A common way of expressing the desirability or otherwise of an investment is through a rate of return. There are several ways in which ‘rate of return’ can be calculated. Comparisons between numbers are only meaningful if the processes producing the numbers are clearly and unambiguously understood. This is where J can be of great help in sorting pathways through a terminological jungle.

Internal Rate of Return (IRR) is a commonly used measure. To explain how it is obtained, consider that say 5% growth in any commodity can be calculated by applying a multiplier 1.05, and compound growth by successive applications of this multiplier. In a similar way a list of monetary values can be adjusted for 5% inflation by successive divisions by this multiplier. The opposite of 5% growth is 5% decay (that is growth of -5%), and successive multiplications of 0.95 result in exponential decay. However the inverse of growth is not exponential decay, but rather the process determined by the multiplier (1.05)<sup>-1</sup>=0.9524, meaning that a post-growth value is 95.24% of its original. In the case of decay of, say 20%, the corresponding inverse multiplier is (0.8)<sup>-1</sup>= 1.25 corresponding to 25% . For example under a 20% tax regime, a pre-tax value is obtained from a net value by adding 25%, a process frequently referred to as ‘grossing up’.

The above discussion is consolidated in the following two verbs:

   ptom=.%@>:@(0.01&*)   NB. %age growth to inverse multiplier
   mtop=.100&*@<:@%  NB. inverse multiplier to %age growth
   ptom 5 _20
0.952381 1.25
   mtop 0.952381 1.25
4.99999 _20

In everyday terms this says that under an assumption of 5% inflation a rational person would be indifferent regarding offers of £95.24 today and a promise of £100 next year. £95.24 is said to be the net present value (often abbreviated to NPV) of £100 delayed until next year.

An income stream is a list such as

    is=._100 20 50 70 80

which describes, say, an initial investment of 100 which produces subsequent returns of 20, 50, 70 and 80. The verb npv provides compounded net present values for a decay rate (discount) given by the left argument:

   npv=.]*(ptom@[)^(i.@#@]) NB. net present value
   5 npv is             NB. 5% decay
_100 19.0476 45.3515 60.4686 65.8162
   10 npv is            NB. 10% decay
_100 18.1818 41.3223 52.592 54.6411

The total of such lists is called the net benefit, and the absolute sum of the negative items is the cost. The net benefit of is before discounting is thus 120, and the cost 100. After discounting by 5% and 10% the net benefits reduce to

   +/"(1) 5 10 npv each <is
90.6839 66.7372

Because exponential decay is not the inverse of growth the NPV discounted streams of is are different from the result of repeated compound reduction:

_100 19 45.125 60.0162 65.1605
_100 18 40.5 51.03 52.488

Internal Rate of Return answers the question what NPV discount rate would reduce net benefit to zero. In algebraic terms, this amounts to finding a positive real-valued multiplier which satisfies the equation

0 = 100 + 20m +{50m}^{2}+{70m}^{3}+{80m}^{4}

This can be obtained using the root finding verb p. which solves polynomials whose argument is coefficients in ascending power order. In the verb irr this is followed by a conversion to a percentage:

   irr=.mtop@pos@real@roots NB. internal rate of return
      roots=.>@{:@p.        NB. list of roots from result of p.
      real=.#~ (= +)        NB. select real values
      pos=.#~ >&0           NB. select positive values
   irr is
_100 15.1158 28.5608 30.2203 26.1031
   +/_100 15.1158 28.5608 30.2203 26.1031
0                           NB. confirms net benefit is zero

In practical terms this means that under an inflation rate of 32.3% the underlying project would finish with no residual financial worth. (Practical note: If irr fails through a limit error in p. use a standard root finding technique such as the following:

   Newton=.adverb :']-x%x D.1'(^:_)("0)  NB. Newton-Raphson
   poly=.|.is                            NB. define a polynomial
   mtop (#.&poly) Newton 0.9
32.3121                                                           )

Applying NPV before computing irr allows prior assumptions to be made about inflation, e.g.

   irr 10 npv is  NB. IRR assuming 10% inflation
   irr 25 npv is  NB. IRR assuming 25% inflation

For the reason given earlier these values are not identical with the irr values which arise on an a priori exponential decay assumption:

   irr is*0.9^i.5       NB. assuming 10% exponential decay
   irr is*0.75^i.5      NB. assuming 25% exponential decay 

If all the inflows are delayed until the final period, the irr value is simply the rate of compound interest which for is would turn 100 into 220 in the course of four periods:

   irr _100 0 0 0 220

Average Growth Rate

A reasonable question to ask is why bother with the relative complexity of IRR when a rate of return can be found by simply calculating (inflows–outflows)÷outflows and averaging over the number of periods. To address this, the numerator (inflows–outflows) is just another way of expressing net benefit, while the positive sum of the outflows are given by the verb costs. This is illustrated with two input streams which differ only in the order of their items:

   costs=./@:(0&>.@-)   NB. outflows (i.e. sum of -ve values)

   is1=._100 20 45 60 _10 55 70 20 10 10
   is2=._100 20 45 60 _10 10 10 20 70 55
   costs each is1;is2
110 110

The benefit-cost ratios associated with these streams are

   (+/ % costs)each is1;is2
1.636 1.636

However the stream in which high returns come later has a lower IRR:

   ,irr each is1;is2
30.19 25.22

showing that averaging loses some of the information implicit in ordering.

Where the term ‘benefit-cost ratio’ is used it is worth checking that the numerator reflects net benefits – effectively that is it is inherently a profit ratio. If only inflows appear in the numerator it is a ‘selling price/cost price’ ratio and thus related to mark-up rather than return. The verb bcr first incorporates net present value discounting into the averaging process with the discount level set as the left argument:

   bcr=.(+/%costs)@:npv     NB. benefit/cost ratio with discounting
   10 bcr each is1;is2
0.80926 0.67133

Since the lengths of is1 and is2 are both 10 these ratios can be converted to an average compound growth rate by taking the 9th root of 1.80926 1.67133

   9%:1+0.80926 0.67133
1.0681 1.0587

that is the average growth rates are roughly 7% and 6% given a discount rate of 10%. This procedure can be generalised as

   ktop=.(100&*@<:)      NB. transform (1+k) to 100k%
   10 agr each is1;is2
6.81 5.873

If discounting is applied at around the irr value then the net average growth rate is more or less zero:

   30 agr is1
0.0527249           NB. growth rate is virtually zero (0.05%)

Using IRR

Where values are money, choices between alternative projects can be guided by IRRs. IRRs can be recalculated dynamically, although they should not be used as a sole criterion since, provided inflows are positive, IRRs carry on increasing as the input stream lengthens, and so increases in the IRR tailing off can be a signal suggesting that other statistics should be investigated as well. In the case of is successive income streams are given by

|_100 20|_100 20 50|_100 20 50 70|_100 20 50 70 80|

and successive IRRs by

   ,irr each }.<\is
_80 _18.5857 15.6152 32.3121

leading to

   irrseq=.,@:(irr each)@}.@(<\)
   irrseq is
_80 _18.5857 15.6152 32.3121

Post-initial negative items in an income stream as in is1 might reflect costs incurred in developing a second version of a product. Since polynomials of even order must have an even number of positive roots, using p. means that spurious values of IRR will necessarily arise, as for example in

   irr _100 20 45 60 _10
_85.04 6.67385

where the massive compound 85% decay is clearly inadmissible. To deal with such circumstances it makes sense to insert a filter at the mtop level to exclude multipliers of greater than, say 3, or, at the other end of the scale, less than 0.5, hence

   filter=.#~ (<&3)*.(>&0.5)
   irrseq is1
0 _22.18 10.22 6.674 19.96 27.96 29.33 29.82 30.19

The initial value of _80 following the first positive flow has been filtered to 0. Following the third inflow of 60 the IRR becomes positive at 10.2%, following which there is a dip to 6.67 due to the negative inflow of 10. Towards the end of the series subsequent increases in IRR tail off as a result of smaller inflows, a signal that benefit is tailing off.

Annuities and Personal Finance

An annuity is a special case of an input stream where the inflows are regular. Annuities in general are a large field within actuarial science, although in the simplest case the calculation has a closed form in which the multiplier <span class="nowrap">{1-(1+p)<sup>-n</sup>}/p</span> converts repayments as a percentage of the initial cost into a multiple of present value assuming repayments continue for n years. In May 2014 this factor for a single whole-of-life level annuity at age 65 was around 17. Compare this with an annuity paid for 21 periods at 2.6% whose multiplier is given by

   ptok=.>:@*&0.01           NB. convert %age to (1+p/100)
   am=.4 :'100*(1-(ptok x)^-y)%x'   NB. annuity multiplier
   2.6 am 21                NB  multiplier for ..
16.026                  NB. .. 21 repayments at 2.6%
   irr _16.026,21#1         NB. confirmed by IRR

Apply a 5% inflation rate and the annuity doesn’t look such a great bargain!

   irr _16.026,5 npv 21#1

In the case of personal investment, IRR can give a rough guide to the true worth of an investment on completion. Consider a share purchase for £33.50 which attracts nine annual dividends of £2, £2, £2, £2.50, £3, £4, £4.45, £5.25 and £4.55, after the last of which it is sold a year later for £65.88.

   is3=._33.5 2 2 2 2.5 3 4 4.45 5.25 4.55 65.88
   irr is3

This indicates that over the period the investor has experienced a discount rate of about 13.5% but has no final asset – the inflows account for all benefit. However the average growth rates for 0 and 5% inflation are:

   0 5 agr each <is3
11.06 6.508

indicating that under 5% inflation is3 has an average profit-growth based return of 6.5%.

There are streams for which growth can be consistently negative as with

   is4=._57.4 1.5 1.5 1.6 1.7 1.8 1.8 1.95 2 27.9
_4.051 _7.289

Nevertheless because only the first item is negative, IRR is still an increasing quantity:

   ;irrseq is4
_65.84 _52.32 _41.93 _34.25 _28.03 _23.22 _4.051

Future Value Analysis

Future Value Analysis (FVA) is a variation of net present value which, for an income stream with a single initial outflow, expresses benefits valued at today’s money:

   ratio=.{:@] % {:@npv     NB. find the scale-up factor which ...
   nfv=.ratio*]@npv         NB. ...leaves final npv value unchanged
   5 nfv is
_121.6 23.15 55.13 73.5 80

Since ratios of successive terms are unchanged there is no change either to IRRs:

   ,irr each (5 nfv is);5 npv is
26.01 26.01

For is and is3 the net benefits valued at the end of the investment period assuming 5% inflation are

   (+/5 nfv is),(+/5 nfv is3)
110.2 47.94

to be compared with values at time of investment of +/each is;is3 which are 120 and 62.13 respectively. For is3, the 47.94 is made up of 36.63 in dividends and 11.31 profit on sale, all at today’s values, representing a gain of 47.94/54.57 or 87.85%. The geometrical average of this is 9 %:1.8785 which equals 7.3%. Three values for growth of this series have been obtained, 13.6% from irr, 6.5% from agr, and now 7.3% from nfv. The need to understand measurement criteria should be clear!

In conclusion, the techniques described here barely begin to scrape the surface both of practical detail and of jargon. Legitimate further questions might concern how the initial outflow was acquired – if by borrowing this leads to further necessary refinement of the irr process. The important message is the usefulness of J in penetrating terminology and in understanding precisely the meanings of quantities which may be subtler than their names suggest.