SCC2301 CH4 L09 Rates of Return Time Value of Money

8/14/2019 SCC2301 CH4 L09 Rates of Return Time Value of Money

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Time Value of Money

Lecture 9

This lecture is part of Chapter 4:

Investing in the Company


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Todays Lecture

Understand the Time Value of Money

Use Excel to calculate the present and future value of a

stream of cash flows

Make time work for you!


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Time Value

The core of this lecture is actually quite similar to what we

have done for bonds.

We say that money has a Time Value because it can be

invested and thus become more. In other words, if we have a

dollar today, we expect/hope that we will have more than a

dollar in the future.

The Time Value of money is an essential concept when

deciding on an investment.


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Time Value

The are a few terms important to know as with regards to the

time value of money:

Present Value: This is just the monetary value of the

investments we have right now

Future Value: This is the value of our investment in the future

Compounding: Reinvesting the interest received, in other

words, receiving interest on interest


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Time Value

The present and future values can easily be calculated in Excel

A B C D E F G H I

2

3 Compounding

4

56 Year

7

8 present 1,000.00

9 1 1,100.00 =C8*(1+0.1)10 2 1,210.00 =C9*(1+0.1)

11 3 1,331.00 =C10*(1+0.1)

12 4 1,464.10 =C11*(1+0.1)

13 5 1,610.51 =C12*(1+0.1)

14

15

1617

14

15

16

We assume 10% interest

So we see that 1000 dollars now

will be 1610 dollars in 5 years


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Time Value

Of course this can easily be expressed mathematically, but lets

do it step by step again by understanding what we are doing:

Or:

Value after two years = (Present Value + Interest) + Interest

1st

year: Value after one year = Present Value + Interest

2nd year: Value after two years = Value after one year + Interest

Substitute Line 1

This interest needs to be on the entire Value after one year


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Time Value

Or:

1st year: Value after one year = PV + PV* r = PV*(1 + r)

2nd year: Value after two years = PV * (1 + r) + PV * (1 + r) * r

Hence we have:

FV2 = PV * ( (1+r) + (1+r)*r) ) = PV * ( 1+r+r+r*r)= PV * ( 1 + 2r + r )

= PV * ( 1 + r)

2

2

FV2 = PV * ( 1 + r)

2


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Time Value

NrPVNFV )1(*)( +=

And thus we obtain the formula:

5)1.01(*1000)5( +=FV

Lets check this for our example

And indeed this is equal to 1610 as before.


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Time Value

Surprise! Theres also an Excel function for this: FV

A B C D E F G H I

2

3 Compounding

4

56 Future Value: 1,610.51

7

8

9

10

11

12

13

=FV(10%,5,0,-1000,0)

As expected, the same as before!

The interest rate

The number of years

The present valueNote the minus!

Unused parameters for this problem


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Compounding

Compounding Interest is powerful .

3 Compounding

4

5

6 Year

7

8 0 1,000.00

9 1 1,100.00 =C8*(1+0.1)10 2 1,210.00 =C9*(1+0.1)

11 3 1,331.00 =C10*(1+0.1)

12 4 1,464.10 =C11*(1+0.1)

13 5 1,610.51 =C12*(1+0.1)

14 6 1,771.56

15 7 1,948.72

168 2,143.59

17 9 2,357.95

18 10 2,593.74 =C17*(1+0.1)

19

1,000.00

1,200.00

1,400.00

1,600.00

1,800.00

2,000.00

2,200.00

2,400.00

2,600.00

2,800.00

0 2 4 6 8 10

One thousand dollars becomes nearly 2600 after 10 years!

That must be too good to be true.


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Compounding

Compounding Interest is powerful .

8 0 1,000.00

9 1 1,100.00 =C8*(1+0.1)10 2 1,210.00 =C9*(1+0.1)

11 3 1,331.00 =C10*(1+0.1)

12 4 1,464.10 =C11*(1+0.1)

13 5 1,610.51 =C12*(1+0.1)

14 6 1,771.56

15 7 1,948.72

16 8 2,143.59

17 9 2,357.95

18 10 2,593.74 =C17*(1+0.1)

19 11 2,853.12

20 12 3,138.43

13 3,452.2714 3,797.50

15 4,177.25

16 4,594.97

1,000.00

1,200.00

1,400.00

1,600.001,800.00

2,000.00

2,200.00

2,400.00

2,600.00

2,800.00

0 2 4 6 8 10

0.00

20,000.00

40,000.00

60,000.00

80,000.00

100,000.00

120,000.00

0 10 20 30 40 50

One thousand dollars becomes nearly 120,000 after 50

years. Note how the curve bend Upwards!

50-year Chart


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Inflation

Inflation is the phenomenon that goods become more expensive

(and hence that thus their price inflates).

As a consequence either one dollar can buy

less goods, or one needs more dollars to pay

for the same item.

The calculation of how much a future dollar is worth is exactly

the same as the one for discounting bonds. Only now we need to

use the inflation rate rather than the interest rate for our

calculation.


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Inflation

This is the almost same spreadsheet as we had for bonds

A B C D E F G H I

23 My first inflation calculations

4

5 Current Value 1,000.00

6 Inflation 3%

7

8 Worth in todays

9 dollars

10 In .. Years

11 1 970.87 =D5/(1+D6)12 2 942.60 =C11/(1+$D$6)

13 3 915.14

=C12/(1+$D$6)14 4 888.49

15 5 862.61

16 6 837.48

17 7 813.09

14 8 789.41

15 9 766.42

16 10 744.09

Same formula as for

Bonds!


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Inflation

Note the subtle point -

Though close, there is a difference between:

1000 * (1-r) and 1000/(1+r)

Eg.

1000 * 0.97 = 970 and 1000/1.03 = 970.87

This may look like a small difference, but differences can add

up!

Note


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Inflation

From this calculation we see that in terms of todays buying

power our original 1000 will only be worth 744 dollars in ten

years.

But we had also seen that our 1000 will grow to 2594 if invested

at 10% a year.

Oh thats only 256 dollars less so

we still should have:

2594 256 = 2338 dollars

Not too bad... WRONG!


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Inflation

If we know that we have 2594 in 10 year then we need to

discount this back to today with the prevailing interest rate in

order to see how much that is in todays dollars.A B C D E F G H I


4


6 Inflation 3%

7

8 Maturing in Future

9 years Value

10

11 1 2,518.19 =D5/(1+D6)

12 2 2,444.85 =C11/(1+$D$6)

13 3 2,373.64

=C12/(1+$D$6)14 4 2,304.50

15 5 2,237.38

16 6 2,172.22

17 7 2,108.95

14 8 2,047.52

15 9 1,987.89

16 10 1,929.99

Hence well only have 1930


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Inflation

For this case it is more useful to combine the two calculations:

A B C D E F G H I


4


6 Inflation 3% Interest 10%

7

8 Value in Future

9 years Value

10

11 1 1,067.96 =D5/(1+$D$6)*(1+$H$6)

12 2 1,140.54 =C11/(1+$D$6)*(1+$H$6)

13 3 1,218.05

=C12/(1+$D$6)*(1+$H$6)14 4 1,300.83

15 5 1,389.24

16 6 1,483.65

17 7 1,584.49

14 8 1,692.17

15 9 1,807.17

16 10 1,929.99

Indeed the same


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Inflation

Couldnt we just say Effective Interest = Interest Inflation?A B C D E F G H I

2


4

5 Current Value 1,000.006 Inflation 3% Interest 10%

7 Effective Interest? 7%

8 Value in Future

9 years Value

10

11 1 1,070.00 =D5*(1+$H$7)12 2 1,144.90 =C11*(1+$H$7)

13 3 1,225.04 =C12*(1+$H$7)

14 4 1,310.80

15 5 1,402.55

16 6 1,500.7317 7 1,605.78

14 8 1,718.19

15 9 1,838.46

16 10 1,967.15

Its different!


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Inflation

Its different because one should first reduce the value by the

inflation rate and then apply the interest.

Or:

1/1.03 * 1.1 = 1.06796 unequal 1.07!

Its small but nevertheless important.


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Discount

A closely related topic especially in the context of the time

value of money is that of discount.

We already used this term in the context of bonds and inflation.

When a business decides to invest a certain sum, it also needs to

discount the expected future cash flows in order to decide

whether the investment is worthwhile.

After all, if your return is too small, it

would not be wise to make the investment.


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Discount

The problem is now that the cash flow is expected to grow over

the years (since the business is hopefully getting better and

better).

Conceptually, this is the same as before, only now we need to

do a separate calculation for each year.

As always, it may be complicated to imagine at first, but if we

have an idea of how to get started we can take it from there.

The obvious starting point is: The Cash Flows


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Discounting uneven Cash

FlowsLet us assume that we have the following cash flows:A B C D E F G H I

2

3 How much is a stream of cash flows worth?

4

5 Present Value ?

6 Discount 10%7

8 In Cash

9 years Flow

10

11 1 1,000.00

12 2 1,300.00

13 3 1,200.00

14 4 1,500.00

15 5 1,400.00

16 6 1,600.0017 7 1,700.00

14 8 1,750.00

15 9 1,900.00

16 10 2,100.00

What would they be worth?


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FlowsWe can of course just sum them up:A B C D E F G H I

2


4

5 Present Value ?

6 Discount 10%

78 In Cash

9 years Flow

10

11 1 1,000.00

12 2 1,300.00

13 3 1,200.00

14 4 1,500.00

15 5 1,400.00

16 6 1,600.00

17 7 1,700.00

14 8 1,750.0015 9 1,900.00

16 10 2,100.00

15,450.00

Is this a reasonable value for the cash flows?

15,450.-


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FlowsNo! we need to have some return (namely 10% in this case):A B C D E F G H I

2


4

5 Present Value ?

6 Discount 10%

78 In Cash

9 years Flow

10

11 1 1,000.00 909.09 =C11/POWER(1+$D$6,B11)

12 2 1,300.00 1,074.38 =C12/POWER(1+$D$6,B12)

13 3 1,200.00 901.58 =C13/POWER(1+$D$6,B13)

14 4 1,500.00 1,024.52

15 5 1,400.00 869.29

16 6 1,600.00 903.16

17 7 1,700.00 872.37

14 8 1,750.00 816.3915 9 1,900.00 805.79

16 10 2,100.00 809.64

Thesum of each years cash flows present values!


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FlowsThus we obtain:A B C D E F G H I

2


4

5 Present Value 8,986.20

6 Discount 10%

78 In Cash

9 years Flow

10

11 1 1,000.00 909.09 =C11/POWER(1+$D$6,B11)

12 2 1,300.00 1,074.38 =C12/POWER(1+$D$6,B12)

13 3 1,200.00 901.58 =C13/POWER(1+$D$6,B13)

14 4 1,500.00 1,024.52

15 5 1,400.00 869.29

16 6 1,600.00 903.16

17 7 1,700.00 872.37

14 8 1,750.00 816.3915 9 1,900.00 805.79

16 10 2,100.00 809.64

=SUM(D11:D20)

Surprisingly little, isnt it!


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FlowsNaturally there also is an Excel function for this: NPVPresumably standing for Net Present Value.

A B C D E F G H I

2


4


6 Discount 10%

7

8 In Cash

9 years Flow

10

11 1 1,000.00

12 2 1,300.00

13 3 1,200.00

14 4 1,500.00

15 5 1,400.0016 6 1,600.00

17 7 1,700.00

14 8 1,750.00

15 9 1,900.00

16 10 2,100.00

=NPV(D6,C11:C20)

Discount Rate

Range of Cash Flows


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Excels NPV

We just used the function NPV with NPV presumably standing

for Net Present Value.

If we look back at our previous notes though it would seem thatwhat we have calculated is the Present Value and that there is

no need for the Net.

Indeed, usually one calls what we have calculated Present

Value.

Net Present Value is when we subtract from this the cost of

acquiring the cash flow in question.


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Rate of Return

Of course, often things work the other way around. We bargain

to get a certain stream of cash flows and then we wonder what

the compounded yield on this asset is going to be.


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Calculating the Yield

A B C D E F G H I

2

3 Purchase Price 10,000.00

4

5 P resent V alue 8,986.20 6 Discount 10%

7

8 In Cash

9 years Flow

1011 1 1,000.00 909.09 =C11/POWER(1+$D$6,B11)

12 2 1,300.00 1,074.38

13 3 1,200.00 901.58

14 4 1,500.00 1,024.52

15 5 1,400.00 869.29

16 6 1,600.00 903.16

17 7 1,700.00 872.37

18 8 1,750.00 816.39

19 9 1,900.00 805.79

20 10 2,100.00 809.64

=SUM(D11:D20)

The key thing to realize is that at the

actual yield, the purchase priceequals the present value. In other

words, the net present value is zero.

Hence we can use the solver.


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A B C D E F G H I

2

3 Purchase Price 10,000.00 Net Present Value 0.00

4


6 Discount 7.84%

7

8 In Cash

9 years Flow10

11 1 1,000.00 927.34

12 2 1,300.00 1,117.94

13 3 1,200.00 956.96

14 4 1,500.00 1,109.28

15 5 1,400.00 960.10

16 6 1,600.00 1,017.53

17 7 1,700.00 1,002.57

14 8 1,750.00 957.06

15 9 1,900.00 963.59

16 10 2,100.00 987.63

=D5-D3


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A B C D E F G H I

2

3 Purchase Price 10,000.004

5 Internal Rate of Return 7.84%

6

7

8 In Cash

9 years Flow

10 0 -10,000.00

11 1 1,000.00

12 2 1,300.00

13 3 1,200.00

14 4 1,500.0015 5 1,400.00

16 6 1,600.00

17 7 1,700.00

14 8 1,750.00

15 9 1,900.00

16 10 2,100.00

There is also a built in Excel function: IRR

Standing for Internal Rate of Return

=IRR(C10:C20)

This is investment in year 0.


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Key Points of the Day

Money has Time Value

Interest can compound

Discount is an important concept

Compound or be poor!

Time is your friend! Be patient

Date post:	30-May-2018
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SCC2301 CH4 L09 Rates of Return Time Value of Money

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