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. 
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not withdrawn), its future value FV will be: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 %
This page last update:
04/06/13 
