Math tools for Time Value of Money (additional help guide)

Slide 4.1

Corporate Financial
Management, 5th edition
Glen Arnold
Mathematical Tools For Time Value of
Money

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.2

Simple Interest and Future Value
• Simple interest
A sum of £10 is deposited in a bank account that pays 12 per cent per annum. At the end of year 1 the investor has £11.20 in the account.
F = P(1 + i)
11.20 = 10(1 + 0.12) where F = Future value, P = Present value, i = Interest rate.
At the end of five years:
F = P(1 + in) where n = number of years.
16 = 10(1 + 0.12 × 5)

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.3

Compound interest
An investment of £10, an interest rate of 12 per cent.
In one year the capital will grow by 12 per cent to
£11.20. At the end of two years:
F = P(1 + i) (1 + i)

F = 11.20 (1 + i)

Alternatively, F = P(1 + i)2

£10 × 1.2544 = £12.544
Over five years the result is:

F = P (1 + i)n
17.62 = 10 (1 + 0.12) 5
Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

F = 12.54

Slide 4.4

The future value of £1

Exhibit 2.27 The future value of £1

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.5

Present values
• Receipt of £17.62 in five years’ time, what is the present value? • Rearrangement of the compound formula:
1
F
P = ––––––– or P = F × –––––––
(1 + i)n
(1 + i)n
17.62
10 = ––––––––––
(1 + 0.12)5
• Alternatively, discount factors may be used:
The factor needed to discount £1 receivable in five years when the discount rate is 12 per cent is 0.5674. Therefore the present value of £17.62 is:
0.5674 × £17.62 = £10

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.6

The present value of £1

Exhibit 2.28 The present value of £1

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.7

Determining the rate of interest
• To be able to calculate i it is necessary to rearrange the compounding formula. Since:
F = P(1 + i)n first, divide both sides by P:
F/P = (1 + i)n
(The Ps on the right side cancel out.)
• Take the root to the power n of both sides and subtract 1 from each side: i= n



[F/P] – 1 or i = [F/P]1/n – 1

• Five-year investment requiring an outlay of £10 and having a future value of £17.62, the rate of return is:
5 17.62 i= –––––– – 1 i = 12%
10
i = [17.62/10]1/5 – 1 i = 12%


5

i =  £10,000/£8,000 – 1 = 0.046
Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.8

The investment period
F = P(1 + i)n

F/P = (1 + i)n log(F/P) = log(1 + i)n

log(F/P) n = –––––––– log(1 + i)
• How many years does it take for £10 to grow to £17.62 when the interest rate is 12 per cent? log(17.62/10) n = ––––––––––––– Therefore n = 5 years log(1 + 0.12)

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.9

Annuities
£10 per year for five years, interest rate is 12 per cent:
• Method 1

A
A
A
A
A
Pan = ––––– + –––––– + –––––– + –––––– + ––––––
(1 + i) (1 + i)2 (1 + i)3 (1 + i)4
(1 + i)5

where A = the periodic receipt.
10
10
10
10
10
P10,5 = ––––– + –––––– + –––––– + –––––– + –––––– = £36.05
(1.12)
(1.12)2 (1.12)3 (1.12)4 (1.12)5

• Method 2

– Using the derived formula:
1 − 1/(1 + i)n
Pan = ––––––––––– × A i 1 − 1/(1 + 0.12)5
P10,5 = ––––––––––– × 10 = £36.05
0.12
Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.10

Annuities (Continued)
• Method 3
– Use the ‘present value of an annuity’ table.
– This refers to the present value of five annual receipts of £1.
– 3.605 × £10 = £36.05

Exhibit 2.29 The present value of an annuity of £1 per annum

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.11

Perpetuities

P =

A
___

i

P =

10
___

=

£83.33

0.12

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.12

Discounting semi-annually, monthly and daily
• £10 in a bank account earning 12 per cent per annum
10(1 + 0.12) = £11.20

• If the interest is compounded semi-annually
10(1 + [0.12/2])(1 + [0.12/2]) = 10(1 + [0.12/2])2 = £11.236

• If the interest is compounded quarterly:
10(1 + [0.12/4])4 = £11.255

• Daily compounding:
10(1 + [0.12/365])365 = £11.2747

• £10 is deposited that compounds interest quarterly and the nominal return per year is 12 per cent. After 8 years:
10(1 + [0.12/4])4x8 = £25.75

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.13

Continuous compounding
• When the number of compounding periods approaches infinity the future value is found by F = Pein where e is the value of the exponential function. This is set as 2.71828.
• The future value of £10 deposited in a bank paying 12 per cent nominal compounded continuously after eight years is:

10 × 2.718280.12×8 = £26.12

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

Slide 4.14

Converting monthly and daily rates to annual rates
• Annual Percentage Rate (APR)
• Effective Annual Rate (EAR)
• If m is the monthly interest or discount rate, then over 12 months: (1 + m)12 = 1 + i

i = (1 + m)12 − 1

• A credit card company charges 1.5 per cent per month, the
APR is: i = (1+ 0.015)12 − 1 = 19.56%

• Finding the monthly rate when you are given the APR:



m = (1 + i) 1/12 − 1 or m = 12 (1 + i) − 1 m = (1+ 0.1956)1/12 − 1 = 0.015 = 1.5%

• Daily rate:
(1 + d)365 = 1 + i

Glen Arnold, Corporate Financial Management, 5th Edition © Pearson Education Limited 2013

PV of Single Cash Flows (1)
• Using a discount factor table:
– £10,000 received in 4 years time
– Discount rate 14%

Present Value of £1
Discount Factor
6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

16%

17%

18%

19%

20%

1

0.952

0.943

0.935

0.926

0.917

0.909

0.901

0.893

0.885

0.877

0.870

0.862

0.855

0.847

0.840

0.833

2

0.907

0.890

0.873

0.857

0.842

0.826

0.812

0.797

0.783

0.769

0.756

0.743

0.731

0.718

0.706

0.694

3

0.864

0.840

0.816

0.794

0.772

0.751

0.731

0.712

0.693

0.675

0.658

0.641

0.624

0.609

0.593

0.579

4

0.823

0.792

0.763

0.735

0.708

0.683

0.659

0.636

0.613

0.592

0.572

0.552

0.534

0.516

0.499

0.482

5

0.784

0.747

0.713

0.681

0.650

0.621

0.593

0.567

0.543

0.519

0.497

0.476

0.456

0.437

0.419

0.402

Ye ar

5%

• Present value = £10,000 x 0.592 = £5,920

PV of Single Cash Flows (2)
• Using a scientific calculator:
– £10,000 received in 4 years time
– Discount rate 14.8%
– Discount factor formula =
• where t = time (years)

1
(1 + discount rate) t

– Discount factor = 1 / (1.1484) or 1.148-4 = 0.576

• Present value = £10,000 x 0.576 = £5,760

PV of Annuity Cash Flows (3)
• Using annuity factor table:
– £2,500 received annually for 4 years (first receipt in one year)
– Discount rate 14%
Annuity Factors

(Cumulative Discount Factors)
6%

7%

8%

9%

10%

11%

12%

13%

14%

15%

16%

17%

18%

19%

20%

1

0.952

0.943

0.935

0.926

0.917

0.909

0.901

0.893

0.885

0.877

0.870

0.862

0.855

0.847

0.840

0.833

2

1.859

1.833

1.808

1.783

1.759

1.736

1.713

1.690

1.668

1.647

1.626

1.605

1.585

1.566

1.547

1.528

3

2.723

2.673

2.624

2.577

2.531

2.487

2.444

2.402

2.361

2.322

2.283

2.246

2.210

2.174

2.140

2.106

4

3.546

3.465

3.387

3.312

3.240

3.170

3.102

3.037

2.974

2.914

2.855

2.798

2.743

2.690

2.639

2.589

5

4.329

4.212

4.100

3.993

3.890

3.791

3.696

3.605

3.517

3.433

3.352

3.274

3.199

3.127

3.058

2.991

Ye ar

5%

• Present value = £2,500 x 2.914 = £7,285

PV of Perpetuity Cash Flows (4)
• Using a calculator:
– £2,500 received annually in perpetuity (first receipt in one year)
– Discount rate 14%
– Perpetuity factor formula =

1
(discount rate)

– Perpetuity factor = 1 / 0.14 = 7.143

• Present value = £2,500 x 7.143 = £17,858

The Time Value of Money: Equations
Simple Interest

According to the simple interest process, the evaluation of the Future Value, for a specific capital (C), for a specific time horizon (T) and a specific interest rate (i) that applies for the examined time period, is given by the following equation:
FV = C + Interest = C + I
FV = C + C i (T-t0)
FV = C (1+i Δt)
FV = PV (1+ i Δt)
Figure 1.1 simple interest t0 T

C: capital

I: interest

C+ I

Compound interest

However, very often the interest at the end of each period is compounded for the next period leading to the compound interest, which is the interest of the interest:
Figure 1.2 compound interest t0 t1

t2

tk

tn-2

tn-1

tn

C

FV

FVn = C ⋅ (1 + j )

n

FVn = PV ⋅ (1 + j )

n

Furthermore, if the compounding frequency (in terms of a fixed period, i.e. a year) is more often (i.e. semi-annual) then we use the following formula: j( m ) 

FVn = C ⋅ 1 +

 m 



m⋅n

j( m ) 

FVn = PV ⋅ 1 +

 m 



where m is the compounding frequency (m=2, every six months)
1

m⋅n

Continuous compound
In the case that the compounding process is continuous, we use the following formula:



m⋅n



 j( m ) 

1 n n
FVt =0 lim  PV ⋅ 1 +
=
FVt =0 lim  PV ⋅ 1 +
=
 

m →∞ m →∞  m  m  


 



j( m )



Annual Percentage Rate (APR) or Effective Annual Rate (EAR) m j( m ) 

APR =  1+
 -1

m 



m

 j( m)






j

 ( m)








⋅n

n
FV= PV ⋅ e t =0

j( m ) ⋅n

where m: compounding frequency example: j(2) = 5%  APR = 5,062% j(4) = 10%  APR = 10,381% example: investments:

A

B

duration:

1 year

1 year

IRR:

j(1 )=7%

j(365 )=6,85%

APRΑ = (1 + 0.07/1)1 – 1 = 0,07
APRΒ = (1 + 0.685/365)365 – 1 = 0,0709

Annuity & Perpetuity
A progress of periodic certain future cash flows. If the duration of the future CF is a finite number, then this process is called annuity. In the case that the duration of future CF is infinite, then this process is called perpetuity.
Figure 2.1 simple annuity (annuity frequency = compounding frequency) t0 t1

C

t2

C

tk

PVannuity =

tn-2

tn-1

tn

C
C
C
C C
+
+ ...+
2
n
(1+ j ) (1+ j )
(1+ j )

C

C

1 
1
PVannuity C  1 −
=
n
 j  (1 + j )
2





Perpetuity: As n goes to infinity then:

 1 
1   C
=
lim ( PV ) = lim  C  1 − n  

n →∞ n →∞ 
  j  (1 + j )    j
Figure 2.2 General case of annuities (annuity frequency < compounding frequency) t0 PVannuity

t2

tk

tn-2

tn-1

tn

C

PVannuity =

t1

C

C

C

C

C

C m +

C
2m

+ ...+

C

j( m )  j( m )  j( m ) 



 1+

 1+

 1+



 m  m  m 






C
C
C
=
+
+ ...+
2
n
(1+ j AER ) (1+ j AER )
(1+ j AER )

nm

 1 

1
= C
PVannuity
1 − n 
 j AER  (1 + j AER )  

Figure 2.3 General case of annuities (annuity frequency > compounding frequency) t0 t1

tn-1_

tn-1_6

tn

C t0 t0_6

C

C

C

C

t1

tn-1_

tn

C*

C*

C*

6

C * =1 + j(1)  + C
C
12 


 1 
1
= C* 
PVannuity
1 − n  j(1)  (1 + j(1) )








Figure 2.4 General case of annuities (annuity and compounding frequency: not analogous)
(i.e.Τcompounding =6, Τannuity =4)

3

t0_4

T0_8

T1

C

t0

C

C

t0

tn-2

tn-1_4_

tn-1_8

tn

C

C

C

T1

tn

C*

C*

2 
6
4


C * = 1 + j(2)  1 + j(2)  + C 1 + j(2)  + C
C
12  
12 
12 



 1 
1
= C* 
PVannuity
1 − n  j AER  (1 + j AER )





2

j( 2 ) 

AER =  1+
 -1

2 



4