Time Value of Money
Chapter 5:The Time Value of Money
5.1 to 5.8
The Time Value of Money
• Money NOW is worth more than money LATER!
•Why?
• dollar received today can be invested to earn interest
• The amount of interest earned depends on the rate of return
Simple Interest
Interest only on Principle
Compounding Interest
Interest on both Principal and Interest.
Interest on interest makes compounding different from simple interest
3
Simple Interest - Interest only on Principal
You invest $500 today for a 1-year term and receive 8% simple interest annually
on your investment. How much will you have after 1 year?
PV = 500 n=1 FV=???
FV1 = P + [n x (P x k) ]
FV1 = $500 + [1 x ($500 x 0.08)]
FV1 = $500 + [1 x ($40)]
FV1= $540
You invest $500 today for a 5-year term and receive 8% simple interest annually
on your investment. How much will you have after five years?
PV = 500 n=5 FV=???
FV5 = P + [n x (P x k) ]
FV5 = $500 + [5 x ($500 x 0.08)]
FV5 = $500 + [5 x ($40)] = $700
4
Compounding Interest - Interest on both Principal and Interest.
We invest $500 today and receive 8% annual compound interest. What
would be the FV at years 1,2,3,4,5? For year 1, we have:
FV1 = P + [n x (P x k) ]
FV1 =$500 + [1 x ($500 x 0.08)]
FV1 = $500 + ($500 x 0.08)
FV1 = $500 x (1 + 0.08)
FV1 = $540
For year 2 – we are gaining interest on interest also, not only on
principal.
FV2 = P + [n x (P x k)]
FV2 = $540 + [1 x ($540 x 0.08)]
FV2 = $540 + [($540 x 0.08)]
FV2 = $540 + ($540 x 0.08)
FV2 = $540 x (1 + 0.08)
FV2 = $500 x (1 + 0.08) x (1 + 0.08)
FV2 = $500 x (1 + 0.08) 2
FV2 = $583.2
5
FV3 = $500 x (1 + 0.08) 3 = 629.856
FV4 = $500 x (1 + 0.08) 4 = 680.244
FV5 = $500 x (1 + 0.08) 5 = 734.664
We notice that FV5 obtained with compound interest ($734.664) is
greater than the FV5 obtained with simple interest ($700).
What is the additional amount of interest you receive due to
compounding in year 5?
734.664 – 700 = 34.664 $
What is the additional amount of interest you receive due to compounding
in year 5?
734.664 – 700 = 34.664 $
6
Simple vs Compounding Example:
Principal = 100 Interest = 10%
Simple Interest:
1st Year Interest : 10 2nd Year Interest : 10
Compounding Interest:
1st Year Interest : 10 2nd Year Interest : 11
Because 2nd year interest is on (100 + 10) = 110.
10% simple interest after 20 years becomes 200% of principal
10% compounding interest after 20 years becomes 672.75% principal
7
Compound vs. Simple Interest
$3,500
$3,000
Value of $100 @10%
$2,500
$2,000
$1,500
$1,000
$500
$0
0 5 10 15 20 25
Years
8
If you invest $1,000 today at an interest rate of 10
percent, how much will it grow to be after 5 years?
$1,000
PV=1000 k=10% n=5 FV=????
• FVn = PV0(1+k)n
• FVn = 1,000(1.10)5
• FVn = $1,610.51
Discounting is a way to compute the present value of future
money
• Assume you will receive an inheritance of $100,000, six years from now.
How much could you borrow from a bank today and spend now, such that
the inheritance money will be exactly enough to pay off the loan plus
interest when it is received? Assume the bank charges an interest rate of 12
percent?
• FV = $100,000 n=6 PV=??? K=12%
PV0 = FVn/(1+k)n
PV0 = 100,000/(1.12)6
PV0 = $50,663
Financial Calculator terminology
• N = number of periods payments were made
• K= I/Y = Interest per year
• PMT = Periodic Payment
• FV = Future Value
• PV = Present Value
• P/Y = Number of payments per year
• C/Y = Number of time compounded per year
• BEG = Beginning of the Period
• END = End of the period
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Finding “n”
• How long will it take for $10,000 to grow to $20,000 at an interest
rate of 15% per year?
• In normal compounding we have no PMT
• PV = $10,000 FV = $20,000 k=15% n=???
• FVn = PV0(1+k)n
• 20,000 = 10,000(1.15)n
• 2 = (1.15)n
• ln(2) = n*ln(1.15) (Multiply both side by Ln)
• n = ln(2)/ln(1.15) = 4.96 years
-10,000 (its –ve because we 20,000 (its +ve because we
assume its an investment) assume we receive that cash)
0 10
Annuity & Perpetuity Tricks
Future value or Present Value?
Cash flow growing or not?
Cash flow or growth in cash flow consistent or not?
Beginning of the period or End of the period
Does the cash flow go on forever or limited time?
Annuity & Perpetuity Tricks
Annuity:
Cash flow paid to someone each time period for a limited time. The
amount can be constant or grow at a constant rate.
Annuity Due:
Annuity with beginning of period and constant cash flow
Ordinary Annuity:
Annuity with end of period and constant cash flow
Growing Annuity:
Annuity with cash flow that grows at a constant rate.
Perpetuity:
Cash flow that goes on forever. Key words – perpetual /forever etc.
Growing Perpetuity:
Cash flow that goes on forever and grows at a constant rate.
Annuity
4 Payments Due
Ordinary
Annuity
Annuity
4 Payments Due
Ordinary
Annuity
Better to use a financial
calculator for these 4
formulas
15
The premiums of an insurance policy are $65, payable at the start of each year. If
interest is 3.3% compounded annually, how much should a policyholder pay in
order to cover 3 Year’s premium
BEG of the period PMT= 65 P/Y = 1 C/Y = 1 PV = ??
N=3 I/Y =k= 3.3%
FV = 0 (since everything will be paid back at the end)
CPT PV = 188.84
Alternate Solution:
= 188.84
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Example: (Question #12) You will receive a 4-year annuity of $500 per year beginning
in year 6. Payments are at the end of year. If k = 10%, what is the present value of this
annuity?
PMT = 500 k=10% n=4 P/Y=1 C/Y=1
END of period (if nothing is said, we assume its END )
Ordinary Annuity
= 1,584.93
PV=1,584.93/(1.10)^6
= 984.12
Ordinary
Annuity n=5
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Example: (Question #12) You will receive a 4-year annuity of $500 per year beginning
in year 6. Payments are at the end of year. If k = 10%, what is the present value of this
annuity?
PMT = 500 k=10% n=4 P/Y=1 C/Y=1 BEG of period
Annuity Due
= 1,743.43
PV=1,743.43/(1.10)^6
= 984.12
10
n=6
Annuity
Due
6 18
Calculate the present value of a growing annuity given the following information:
current cash flows: $90,000; cash flow growth rate = 2%; timeframe = 20 years;
required rate of return = 5%
PMT = 90,000 g= 2% n = 20 k=5%
90000/(5% - 2%) [1-(1+ 2%/1 + 5%)20] It does not go forever.
Hence its annuity not
=3000000 [1-0.56003] perpetuity
= 1,319,886
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Example Question #14 Your 6 year old wants to go to Princeton. This will cost you
$30,000 per year for 4 years. On her next birthday you start putting money into an
account paying 14%
annually and continue to deposit the same amount every year until her 17th birthday.
The first tuition payment will be payable on her 18th birthday, the second on her 19th
birthday etc. How large are your annual deposits to this account?
PV11 = 30k 30k 30k 30k
87,411.37
1 1
PV11 = $30, 000 1 −
0.14 (1.14 )
4
PV11
PV0 = 11
= $20,683.05 = $87, 411.37
1.14
PMT = 30,000 k=14% n=4 END
P/Y = 1 C/Y =1 FV = 0
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Your uncle has decided to give you $500 every year. You will get the 1st $500
3 years from now. If the bank is paying you 4% interest today what is the
value of the cash stream for you today?
We assume the PMT goes
$500 $500 $500 on forever
PV0 = 11,557 PV2 = 12,500
PMT = 500 k=0.04
PV0 = 12,500/(1.04)^2
PV2 = PMT/k = 500/0.04
PV0 = 11,557
PV2 = 12,500
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You sell an invention in exchange for a perpetual payment that will start at
$10,000 and grow at a rate of 3% each year. The discount rate is 7%.
What is the present value of this invention?
PMT1 = 10,000 k= 0.07 g= 0.07
PV0 = 10,000 / (0.07 – 0.03) = 250,000
10,000 10,000 (1.03) 10,000 (1.03)^2 Goes on forever
PV0 = 250,000
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