Scc2301 Ch4 L09 Rates Of Return Time Value Of Money

Time Value of Money Lecture 9 This lecture is part of Chapter 4: Investing in the Company

Today’s 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!

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.

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

Time Value The present and future values can easily be calculated in Excel A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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Compounding

Year present 1 2 3 4 5

1,000.00 1,100.00 1,210.00 1,331.00 1,464.10 1,610.51

    

=C8*(1+0.1) =C9*(1+0.1) =C10*(1+0.1) =C11*(1+0.1) =C12*(1+0.1)

So we see that 1000 dollars now will be 1610 dollars in 5 years

We assume 10% interest

Time Value Of course this can easily be expressed mathematically, but let’s do it step by step again by understanding what we are doing: 1st year: Value after one year = Present Value + Interest 2nd year: Value after two years = Value after one year + Interest Substitute Line 1

Or: Value after two years = (Present Value + Interest) + Interest This interest needs to be on the entire “Value after one year”

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

FV2 = PV * ( 1 + r)

Time Value And thus we obtain the formula:

FV ( N ) = PV * (1 + r )

N

Let’s check this for our example

FV (5) = 1000 * (1 + 0.1)

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And indeed this is equal to 1610 as before.

Time Value Surprise! There’s also an Excel function for this: FV A 2 3 4 5 6 7 8 9 10 11 12 13

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Compounding

Future Value: 1,610.51

Unused parameters for this problem

=FV(10%,5,0,-1000,0) The interest rate

The present value Note the minus!

The number of years As expected, the same as before!

Compounding Compounding Interest is powerful …. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Compounding

Year 0 1 2 3 4 5 6 7 8 9 10

1,000.00 1,100.00 1,210.00 1,331.00 1,464.10 1,610.51 1,771.56 1,948.72 2,143.59 2,357.95 2,593.74

    

=C8*(1+0.1) =C9*(1+0.1) =C10*(1+0.1) =C11*(1+0.1) =C12*(1+0.1)

2,800.00 2,600.00 2,400.00 2,200.00 2,000.00 1,800.00 1,600.00 1,400.00 1,200.00 1,000.00 0

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 =C17*(1+0.1)

One thousand dollars becomes nearly 2600 after 10 years! That must be too good to be true.

Compounding Compounding Interest is powerful …. 8 9 10 11 12 13 14 15 16 17 18 19 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1,000.00 1,100.00 1,210.00 1,331.00 1,464.10 1,610.51 1,771.56 1,948.72 2,143.59 2,357.95 2,593.74 2,853.12 3,138.43 3,452.27 3,797.50 4,177.25 4,594.97

    

=C8*(1+0.1) =C9*(1+0.1) =C10*(1+0.1) =C11*(1+0.1) =C12*(1+0.1) 120,000.00 100,000.00

50-year Chart 2,800.00 2,600.00 2,400.00 2,200.00 2,000.00 1,800.00 1,600.00 1,400.00 1,200.00 1,000.00 0

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80,000.00 60,000.00

 =C17*(1+0.1) 40,000.00 20,000.00 0.00 0

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One thousand dollars becomes nearly 120,000 after 50 years. Note how the curve bend Upwards!

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.

Inflation This is the almost same spreadsheet as we had for bonds… A B C D E F 2 3 My first inflation calculations 4 5 Current Value 1,000.00 6 Inflation 3% 7 8 Worth in todays 9 dollars 10 In .. Years  =D5/(1+D6) 11 1 970.87  =C11/(1+$D$6) 12 2 942.60  =C12/(1+$D$6) 13 3 915.14 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

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Same formula as for Bonds!

Inflation Note the subtle point Though close, there is a difference between: 1000 * (1-r) and 1000/(1+r)

e Not

Eg. 1000 * 0.97 = 970 and 1000/1.03 = 970.87 This may look like a small difference, but differences can add up!

Inflation From this calculation we see that in terms of today’s 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 that’s only 256 dollars less so we still should have: 2594 – 256 = 2338 dollars Not too bad...

WRONG!

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 today’s dollars. A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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My first inflation calculations Current Value Inflation Maturing in years 1 2 3 4 5 6 7 8 9 10

2,593.74 3% Future Value 2,518.19 2,444.85 2,373.64 2,304.50 2,237.38 2,172.22 2,108.95 2,047.52 1,987.89 1,929.99

 =D5/(1+D6)  =C11/(1+$D$6)  =C12/(1+$D$6)

Hence we’ll only have 1930

Inflation For this case it is more useful to combine the two calculations: A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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My first inflation calculations Current Value Inflation Value in years 1 2 3 4 5 6 7 8 9 10

1,000.00 3%

Interest

10%

Future Value 1,067.96 1,140.54 1,218.05 1,300.83 1,389.24 1,483.65 1,584.49 1,692.17 1,807.17 1,929.99

 =D5/(1+$D$6)*(1+$H$6)  =C11/(1+$D$6)*(1+$H$6)  =C12/(1+$D$6)*(1+$H$6)

Indeed the same

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Inflation Couldn’t we just say Effective Interest = Interest – Inflation? A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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My first inflation calculations Current Value Inflation Value in years 1 2 3 4 5 6 7 8 9 10

1,000.00 3%

Interest Effective Interest?

Future Value 1,070.00 1,144.90 1,225.04 1,310.80 1,402.55 1,500.73 1,605.78 1,718.19 1,838.46 1,967.15

 =D5*(1+$H$7)  =C11*(1+$H$7)  =C12*(1+$H$7)

It’s different!

10% 7%

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Inflation It’s 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! It’s small but nevertheless important.

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.

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

Discounting uneven Cash Flows Let us assume that we have the following cash flows: A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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How much is a stream of cash flows worth? Present Value Discount In years

? 10% Cash Flow

1 2 3 4 5 6 7 8 9 10

1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00

What would they be worth?

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Discounting uneven Cash Flows We can of course just sum them up: A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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How much is a stream of cash flows worth? Present Value Discount In years

? 10% Cash Flow

1 2 3 4 5 6 7 8 9 10

1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00 15,450.00

15,450.-

Is this a reasonable value for the cash flows?

Discounting uneven Cash Flows No! we need to have some return (namely 10% in this case): A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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How much is a stream of cash flows worth? Present Value Discount In years

? 10% Cash Flow

1 2 3 4 5 6 7 8 9 10

1,000.00 909.09 1,300.00 1,074.38 1,200.00 901.58 1,500.00 1,024.52 1,400.00 869.29 1,600.00 903.16 1,700.00 872.37 1,750.00 816.39 1,900.00 805.79 2,100.00 809.64

 =C11/POWER(1+$D$6,B11)  =C12/POWER(1+$D$6,B12)  =C13/POWER(1+$D$6,B13)

The sum of each year’s cash flow’s present values!

Discounting uneven Cash Flows Thus we obtain: A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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How much is a stream of cash flows worth? Present Value Discount In years

8,986.20 10% Cash Flow

1 2 3 4 5 6 7 8 9 10

1,000.00 909.09 1,300.00 1,074.38 1,200.00 901.58 1,500.00 1,024.52 1,400.00 869.29 1,600.00 903.16 1,700.00 872.37 1,750.00 816.39 1,900.00 805.79 2,100.00 809.64

 =SUM(D11:D20)

Surprisingly little, isn’t it!  =C11/POWER(1+$D$6,B11)  =C12/POWER(1+$D$6,B12)  =C13/POWER(1+$D$6,B13)

Discounting uneven Cash Flows Naturally there also is an Excel function for this: NPV Presumably standing for Net Present Value. A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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How much is a stream of cash flows worth? Present Value Discount In years

8,986.20 10%

 =NPV(D6,C11:C20)

Cash Flow 1 2 3 4 5 6 7 8 9 10

1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00

Discount Rate Range of Cash Flows

Excel’s 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 that what 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.

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.

Calculating the Yield A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Purchase Price

10,000.00

Present Value Discount

8,986.20 10%

 =SUM(D11:D20)

909.09 1,074.38 901.58 1,024.52 869.29 903.16 872.37 816.39 805.79 809.64

 =C11/POWER(1+$D$6,B11)

In years

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Cash Flow 1 2 3 4 5 6 7 8 9 10

1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00

The key thing to realize is that at the actual yield, the purchase price equals the present value. In other words, the net present value is zero.

Hence we can use the solver.

Calculating the Yield A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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Purchase Price

10,000.00

Present Value Discount

10,000.00 7.84%

In years

Cash Flow 1 2 3 4 5 6 7 8 9 10

1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00

927.34 1,117.94 956.96 1,109.28 960.10 1,017.53 1,002.57 957.06 963.59 987.63

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Net Present Value

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=D5-D3

Calculating the Yield There is also a built in Excel function: IRR Standing for Internal Rate of Return A 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 14 15 16

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Purchase Price Internal Rate of Return

In years 0 1 2 3 4 5 6 7 8 9 10

Cash Flow -10,000.00 1,000.00 1,300.00 1,200.00 1,500.00 1,400.00 1,600.00 1,700.00 1,750.00 1,900.00 2,100.00

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10,000.00 7.84%

=IRR(C10:C20) This is investment in year 0.

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.