Compound rate of return and present value

What does an investor's return include?

Milking the cow.

An investor return includes all the "cash flows" he/she gets for its investment.

Positive cash flows: rents, interests, dividends, capital gain

(positive difference between buying / subscription price and resale value)

Negative cash flows: expenses (fees, taxes...), capital loss.

Normally, the more risky the investment is, the higher should the rate of return be,
so as to convince him/her to invest.

In other words it includes a "risk premium".

Compound rate

Snowball arithmetic" under "geometric progression".

Purpose

Compound rate calculations allow to know the return of assets / investments .
over several years.

They are used notably in finance, thus for investing and borrowing (rates of return,
interest rates) and by extension in economics (growth rates, inflation effects).

For savers, investors, borrowers, compound rates are used to:

Know the rate of return of an investment or the real financial cost

("actuarial interest rate") of a debt

Know what an investment, a cash flow, an asset (and also a debt)

will be worth after several years (future value FV).

Give the present value PV of an asset or a future cash flow

Definition / calculation

The compound annual rate of return of an asset (or of a project) is:

An annual return (or growth percentage) R

which, compounded (*) year after year,

brings a total return TR at the end of the period.

(*) Compounded = cumulated by using the hypothesis that:

Every year the asset value is increased

   ("capitalized") thanks to the annual return.

In the next year the return rate is applied to that increased value.

F
or example if the rate is 6%, an asset which value is today 1 Euro will be worth:

1.06 Euro after 1 year,

1.06 x 1.06 after 2 years, and so on..

Future value FV

Parameters

 R = the annual "compounded" rate of return (or, in other areas than finance,

          any other annual increase / decrease: economic growth, inflation...)

Y = the number of years

FV = future value at the end of the period

TR = total return of that period

 

Example

If 1 Euro is invested in an account with R = 6%, Y = 4 years (and if the interests are
not withdrawn), its future value FV will be:

1 Euro x 1.06 x 1.06 x 1.06 x 1.06 = 1.262 Euro (this is the future value FV)

The total return TR (relative increase in value) is therefore 26,2 %

Present value PV

Such asset valuation is done by "discounting" the future expected asset worth
(and/or its expected cash flows) with a return rate.

If we use the data above, 10,000 Euros that we will receive in 4 years

are theoretically worth now (PV / present value):

10,000 / 1.262 = 7,924 Euros (present value PV)

Chains of annual cash flows and probabilities

Financial assets bring often, not a only a final worth (at maturity or at resale) but also
a chain of cash flows year after year.

You get a fresh egg everyday and the hen for a soup at the end.

The calculations must take that into account, for example to determine the VA of the asset.

In that case, every cash flow is discounted according to the number of years
in which it will take place
in order to obtain its present value.

=> Then all those PV are summed up to give the asset VA.

Also, we can have several scenarios, each one with its probability.

=> The asset VA is then the sum of the PV of every scenario multiplied by its
     probability coefficient.

<Short table of compound rates and future values

Horizontal scale: rate. Vertical scale: years,

At the crossing: future value for 1 Euro invested

-5%

-3%

-2%

-1%

R

1%

2%

3%

5%

6%

7%

8%

10%

12%

15%

20%

30%

 

Y

 

0.95

0.97

0.98

0.99

1

1.01

1.02

1.03

1.05

1.06

1.07

1.08

1.100

1.12

1.15

1.20

1.30

.903

.941

.960

.980

2

1.020

1.040

1.061

1.103

1.124

1.145

1.166

1.210

1.254

1.323

1.440

1.690

.857

.913

.941

.970

3

1.030

1.061

1.093

1.158

1.191

1.225

1.260

1.330

1.405

1.521

1.728

2.197

.815

.885

.922

.961

4

1.041

1.082

1.126

1.216

1.262

1.311

1.360

1.464

1.574

1.749

2.074

2.856

.774

.859

.904

.951

5

1.051

1.104

1.159

1.276

1.338

1.403

1.469

1.611

1.762

2.011

2.488

3.713

.735

.833

.886

.941

6

1.062

1.126

1.194

1.340

1.419

1.501

1.587

1.772

1.974

2.313

2.986

4.827

.698

.808

.868

.932

7

1.072

1.149

1.230

1.407

1.504

1.606

1.714

1.949

2.211

2.660

3.583

6.275

.663

.784

.851

.923

8

1.083

1.172

1.267

1.477

1.594

1.718

1.851

2.144

2.476

3.059

4.300

8.157

.630

.760

.834

.914

9

1.094

1.195

1.305

1.551

1.689

1.838

1.999

2.358

2.773

3.518

5.160

10.60

.599

.737

.817

.904

10

1.105

1.219

1.344

1.629

1.791

1.967

2.159

2.594

3.106

4.046

6.192

13.79

.569

.715

.801

.895

11

1.116

1.243

1.384

1.710

1.898

2.105

2.332

2.853

3.479

4.652

7.430

17.92

.540

.694

.785

.886

12

1.127

1.268

1.426

1.796

2.012

2.252

2.518

3.138

3.896

5.350

8.916

23.30

.513

.673

.769

.878

13

1.138

1.294

1.469

1.886

2.133

2.410

2.720

3.452

4.363

6.153

10.70

30.29

.488

.653

.754

.869

14

1.149

1.319

1.513

1.980

2.261

2.579

2.937

3.797

4.887

7.076

12.84

39.37

.463

.633

.739

.860

15

1.161

1.346

1.558

2.079

2.397

2.759

3.172

4.177

5.474

8.137

15.40

51.19

.440

.614

.724

.851

16

1.173

1.373

1.606

2.183

2.540

2.952

3.426

4.595

6.130

9.358

18.49

66.54

.397

.578

.695

.835

18

1.196

1.428

1.702

2.407

2.854

3.380

3.996

5.560

7.690

12.38

26.62

112.5

.358

.544

.668

.818

20

1.220

1.486

1.806

2.653

3.207

3.870

4.661

6.727

9.646

16.37

38.34

190.0

.323

.512

.641

.802

22

1.245

1.546

1.916

2.925

3.604

4.430

5.437

8.140

12.10

21.64

55.21

320.2

.277

.467

.603

.778

25

1.282

1.641

2.094

3.386

4.292

5.427

6.848

10.83

17.00

32.92

95.40

705.6

.215

.401

.545

.740

30

1.348

1.811

2.427

4.322

5.743

7.612

10.06

17.45

29.96

66.21

237.4

2620

.166

.344

.493

.703

35

1.417

2.000

2.814

5.516

7.686

10.68

14.79

28.10

52.80

133.2

590.7

9728

.129

.296

.446

.669

40

1.489

2.208

3.262

7.040

10.29

14.97

21.72

45.26

93.05

267.9

1470

36 k

.077

.218

.364

.605

50

1.645

2.692

4.384

11.47

18.42

29.46

46.90

117.4

323.7

1084

9100

498 k

The case of real rates

Just an example, in which:

* Inflation is 4%

* The return rate is 10%

The "real" return rate is

1,10 / 1,04 = 1,058 thus 5,8 %

separ

 This page last update: 04/06/13
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