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not withdrawn), its future value FV will be: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 %