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FNCE10002 Principles of Finance

Semester 1, 2018

Department of Finance FNCE10002 Principles of Finance Teaching Note 1 Introduction to Financial Mathematics* Asjeet S. Lamba, Ph.D., CFA Associate Professor Department of Finance Faculty of Business and Economics Room 12.043, 198 Berkeley Street The University of Melbourne Carlton, Victoria 3053 [email protected] Outline 1.

Future and present values of single cash flows

2

2.

Future and present values of a series of cash flows

3

3.

Present and future values of equal, periodic cash flows 3.1 Present value of a perpetuity 3.2 Present value of a deferred perpetuity 3.3 Present value of an ordinary annuity 3.4 Future value of an ordinary annuity 3.5 Present and future values of annuities due 3.6 Present value of a growing perpetuity 3.7 Present and future values of a growing annuities

5 5 6 7 8 9 10 12

4.

Suggested answers to practice problems

15

* This teaching note has been prepared for use by students enrolled in FNCE10002 Principles of Finance. This material is copyrighted by Asjeet S. Lamba and reproduced under license by the University of Melbourne (© 2018). This document was last revised in February 2018. Please email me if you find any typos or errors. Teaching Note 1: Introduction to Financial Mathematics

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This teaching note deals with the area of Principles of Finance that are central to the understanding of the investment and financing decisions of investors and companies, namely financial mathematics. After reading this note you should be able to:      1.

Calculate the future and present values of a series of cash flows Calculate the future and present values of ordinary annuities Calculate the future and present values of annuities due Calculate the present value of ordinary and deferred perpetuities Calculate the future and present values of growing annuities Future and present values of a single cash flow

Given the time value of money, a cash flow received today (that is, time 0) is more valuable than the same cash flow received some time in the future. Accordingly, a cash flow of PV0 today that earns interest at a rate of r percent per period for n periods has a future value of:

FVn  PV0 (1  r)n ,

(1)

where (1+ r)n is the future dollar value of $1 today earning an interest rate of r percent per period for n time periods. This amount is then multiplied by PV0 to obtain its future value at the end of time period n. On a timeline, the cash flow can be represented as follows: Figure 1: Present and Future Values of Single Cash Flows 0

1

2

PV0

3

n

FVn

Note that in the above timeline we’re assuming that the cash flow occurs at the end of a particular period. For example, PV0 occurs at the end of period 0 while FVn occurs at the end of period n. This is the convention that we use throughout this subject to simplify the calculations. Where a cash flow does not occur at the end of a period it would need to be specified as such. (For example, in our discussion of annuities due later in this note we will assume that the cash flows occur at the beginning of the period.) Example 1: Future value of a single cash flow What is the future value of $1,000 invested at an interest rate of 10 percent per annum at the end of: (a) 3 years and (b) 20 years? Solution a)

The future value at the end of 3 years is: FV3 = 1000(1 + 0.10)3 = $1,331.00.

b)

The future value at the end of 20 years is: FV20 = 1000(1 + 0.10)20 = $6,727.50.

A similar principle applies in the determination of a present value equivalent of an expected cash flow in a future period. Formally, a cash flow of FVn that is due in n periods and is discounted at a rate of r percent per period has a present value today of:

Teaching Note 1: Introduction to Financial Mathematics

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FNCE10002 Principles of Finance

PV0 

Semester 1, 2018

FVn , (1  r ) n

(2)

where 1/(1+ r)n is the present dollar value of $1 to be received n time periods from today at an interest rate of r percent per period. Note that the term 1/(1+ r) is referred to as the one-period discount factor. It is the dollar value today of $1 occurring at the end of one period discounted at an interest rate of r. Example 2: Present value of a single cash flow Calculate the present value today of $1,331.00 occurring at the end of year 3 assuming an interest rate of 10 percent per annum. Calculate the present value of $6,727.50 occurring at the end of year 20 assuming an interest rate of 10 percent per annum. Solution The present value today of $1,331 at the end of year 3 is: PV0 = 1331/1 + 0.10)3 = $1,000.00. The present value today of $6,727.50 at the end of year 20 is: PV0 = 6727.50/1 + 0.10)20 = $1,000.00. Note that, as expected, the present values calculated here are the same as those in the previous example. 2.

Future and present values of a series of cash flows

If we are given a series of cash flows that occur over different time periods their future value can be calculated as the sum of the future values of the individual cash flows. This is also known as the value additivity principle. On a timeline, the cash flow can be represented as follows: Figure 2: Present and Future Values of Multiple Cash Flows with no Cash Flow at Time 0 0

1

2

3

n

PV0

C1

C2

C3

Cn

FVn As the future value of a sum of a series of cash flows earning an interest rate of r percent per period at the end of n periods is equal to the sum of their individual future values, we have:

FVn  C1(1 r)n1  C2 (1 r)n2  ... Cn .

(3a)

Note that in the above expression we assume that the first cash flow occurs at the end of time 1 and the last cash flow occurs at the end of time n. Clearly, the cash flow occurring at the end of time n will not earn any interest. If we assume that the first cash flow occurs at the end of time 0 (that is, today) then the cash flows will look a little different, as follows:

Teaching Note 1: Introduction to Financial Mathematics

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Figure 3: Present and Future Values of Multiple Cash Flows with a Cash Flow at Time 0 0

1

2

3

n

C0

C1

C2

C3

Cn

PV0

FVn

The future value at the end of time n will now be:

FVn  C0 (1 r)n  C1 (1 r)n1  C2 (1 r)n2  ... Cn .

(3b)

This amount is higher than what we got in expression (3a) because there is now one additional cash flow at time 0 (C0) which earns interest over n time periods. Example 3: Future value of a series of cash flows Calculate the future value at the end of year 4 of investing the following cash flows: C1 = $1,000, C2 = $2,000, C3 = $3,500, C4 = $3,000. Assume that the applicable interest rate is 10 percent per annum. Recalculate the future value above if an additional cash flow of $2,000 were invested today (that is, C0 = $2,000). Solution In the first case, we have: FV4 = 1000(1.10)3 + 2000(1.10)2 + 3500(1.10)1 + 3000. FV4 = 1331 + 2420 + 3850 + 3000 = $10,601.00. In the second case, we have: FV4 = 2000(1.10)4 + 1000(1.10)3 + 2000(1.10)2 + 3500(1.10)1 + 3000. FV4 = 2928.20 + 1331 + 2420 + 3850 + 3000 = $13,529.20. If we are given a series of cash flows occurring over different time periods their present value can be calculated as the sum of the present values of each individual cash flow. That is, the present value of a sum of a series of cash flows discounted at r percent per period over n periods is equal to the sum of their individual present values, which is:

PV0 

C1 C2 Cn . 1  2  ...  (1 r ) (1  r ) (1  r )n

(4a)

Note again that in the above expression we assume that the first cash flow occurs at the end of time 1 and the last cash flow occurs at the end of time n (see the timeline in figure 2). If we assume that the first cash flow occurs at the end of time 0 (that is, today, as in the timeline in figure 3) then the present value today will be:

PV0  C0 

C1 C2 Cn . 1  2  ...  (1  r ) (1  r ) (1 r )n

Teaching Note 1: Introduction to Financial Mathematics

(4b)

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Note that the difference between the present value in expression (4b) and (4a) is the additional cash flow at time 0 (C0) which does not need to be discounted as it occurs at time 0. Example 4: Present value of a series of cash flows Calculate the present value of the following future cash flows: C1 = $1,000, C2 = $2,000, C3 = $3,500, C4 = $3,000. Assume that the applicable interest rate is 10 percent per annum. Recalculate the present value if an additional cash flow of $2,000 were invested today (that is, C0 = $2,000). What is the relation between the future values calculated in each case of the previous example and the present values calculated here? Solution In the first case, we have: PV0 = 1000/(1.10)1 + 2000/(1.10)2 + 3500/(1.10)3 + 3000/(1.10)4. PV0 = 909.10 + 1652.89 + 2629.60 + 2049.04 = $7,240.63. In the second case, we have: PV0 = 2000 + 1000/(1.10)1 + 2000/(1.10)2 + 3500/(1.10)3 + 3000/(1.10)4. PV0 = 2000 + 909.10 + 1652.89 + 2629.60 + 2049.04 = $9,240.63. Note the relation between the future values calculated in each case of the previous example and the present values calculated above. Given the interest rate of 10 percent per annum, if we know the future value at the end of year 4 we can directly calculate the present value today using the future value amount, as follows: In the first case, we have: PV0 = FVn/(1 + r)n = 10601/(1 + 0.10)4 = $7,240.63. In the second case, we have: PV0 = 13529.20/(1 + 0.10)4 = $9,240.63. 3.

Present and future values of equal, periodic cash flows

3.1

Present value of a perpetuity

The simplest type of equal, periodic cash flow is a perpetuity where the cash flow recurs forever. On a timeline, a perpetual cash flow (C) can be shown as follows: Figure 4: Present Value of a Perpetuity 0

1

2

n

n+1

n+2

PV0

C

C

C

C

C

The present value of a perpetuity is calculated by discounting each cash flow to time period 0 as follows: PV0 = C/(1+r) + C/(1+r)2 +…+ C/(1+r)n + C/(1+r)n+1 + … The above expression can be rewritten as: Teaching Note 1: Introduction to Financial Mathematics

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PV0 = C[1/(1+r) + 1/(1+r)2 + … + 1/(1+r)n + 1/(1+r)n+1 + …] As n approaches infinity, the right-hand side expression [1/(1+r) + 1/(1+r)2 +…+ 1/(1+r)n + 1/(1+r)n+1 +…] approaches 1/r. So, in the limit, the present value of a perpetuity is:

PV0 

C . r

(5)

Note that the above expression requires that the first cash flow occurs at the end of time 1, and not time 0. Example 5: Present value of a perpetuity Your company can lease a computer system for equal annual payments of $2,000 forever, or purchase it today for $23,000. The first payment is to be made at the end of year 1 with subsequent payments being made at the end of each year. Ignoring taxes and other complications, what should the company do if the interest rate is 10 percent per annum? Solution We need to compare the purchase price today of $23,000 with the present value of equal annual payments of $2000 forever. The present value of this perpetuity is:

PV0 

2000  $20, 000. 0.10

So, the company would prefer to lease the computer system as it has the lower present value of cost. 3.2

Present value of a deferred perpetuity

A deferred perpetuity is a series of equal, periodic cash flows that recur forever but with the first cash flow occurring at some point in the future. For example, the following timeline shows a perpetual cash flow which is deferred until the end of year n+1. Figure 5: Present Value of a Deferred Perpetuity 0

1

2

PV0

n

n+1

n+2

C

C

The present value of a deferred perpetuity can be calculated by first obtaining the present value of the perpetuity at time n using expression (5), as follows:

PVn 

C . r

Next, we get the present value of the above cash flow at time 0 by discounting it over n time periods, as follows:

 C  1  PV0     n.  r   1 r   Teaching Note 1: Introduction to Financial Mathematics

(6)

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Again, note that in the above expression we assume that the first cash flow of the deferred annuity occurs at the end of time n + 1, and not time n. The first term in the above expression (C/r) is the present value of this deferred perpetuity at the end of time n. We then discount this future value over n time periods to get the present value at time 0, as shown in expression (6). Example 6: Present value of a deferred perpetuity Your company can lease a computer system for equal annual payments of $2,000 forever, or purchase it today for $14,500. The company has been able to enter a deal with the supplier where the first lease payment has been deferred to the end of year 4 with subsequent payments being made at the end of each of the following years forever. Ignoring taxes and other complications, what should the company do if the interest rate is 10 percent per annum? Solution We need to compare the purchase price today of $14,500 with the present value of the deferred perpetuity of $2,000 forever where the cash flow is deferred until the end of year 4. The present value of this deferred perpetuity is:

 1  2000      $15,026.30. PV0    3  0.10   1  0.10   Again, note that the first amount (2000/0.10 = $20,000) is the present value of the deferred perpetuity at the end of year 3. This amount is then discounted over 3 years to get the present value at the end of year 0. The company would prefer to purchase the computer system as it has the lower present value of cost. 3.3

Present value of an ordinary annuity

An annuity is a series of equal, periodic cash flows occurring over n periods. Ordinary annuities occur at the end of each period. In valuing an ordinary annuity, it is assumed that the first cash flow of the annuity occurs at the end of the first period, and the last cash flow occurs at the end of period n. On a timeline, an n-period ordinary annuity is as follows: Figure 6: Present Value of an Ordinary Annuity 0

1

2

n

PV0

C

C

C

Note that the present value of an n-period annuity can be obtained as the difference between the present value of a perpetuity that starts at the end of time period 1, or expression (5) above, and the present value of a deferred perpetuity that starts at the end of time period n+1, or expression (6) above. That is:

 C   C  1  PV0       . n   r   r   1  r   Simplifying the above expression, we get:

Teaching Note 1: Introduction to Financial Mathematics

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FNCE10002 Principles of Finance

C PV0   r

1  1  n  1  r

Semester 1, 2018

 .  

(7)

Example 7: Present value of an annuity You have won a contest and have been given the choice between accepting $32,000 today or an equal annual cash flow of $5,000 per year at the end of each of the next 10 years. What should you do if the interest rate is 10 percent per annum? Solution We need to compare the lump sum amount of $32,000 available today with the present value of the tenyear annuity of $5,000 per year. The present value of this annuity is:

 1  5000  PV0    $30,722.84.  1  10   0.10  1  0.10   So, you would prefer the lump sum amount today as it has a higher present value compared to the annuity of $5,000 per year for ten years. 3.4

Future value of an ordinary annuity

The future value at the end of period n of an n-period ordinary annuity is the sum of the future values of each of these n cash flows. On a timeline, the future value of an n-period ordinary annuity can be depicted as follows: Figure 7: Future Value of an Ordinary Annuity 0

1

2

n

C

C

C FVn

The future value of an n-period ordinary annuity is easily obtained from its present value in expression (7) by compounding the present value over n time periods, as follows:

C F Vn   r

1   n  1 r  . 1  n    1  r 

Simplifying the above expression, we get: n C FVn    1  r   1 .   r 

(8)

Example 8: Future value of an annuity You have won a contest which pays an equal annual cash flow of $5,000 per year at the end of each of the next 10 years. What is the future value of this prize if the interest rate is 10 percent per annum?

Teaching Note 1: Introduction to Financial Mathematics

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FNCE10002 Principles of Finance

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Solution Using the future value of an ordinary annuity expression, we get: 10  5000  1  010  1  $79,687.12. FV10       0.10  

Note that we had calculated the present value of this cash flow in the previous example at $30,722.84. So, we could also calculate the future value of this cash flow at the end of ten years as: FV10 = 30722.84(1 + 0.10)10 = $79,687.12. 3.5

Present and future values of annuities due

So far, the series of cash flows we have valued have been assumed to occur at the end of each period. In some cases, these cash flows may occur at the beginning, rather than the end, of each period. Such an annuity is referred to as an annuity due. Using the convention that we have been using above and treating cash flows as occurring at the end of a particular time period this implies that if a cash flow now occurs at the beginning of time n then that is the same as the cash flow occurring at the end of time n – 1. This is because the beginning of time n is the same as the end of time n – 1. For example, a cash flow occurring at the beginning of 2017 (that is, 1 January 2017) is the same as if the cash flow occurs at the end of 2016 (that is, 31 December 2016) and so on, as shown below. End of previous year 31 Dec 2013 31 Dec 2014 31 Dec 2015 31 Dec 2016

Beginning of year 1 Jan 2014 1 Jan 2015 1 Jan 2016 1 Jan 2017

If the cash flows were monthly, we would have the following equivalence: Beginning of month 1 Dec 2016 1 Nov 2016 1 Oct 2016 1 Sep 2016

End of previous month 30 Nov 2016 31 Oct 2016 30 Sep 2016 31 Aug 2016

On a timeline, an n-period annuity due can be shown as follows: Figure 8: Present and Future Values of Annuities Due 0

1

2

n–1

C PV0

C

C

C

n

FVn

As the figure shows, an annuity due is essent...