Compound rate of return
and present value

## Financial return and financial value

For savers, investors, borrowers, "compound rates" are used to
calculate

* the annual return or annual interest rate
* what an investment can be worth after several years ("future value")
* and also the "present value" of an asset or a future cash flow,

by applying to future incomes a discount rate of return that is
considered necessary.

Snowball effect in    financial returns

(Extract from the

stock valuation and behavioral finance site)

## 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:
incomes, rents, interests, dividends, capital gains as positive
differences between resale values (**) and buying / subscription
prices
• and Negative cash flows: expenses (fees, taxes...), capital loss (*).
Well, a full calculation would also include

* the monetary investment itself as a negative cash flow
* its recuperation at the end as a positive one.

(*) 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

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 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

Calculations that use compound rate allow to know

* 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.
It is what an investment, a cash flow, an asset (and also a
debt) will be worth after several years .
• Or give the present value PV, aka discounted value
of an asset or a future cash flow

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

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

(or, in other areas than finance, any annual increase / decrease:
•  economic growth, inflation...)

Y = the number of years

FV = the future value at the end of the period

• TR = the 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
(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
are
theoretically worth now (PV / present value, aka discounted
value):

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

• Chains of annual cash flows
and use of probabilities

Getting 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
.

=>
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 place
to obtain its
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