smiths expect their income will be $48,000. and the mortgage interest rate will be 9% when they buy the house.
They now have $10,000 in a bank account which pays 6% compounded quarterly.
How much will they have to add to the account each quarter to buy the house in three years?
Mortgage: k = 9%/12 = .75%, n = 360
PMT = ($48,000/12) x .25 = $1,000
PVA = PMT [PVFAk,n]
= $1,000 [PVFA.75,360]
= $1,000 [124.2819]
= $124,282
Future value of the $10,000 already in bank:
k = 6%/4 = 1.5%, n = 12
FV12 = $10,000 [FVF1.5,12]
= $10,000 [1.1956]
= $11,956
Savings requirement:
$200,000 - $124,281.90 - $11,956.00 = $63,762.10
= the FVA of savings deposits
FVAn = PMT [FVFAk,n]
$63,762.10 = PMT [FVFA1.5,12]
$63,762.10 = PMT [13.0412]
PMT = $4,889
Solution:
Payment 1: PV = FV1[PVF12,1] = $5(.8929) = $4.46
Payment 2: PV = FV2[PVF12,2] = $7(.7972) = $5.58
Payment 7: PV = FV1[PVF12,7] = $6(.4523) = $2.71
Payment 8: PV = FV1[PVF12,8] = $7(.4039) = $2.83
Annuity:
PVA = PMT [PVFA12,4] = $3(3.0373) = $9.11
and
PV = FV2[PVF12,2] = PVA(.7972) =
$9.11(.7972)= $7.26
$22.84
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