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Time Value of Money
by Binam Ghimire
Learning Objectives
Concept of TVM
Represent the cash flows occurred in different time periods using cash flow time line
Calculate the present value and future value of given streams of cash flows with and without using table
Identify the impact of time period and required rate of return on present value and future value
Prepare amortised schedule for amortised term loan
Compare interest rates quoted over different time intervals
Understand the real and nominal cash flows
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Source: Total Access http://www.totalaccess.co.uk/Case_Studies/big_ben
Warm Up: The School Maths
Adding a percent to an amount
A computer costs £ 420 + VAT 15%. What is the costafter VAT is added?
Reverse percentage
The price of a computer is £230 after VAT of 15%. Whatis the price before VAT is added?
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Time value of Money:
Concept and Significance
Theory behind TVM: “One pound today is worth morethan one pound tomorrow”
Deals with values of cash flows occurred at differentpoints in time
Every sum of money received now has a value forreinvestment
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Time value of Money:
Concept and Significance
Why is APR a legal requirement?
6Image source: google
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Time value of Money:
Concept and Significance
Having Cash NOW is valuable
Inflation
People’s preference to current consumption
TVM Usage: Firms, Households, Banks, Insurance Cos.
TVM in Corporate Finance: Economic Welfare of Shareholders
Investment Decisions
Financing Decisions
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Cash Flow Time Line
Cash Flow Time Line is an important tool used tounderstand the timing of cash flows. It is a graphicalpresentation of cash flows occurring at different pointsof time, and is helpful for analysing the time value ofcash flows
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Time 0 1 2 3 4 5
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Cash Flow Time Line
The corresponding cash flows are placed below thescale as shown in the following time line of cash flows:
The time line of cash flow is also used to denote theinterest rate that each cash flow earns
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Time 0 1 2 3 4 5
Cash Flows -100 10 50 70 100 90
Time 0 1 2 3 4 5
Cash flow – 100
8%
Future Value
Future Value (FV) is the sum of investment amount andinterest earned on the investment
Interest may be earned simple and compound
Accordingly FV can be calculated using simple andcompound interest rate methods
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Future Value:
Simple Interest
Interest earned only on the original investment isSimple Interest. For example, a savings account witha bank in which bank posts the interest every year. Theinterest for principle £P deposited at r % interest p.a.for t number of years will equal to I (where I = p x n x r÷ 100). So the FV of an investment will be P + I
We aim to maximise the benefit from any monetarytransaction. So in real world, interest is paid (savings) orcharged (loan) more than once during the tenure of thesavings or loan. As such the Simple Interest does notexist
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Future Value :
Compound Interest
Compound Interest is interest earned on principleplus interest from the previous period
FV of £ 100,000 investment for one year, rate ofinterest: 12% and when interest is 1) simple interest 2)compounded monthly and daily are shown in figure next
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FV=P x 1+r
100
t
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112,747
112,000
100,000
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
2011 Beginning 2011 End
Simp
le In
tere
st
Inte
rest co
mp
ou
nd
ed
mo
nth
ly
Inte
rest co
mp
ou
nd
ed d
aily
112,683
Future Value :
Compound Interest Effect
Hence, with Compound interest, FV grows when thefrequency of interest payment is higher. (i.e. totalamount of interest is growing every next period).
Note that when interest is per annum and to becalculated for mth times within a year the formulachanges to:
And if it is for more than one year, the formula is
Where, n=no. of years
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Future Value :
Compound Interest different intervals
m
100
mr
1xPFV
n x m
100
mr
1xPFV
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has offered 5 %
interest charged half
yearly
has offered 4.5%
interest charged every
3 monthsThe name of the banks are used only to illustrate the APR and the numbers are imaginary. Logo source: website of respective banks.
Compound Interest different intervals
You want to borrow money
Below FV of £ 1 at the end of various numbers of years have been calculated for different rates of interest:
The table is known as FV Compound Interest table [(1+r)t]
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Future Value :
Compound Interest Table
12% 13% 14% 15% 16%
1 1.12 1.13 1.14 1.15 1.16
2 1.25 1.28 1.30 1.32 1.35
3 1.40 1.44 1.48 1.52 1.56. . . . . .. . . . . .. . . . . .
10 3.11 3.39 3.71 4.05 4.41
20 9.65 11.52 13.74 16.37 19.46
50 289.00 450.73 700.23 1083.66 1670.70
Interest Rate per annumYear
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Future Value :
Graphical View
Future value of £ 1
i = 0%
i = 5%
i = 10%
i = 15%
0 Time periods
2 4 6 8 10
1.0
2.0
3.0
4.0
FV of a sum of money has positive relation with the time period and interest rate
Future Value:
FV, Time Period and Interest rate
FV increases with rate of interest and time
The higher the interest rate higher will be the FV
More the number of years the more will be the FV
Higher the no. of frequency of payment (within the year) higher will be the FV
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In Microsoft Excel, FV function wizard can be used tocalculate FV
Check the relationship between FV and r and t bymaking line chart in Microsoft Excel for r=5, 10 & 15percent for t= 0,2,4,6,8,10,12 & 14 years
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Future Value :
Compound Interest in Excel
Present Value
Present value (PV) is today’s value of a future cash flow
FV is only available in the future
Can we get the FV now? i.e. can we convert the FV intoPV?
You may if you sacrifice 1) the interest that you couldhave earned had you invested the money and 2) adjustfor the time period that you don’t want to wait. Hence,
The sacrifice is reflected in the formula above. When tand r get bigger PV gets smaller
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PV=FV
1+r t
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You bought a television. The payment term is singlepayment of £ 2,000 to be made after 2 years. If youcan earn 8% on your money how much moneyshould you set aside today in order to make finalpayment when due in 2 years?
Note that to calculate PV, we discounted FV at theinterest rate r. The calculation is therefore termed adiscounted cash flow. The interest rate is known asDiscount rate
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PV= £ 2,000
1+0.08 2= £1,714.68
Present Value:
Discount Rate
The PV formula may be converted as follows
The expression 1/(1+r)t basically measures the PV of£1 received in year t. The expression is known asDiscount Factor. Using the PV of £ 1, a table canbe constructed for FV to be received after t years atdifferent discount rates
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Present Value:
Discount Factor
PV=FV x 1
1+r t
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Below PV of £ 1 at the end of various numbers of years have been calculated for different rates of interest:
The table is known as PV Discount Factor table [1/(1+r)t]
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Present Value :
Discount Factor Table
12% 13% 14% 15% 16%
1 0.893 0.885 0.877 0.870 0.862
2 0.797 0.783 0.769 0.756 0.743
3 0.712 0.693 0.675 0.658 0.641. . . . . .. . . . . .. . . . . .
10 0.322 0.295 0.270 0.247 0.227
20 0.104 0.087 0.073 0.061 0.051
50 0.003 0.002 0.001 0.001 0.001
Interest / Discount Rate per annumYear
Present Value:
Graphical View
The PV of a sum of money has inverse relation with the time period and interest rate
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Present Values of £ 1
i = 5%
i = 0%
0 Periods
2 4 6 8 10
0.25
0.50
0.75
1.00
i = 10%
i = 15%
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Present Value:
PV, Time Period and Interest rate
PV increases when rate of interest falls. The lower the interest rate higher will be the PV
Similarly, PV increases when the time period decreases
In the example of TV purchase above, note that £1,714.68 invested for 2 years at 8% will prove just enough to buy your computer
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FV=£ 1,714.68 x 1.08 2=£ 2,000
Present Value:
PV, Time Period and Interest rate
The longer the time period available for payment theless you need to invest today (i.e. lower PV value). Forexample, if the payment for TV was only required in thethird year, the PV would be:
The relationship between PV and Interest Rate is alsothe same i.e. higher interest rate lower PV
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PV= £2,000
1+0.08 3 = £1,587.66
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Multiple Cash Flow:
FV and PV for uneven Cash Flows
Most real word investments will involve many cash flowsover time. This is also known as Stream of cashflows. Calculation of FV and PV of a stream of cashflows is more important and common in finance
To calculate the FV of a stream of uneven cash flows,we may calculate what each cash flow is worth at thatfuture date, and then add up the values
To find the PV of a stream of uneven cash flows, wemay calculate what each FV is worth today and then addup the values
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Multiple Cash Flow:
Future Value of Uneven Cash Flow
A security provides the following Cash Flows
If the interest rate is 10% the FV will be
The third column above is the factor from FV of £ 1table
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End of Year 1 2 3 4 5
Cash Flow (£ ) 100 150 200 250 400
Year Cash Flows (£) 10% FV £1 FV
1 100 (1.1)4 = 1.4641 £ 146.41
2 150 (1.1)3 = 1.3310 199.65
3 200 (1.1)2 = 1.2100 242.00
4 250 (1.1)1 = 1.1000 275.00
5 400 (1.1)0 = 1.0000 400.00
FV of uneven CF stream £ 1,263.06
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Multiple Cash Flow:
Present Value of Uneven Cash Flow
The PV for the same Cash Flow above assuming samerate as discount rate will then be
The third column above is the factor from PV of £ 1table
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Year Cash Flows PV 10% PV
1 100 0.9091 90.91
2 150 0.8264 123.96
3 200 0.7513 150.26
4 250 0.6830 170.75
5 400 0.6209 248.36
PV of uneven CF stream £ 784.24
Multiple Cash Flow:
FV and PV for even Cash Flows
In finance, it is more common to find a stream ofEQUAL cash flows at same time intervals. For example,a home mortgage or a car loan might require to makeequal monthly payments for the life of the loan. Anysuch sequence of equally spaced, level cash flows iscalled an Annuity
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Annuity may be Ordinary Annuity and Annuity Due
In case of an ordinary annuity, each equal payment ismade at the end of each interval of time throughout theperiod
In case of Annuity Due, equal payments are made at thebeginning of each interval throughout the period
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Multiple Cash Flow:
Annuity and Annuity Due
For example, if an individual promises to pay £ 1,000 atthe end of each of three years for amortisation of aloan, then it is called an ordinary annuity. If it were theannuity due, each payment would be made at thebeginning of each of the three years
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Multiple Cash Flow:
Annuity and Annuity Due
Time 0 1 2 3
Ordinary annuity 1,000
8%
1,000 1,000
Time 0 1 2 3
Annuity due 1,000
8%
1,000 1,000
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Future Value for the Ordinary Annuity (FVA) will be
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Multiple Cash Flow:
Future Value of an Annuity
Time 0 1 2 3
Ordinary annuity 1,000
8%
1000 1,000
1,080
1,166.4
FVA3 = £ 3,246.4
1000 × 1.08
1,000 × 1.082
Formula,
where C is the Cash payment from annuity
If payment value is £ 1, it would be:
The table showing future value of £ 1 for various years at different interest rates is called FV Annuity table
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Multiple Cash Flow:
Future Value of an Annuity
r
1]r)C[(1FVA
t
t
r
1]r)[(1FVA
t
t
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Future value for the Annuity Due (FVAD) will be
Hence,
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Multiple Cash Flow:
Future Value of an Annuity Due
r)(1xr
1]r)C[(1FVAD
t
t
FVA3 = £ 3,506.11
Time 0 1 2 3
Annuity due 1,000
8%
1,000 1,000
1,080
1,166.4
1,259.71
1000 × 1.08
1000 × 1.082
1000 × 1.083
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Multiple Cash Flow:
Perpetuity, Delayed Annuity &Annuity
Year 1 2 3 4 5 6
Investment A £1 £1 £1 £1 £1 £1…
Investment B £1 £1 £1…
Investment C £1 £1 £1
Cash Flow
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If the payment stream lasts forever, it is Perpetuity
Example: Consol
For a perpetual stream of £C payment every year the PV= C/r
So a perpetual stream of £1 payment every year the PV= 1/r
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Multiple Cash Flow:
Perpetuity
Delayed perpetuity is one in which the payment (C)only starts after some time t
PV of a delayed perpetuity for £ 1 payment starting aftertime t will be
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Multiple Cash Flow:
Delayed Perpetuity
tr)(1
1x
r
1)1(£PerpetuityDelayedofPV
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Annuity is basically a difference between an Annuity(immediate) and a delayed perpetuity
PV of an Annuity (t year) will therefore be PV ofimmediate Perpetuity – PV of Delayed Perpetuitystarting after t year
This is also called t-year annuity factor. The PV tableconstructed for various time t and interest rate r forpayment £1 each year is called PV Annuity factortable
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Multiple Cash Flow:
Present Value of an Annuity
r
r11or
r)r(1
1
r
1)1(£AnnuityofPV
t
t
For Payment £ C every year, PV of t year annuity istherefore Payment x annuity factor or
Or:
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Multiple Cash Flow:
PV of an Annuity and Annuity Due
c 1
r -
1
r 1+r t
r
r11c
t
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For Payment £ C every year, PV of t year annuity istherefore Payment x annuity factor or
Present value of Annuity Due simply requires the use ofthe formula:
PV of ordinary annuity x (1+r)
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Multiple Cash Flow:
PV of an Annuity and Annuity Due
c 1
r -
1
r 1+r t
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Multiple Cash Flow:
PV of an Annuity and Annuity Due
Year 1 2 3 4 5 6 PV
Investment A £1 £1 £1 £1 £1 £1… 1/r
Investment B £1 £1 £1… 1/r(1+r)3
Investment C £1 £1 £1 1/r - 1/r(1+r)3
Cash Flow
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Loan that is to be repaid in equal periodic installmentsincluding both principal and interest is known asAmortised Loan
Let us suppose a loan of £ 10,000 is to be repaid in fourequal installments including principal and 10 percentinterest per annum
The lender needs to set the payments so that a presentvalue of £ 10,000 is received
Present Value = Mortgage Payment x t yearannuity Factor
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Multiple Cash Flow:
Amortised Loans
Mortgage Payment =
The loan amortisation schedule is shown next
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Multiple Cash Flow:
Amortised Loans
155,3£
)10.1(10.
1
10.
1
000,10£
4
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Multiple Cash Flow:
Amortised Loan Schedule
Year
(1)
Beginning Amount
(2)
Payment
(3)
Interest
(4) = (2) x 0.10
Repayment of Principal
(5) = (3) – (4)
Ending Balance
(6) = (2) – (5)
1 £ 10,000.00 £ 3,154.67 £ 1,000.00 £ 2,154.67 £ 7,845.33
2 7,845.30 3,154.67 784.53 2,370.14 5,475.16
3 5,475.16 3,154.67 547.52 2,607.15 2,868.01
4 2,868.01 3,154.67 286.66* 2,868.01 0
Note that cash flows are affected by the level ofinflation in the country
Generally, the rate of interest quoted in the market arenominal interest rate which does not take intoconsideration the effect of price changes. The realinterest rate can be calculated using the formula:
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Cash Flow and Inflation
RateInflation1
RateInterestNominal1RateInterestReal1
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What is the real interest rate when you deposit £ 1,000in a bank at interest rate of 5%. Suppose that theInflation is 5% as well
What will it be if interest rate is 6% and inflation is only2 %?
Discounting real payment by real interest rate andNominal Payment by Nominal interest rate will alwaysgive the same answer. We must not mix up real andnominal
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Cash Flow and Inflation:
Example
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Thank You
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