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Time Value of Moneyby 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 2

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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

cost after VAT is added? Reverse percentage The price of a computer is £230 after VAT of 15%.

What is 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 more than one pound tomorrow”

Deals with values of cash flows occurred at different points in time

Every sum of money received now has a value for reinvestment

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Time value of Money: Concept and Significance

Why is APR a legal requirement?

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36.5% p.a.*

buy today pay

after 12

months

* Condition apply see page 115 for

detail

On page 115: payable 0.1 % interest

per day

Image source: google

Time value of Money:Concept and Significance

Having Cash NOW is valuableInflationPeople’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 to understand the timing of cash flows. It is a graphical presentation of cash flows occurring at different points of time, and is helpful for analysing the time value of cash flows

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Time

0 1 2 3 4 5

Cash Flow Time Line

The corresponding cash flows are placed below the scale as shown in the following time line of cash flows:

The time line of cash flow is also used to denote the interest 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 and interest earned on the investment

Interest may be earned simple and compound Accordingly FV can be calculated using simple

and compound interest rate methods

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Future Value:Simple Interest Interest earned only on the original investment is

Simple Interest. For example, a savings account with a bank in which bank posts the interest every year. The interest 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 monetary transaction. So in real world, interest is paid (savings) or charged (loan) more than once during the tenure of the savings or loan. As such the Simple Interest does not exist

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Future Value :Compound Interest

Compound Interest is interest earned on principle plus interest from the previous period

FV of £ 100,000 investment for one year, rate of interest: 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 ቀ1+ r100ቁt

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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

Simple Interest

Interest compounded

monthly

Interest compounded daily

112,683

Future Value :Compound Interest Effect

Hence, with Compound interest, FV grows when the frequency of interest payment is higher. (i.e. total amount of interest is growing every next period).

Note that when interest is per annum and to be calculated for mth times within a year the formula changes to:

And if it is for more than one year, the formula is

Where, n=no. of years14

Future Value :Compound Interest different intervals

m

100m

r1xPFV

n x m

100m

r1xPFV

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has offered 5 % interest charged half yearly

has offered 4.5% interest charged every 3 months

The 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

Which bank will you prefer and why?

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 1.162 1.25 1.28 1.30 1.32 1.353 1.40 1.44 1.48 1.52 1.56. . . . . .. . . . . .. . . . . .

10 3.11 3.39 3.71 4.05 4.4120 9.65 11.52 13.74 16.37 19.4650 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 to calculate FV

Check the relationship between FV and r and t by making line chart in Microsoft Excel for r=5, 10 & 15 percent 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 into PV? You may if you sacrifice 1) the interest that you

could have earned had you invested the money and 2) adjust for the time period that you don’t want to wait. Hence,

The sacrifice is reflected in the formula above. When t and r get bigger PV gets smaller

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PV= FVሺ1+rሻt

You bought a television. The payment term is single payment of £ 2,000 to be made after 2 years. If you can earn 8% on your money how much money should you set aside today in order to make final payment when due in 2 years?

Note that to calculate PV, we discounted FV at the interest rate r. The calculation is therefore termed a discounted cash flow. The interest rate is known as Discount rate

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PV= £ 2,000ሺ1+0.08ሻ2 = £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 as Discount Factor. Using the PV of £ 1, a table can be constructed for FV to be received after t years at different discount rates

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Present Value:Discount Factor

PV=FV x 1ሺ1+rሻt

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 0.8622 0.797 0.783 0.769 0.756 0.7433 0.712 0.693 0.675 0.658 0.641. . . . . .. . . . . .. . . . . .

10 0.322 0.295 0.270 0.247 0.22720 0.104 0.087 0.073 0.061 0.05150 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%

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 ሺ1.08ሻ2=£ 2,000

Present Value:PV, Time Period and Interest rate

The longer the time period available for payment the less you need to invest today (i.e. lower PV value). For example, if the payment for TV was only required in the third year, the PV would be:

The relationship between PV and Interest Rate is also the same i.e. higher interest rate lower PV

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PV= £2,000ሺ1+0.08ሻ3 = £1,587.66

Multiple Cash Flow:FV and PV for uneven Cash Flows

Most real word investments will involve many cash flows over time. This is also known as Stream of cash flows. Calculation of FV and PV of a stream of cash flows 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 that future date, and then add up the values

To find the PV of a stream of uneven cash flows, we may calculate what each FV is worth today and then add up 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 £ 1 table

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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

Multiple Cash Flow:Present Value of Uneven Cash Flow

The PV for the same Cash Flow above assuming same rate as discount rate will then be

The third column above is the factor from PV of £ 1 table

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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 of EQUAL cash flows at same time intervals. For example, a home mortgage or a car loan might require to make equal monthly payments for the life of the loan. Any such sequence of equally spaced, level cash flows is called an Annuity

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Annuity may be Ordinary Annuity and Annuity Due

In case of an ordinary annuity, each equal payment is made at the end of each interval of time throughout the period

In case of Annuity Due, equal payments are made at the beginning 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 at the end of each of three years for amortisation of a loan, then it is called an ordinary annuity. If it were the annuity due, each payment would be made at the beginning 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

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

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

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 after time t will be

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Multiple Cash Flow:Delayed Perpetuity

tr)(1

1x

r

1)1(£PerpetuityDelayedofPV

Annuity is basically a difference between an Annuity (immediate) and a delayed perpetuity

PV of an Annuity (t year) will therefore be PV of immediate Perpetuity – PV of Delayed Perpetuity starting after t year

This is also called t-year annuity factor. The PV table constructed for various time t and interest rate r for payment £1 each year is called PV Annuity factor table

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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 is therefore Payment x annuity factor or

Or:

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Multiple Cash Flow:PV of an Annuity and Annuity Due

c1r - 1rሺ1+rሻt ൨

r

r11c

t

For Payment £ C every year, PV of t year annuity is therefore Payment x annuity factor or

Present value of Annuity Due simply requires the use of the formula:

PV of ordinary annuity x (1+r)

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Multiple Cash Flow:PV of an Annuity and Annuity Due

c1r - 1rሺ1+rሻt ൨

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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

Loan that is to be repaid in equal periodic installments including both principal and interest is known as Amortised Loan

Let us suppose a loan of £ 10,000 is to be repaid in four equal installments including principal and 10 percent interest per annum

The lender needs to set the payments so that a present value of £ 10,000 is received

Present Value = Mortgage Payment x t year annuity 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 of inflation in the country

Generally, the rate of interest quoted in the market are nominal interest rate which does not take into consideration the effect of price changes. The real interest rate can be calculated using the formula:

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Cash Flow and Inflation

RateInflation1

RateInterestNominal1RateInterestReal1

What is the real interest rate when you deposit £ 1,000 in a bank at interest rate of 5%. Suppose that the Inflation is 5% as well

What will it be if interest rate is 6% and inflation is only 2 %?

Discounting real payment by real interest rate and Nominal Payment by Nominal interest rate will always give the same answer. We must not mix up real and nominal

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Cash Flow and Inflation:Example

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Thank You