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
TIME VALUE OF MONEY
DISCOUNTED CASH FLOW
Time Lines
0
1
2
3
4
5
6
k=5 %
0
$952.38
Outline of Approach:
1
$1,000.00
amount - present value
amount - future value
annuity - present value
annuity - future value
TM 5-1 Slide 2 of 2
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 A1
The Future Value Factor for k and n
FVFk,n = (1+k) n
k
n
1
2
3
4
5
6
7
.
1%
1.0100
1.0201
1.0303
1.0406
1.0510
1.0615
.
.
2%
1.0200
1.0404
1.0612
1.0824
1.1041
1.1262
.
.
3%
1.0300
1.0609
1.0927
1.1255
1.1593
1.1941
.
.
4%
1.0400
1.0816
1.1249
1.1699
1.2167
1.2653
.
.
5%
1.0500
1.1025
1.1576
1.2155
1.2763
1.3401
.
.
6% ...
1.0600 ...
1.1236 ...
1.1910 ...
1.2625 ...
1.3382 ...
1.4185 ...
.
.
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.
THE PRESENT VALUE OF AN AMOUNT
F V n  P V (1 + k ) n
PV = FVn
1
(1 + k ) n
P V  F V n (1 + k ) - n
PVFk,n
Present Value Factor for k and n (Appendix A-2)
PV = FVn[PVFk,n]
The Reciprocal Relation Between FVF and PVF
FVFk, n 
1
PVFk, n
TM 5-5
Slide 1of 2
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)
ANNUITIES
A stream of equal payments, made or
received, separated by equal intervals of time.
0
1
2
3
$1,000
$1,000
$1,000
4
$1,000
Figure 5-1 Ordinary Annuity
0
1
$1,000
$1,000
2
3
$1,000
Figure 5-2
4
$1,000
Annuity Due
TM 5-7
THE FUTURE VALUE OF AN ANNUITY
DEVELOPING A FORMULA
0
1
2
3
PMT
PMT
PMT
Each PMT earns interest at rate k
from the time it appears on the time line
until the end of the last period
The future value of the annuity
is the sum of all the payments and all the interest
Equivalent to summing the future value of each PMT
treated as an amount
TM 5-8 Slide 1 of 3
Future Value of a Three Year Ordinary Annuity
Figure 5-4
0
1
PMT
2
PMT
3
PMT
Future Values
PMT
PMT(1+k)
PMT(1+k)2
FVA3 = PMT + PMT(1+k) + PMT(1+k)2
TM 5-8 Slide 2 of 3
Future Value of a Three Year Ordinary Annuity
The Three Year Formula
FVA = PMT(1+k)0 + PMT(1+k)1 + PMT(1+k)2
Generalizing the Expression
n
FVAn = PMT
i1
(1+k)n-i
The Future Value Factor for an Annuity
n
FVAn = PMT

i 1
(1 + k)n-i
= FVFAk,n (Appendix A-3)
FVAn = PMT [FVFAk,n]
TM 5-8 Slide 3 of 3
THE FUTURE VALUE OF AN ANNUITY
SOLVING PROBLEMS
Example 5-5
The Brock Corporation will receive fees of $100,000 a year
for ten years and will invest each payment at 7%. How much will it have
after the last payment?
Solution:
0
1
$10K
years @ k=7%
2
3
8
$10K $10K
$10K
9
$10K
10
$10K
FVA10
FVAn = PMT [FVFAk,n]
FVA10 = $100,000 [FVFA7,10]
FVFA7,10 = 13.8164 (from Appendix A-3)
FVA10 = $100,000 [13.8164]
= $1,381,640
TM 5-9 Slide 1 of 2
COMPOUND INTEREST AND NON-ANNUAL COMPOUNDING
Compounding Periods
Frequency with which interest is credited for calculating future interest,
usually annually, semiannually, quarterly, or monthly.
The shorter the period, the more interest is earned on interest
Annually
12%
$100
$112
Semiannually
6%
6%
$100
$106
$112.36
Quarterly
3%
$100
3%
$103
3%
$106.09
3%
$109.27
$112.55
Quote the annual (nominal) rate (knom) stating the compounding period immediately
afterward
"12% compounded quarterly"
TM 5-10 Slide 2 of 2
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
Annual
Semiannual
Quarterly
Monthly
Table 5-2
Final balance
$112.00
$112.36
$112.55
$112.68
Year End Balances at Various Compounding Periods
$100 Initial Deposit and knom = 12%
In general:
EAR  (1 
k m
)
m
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
Quarterly:
k = knom / 4
Monthly:
k = knom / 12
n = years  2
n = years  4
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
THE PRESENT VALUE OF AN ANNUITY
0
1
PMT
2
3
PMT
PMT
PVs
PMT/(1+k)
PMT/(1+k)2
PMT/(1+k)3
PVA = PMT/(1+k) + PMT/(1+k)2 + PMT/(1+k)3
Figure 5-6 Present Value of a Three Period Ordinary Annuity
TM 5-13 Slide 1 of 2
Generalizing:
PVA = PMT(1+k)-1 + PMT(1+k)-2 + . . . + PMT(1+k)-n
PVA = PMT
PVA = PMT [PVFAk,n]
Appendix A-4
THE PRESENT VALUE OF AN ANNUITY SOLVING PROBLEMS
Example 5-9 The Shipson Company will receive payments of $5,000 every
six months (semiannually) for ten years on a sales contract which the bank
will discount at 14% compounded semiannually. How much will Shipson
receive?
Solution:
k = knom/2 = 14%/2 = 7%
0
n = 10 yrs  2 = 20
Half years @ k=7%
1
2
3
18
19
$5K $5K $5K
20
$5K $5K $5K
PVA
PVA = PMT [PVFAk,n],
PVA = $5,000 [PVFA 7,20]
PVA = $5,000 [10.5940]
PVA = $52,970
TM 5-14
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
Beginning
Period Balance
1
2
3
4
$15,053.75
$14,704.29
_________
_________
.
.
.
.
.
.
LOAN AMORTIZATION SCHEDULES
Interest Principal Ending
Payment @ 1% Reduction Balance
$500.00
$500.00
$500.00
$500.00
.
.
.
.
.
.
$150.54
$147.04
_______
_______
.
.
.
.
.
.
$349.46 $14,704.29
$352.96 $14,351.33
_______ __________
_______ __________
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
THE ANNUITY DUE
Payments occur at the beginning of time periods
The Future Value of an Annuity Due
0
1
2
PMT
PMT
PMT
3
PMT
PMT(1+k)
PMT(1+k) (1+k)
PMT(1+k)2 (1+k)
FVAd3 = [PMT + PMT(1+k) + PMT(1+k)2](1+k)
Figure 5-7
Future Value of a Three Period Annuity Due
FVAd3 = PMT(1+k) + PMT(1+k)(1+k) + PMT(1+k)2(1+k)
FVAdn = PMT [FVFAk,n] (1+k)
Recognize by: starting now, today, or immediately
TM 5-18 Slide 1 of 2
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
RECOGNIZING TYPES OF ANNUITY
PROBLEMS
0
1
2
n -2
PVA
Transaction
Here
n -1
n
FVA
Transaction
Here
A loan is always a present value of an annuity problem
The annuity is the stream of loan payments
The transaction is the transfer of the amount borrowed from the
lender to the borrower
Saving up is always a future value of an annuity problem
TM 5-19 Slide 1 of 3
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.
b.
FVn = PV (ekn)
FV3.5 = $5,000 (e(.065)(3.5))
= $5,000 (e.2275)
= $5,000 (1.255457)
FV3.5 = $6,277.29
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%
MULTI-PART PROBLEMS
Example 5-16 Exeter Inc. has $75,000 earning 16% compounded quarterly.
The company needs $500,000 in two years. How much should it deposit each
month in an account paying 12% compounded monthly?
Solution:
Quarters @ 4%
0
1
2
3
4
5
6
7
8
$75,000
FV
Months @ 1%
0
1
2
3
PMT PMT PMT
22
23
24
Product
Launch
$500,000
PMT PMT
FVA
TM 5-21 Slide 1 of 2
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?
Solution: Required savings will be $200,000 less amount borrowed on
mortgage less future value of the $10,000
Qtrs @ 1.5%
0
1
2
10
11
$10K
12
FV12
Amount
Months @ .75%
0
1
2
PVA $1K $1K
359 360
$1K
$1K
$1K
Qtrs @ 1.5%
0
1
2
10
PMT PMT
11
12
PMT PMT PMT
FVA12
Savings
Payments
Mortgage
Payments
Time of Purchase
$200K Required
Problem is focused around the date of purchase
TM 5-22 Slide 2 of 2
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
UNEVEN STREAMS
Require treatment as individual amounts
Solving for k requires an iterative approach
IMBEDDED ANNUITIES
Example 5-19
0
1
$5
Calculate present value:
2
3
4
5
6
7
8
$7
$6
$7
$3
$3
$3
$3
PV
PV
PVA
PV
PV
PV
TM 5-24 Slide 1 of 2
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