**What does an investor's return include?**

** Milking the cow but feeding it also.
**

An investor return includes **all the "cash flows"** (*) he/she gets from

its investment (whether an asset, a business project, even a professional

course...) :

**Positive**cash flows:

differences between resale values (**) and buying / subscription

prices

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

* its recuperation at the end as a positive one.

(*) Past cash flows when calculating historical returns, or expected

cash

(*) Past cash flows when calculating historical returns, or expected

cash

*flows*

*when addressing*

*a present investment.*

(**) Based on the past, actual or expected resale transaction.

If there is a market for the asset, its current price shows a possible

(**) Based on the past, actual or expected resale transaction.

If there is a market for the asset, its current price shows a possible

*resale*

*value.*

Normally, * the more risky the investment is * (and the more risk averse investors are), , * the higher should the rate of return be so as to convince investors to put money in the financial pot. |

In other words, the rate of return used in the calculation should include

a

**risk premium**adapted to the type of investment

A similar calculation can be applied to the financial

weight of a loan on the borrower.

(the loan received itself is a positive cash flow; refunds, interests

and other costs are negative ones).

**Compound rate**

** "Snowball arithmetic" under "geometric progression".**

**Purpose**

* the past or future return of assets / projects / investments over

several years,

* and the present financial value of those assets / projects / investments.

Compound rate calculations are used notably

- in

**finance, investing / borrowing**(rates of return, interest rates)

- and by extension in economics (growth rates, inflation effects).

For savers, investors, entrepreneurs, borrowers, compound rates are

used to:

- Find out:
- the
**rate of return**(*) of an investment, a business project - or the real
**financial cost**("actuarial interest rate) of a debt

- Or find the
**future value FV**.

debt) will be worth after several years .

- Or give the
**present value PV**, aka discounted value

*(*) to compute an expected future return (and a present value),*

to make cash flow previsions is needed.

to make cash flow previsions is needed.

**Definition / calculation**

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

- An
**annual return**(*) R - which,
**compounded**(**)**year after year**, - brings a
**total return**TR at the end of the period.

**(*) Extra value**after one year, in percentage, compared to the initial

value

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

Every year the asset value is increased ("capitalized") by the annual

return.

In the next year

**the return rate is applied to that increased**

value.

value.

For 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...

*(the table below shows that "snowballing effect / hyperbolic
progression")*

**Future value FV**

**Parameters**

R= the annual "compounded" rate of return

Y= the number of years

FV= the future value at the end of the period

TR= the total return of that period

**Example**

**future value FV**

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 % |

*See also the table below*

**Present value PV**

Such asset valuation is done by **"discounting"**the future expected

asset worth (or its expected cash flows) with a return rate that is

considered necessary when investing money.

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

aretheoretically worth now (

**PV / present value,**aka discounted

value):

10,000 / 1.262 = 7,924 Euros ( |

**Chains of annual cash flows**

**and use of probabilities**

Getting fresh eggs day after day,

and the hen for your soup at the endGetting fresh eggs day after day,

and the hen for your soup at the end

Most financial assets bring not only a

**final worth**at the end of a

period (for example at maturity) but also a

**chain of cash flows**

year after year.

year after year

**The calculations must take that into account, for example to**

=>

=>

determine the PV of the asset

In that case,

**every cash flow is discounted according to the**

number of years in which it will take placeto obtain its

number of years in which it will take place

present value.

**Then all those PV are**

=>

=>

**added together**to give the asset PV.

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

**=>**The asset PV is then the

**sum of the PV**

**of every scenario**

**multiplied**

**by its**

**probability coefficient**.

To mention what the extreme scenarios would bring is useful also

to make decisions

**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 |

**Annex: calculation of real (*) rates**

**(*) = inflation adjusted**- The inflation rate is 4%
- The return rate is 10%

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