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In our analysis of the internal rate of return (IRR) method, we found that the central IRR rate is determined using the discounting method. Here, one looks for the interest rate that makes the balance of the discounted return flows (usually the return flows are positive) with the IRR investment expenditure (usually equity and premium flow out to the investor; are therefore negative) become zero. The IRR interest result now exactly matches the effective interest rate e of the capital commitment method (KBM) in our sample. Is that a coincidence or not? It’s not a coincidence. Therefore, prospectus issuers are not afraid to �cross� the IRR result with KBM’s interest rate statements in the prospectuses. The prospectuses and the audited statements on this point refer without exception to the fact that the IRR interest rate r and the KBM effective interest rate e are not comparable with fixed-interest securities such as federal bonds and so on. Weak reasoning: In the case of fixed-interest securities, the capital commitment is constant, whereas this is quite different in the case of capital investments in participations.1

In KBM, the effective interest rate e = r is the interest rate on the respective mathematical capital commitment or, in its second interpretation variant, the interest rate on the average mathematically committed total capital (DDGK). The emphasis is on the word �computational�. We’ll see why in the post.

The KBM originally comes from the securities area for calculating the effective interest rate. In the following article, we therefore present the KBM on the basis of a fictitious fixed-interest security with a nominal interest rate of 4 percent per annum with annual payments and a 10-year term, which is repaid at 100 percent at maturity.

How does KBM work mathematically? What does the KBM return on investment say as KBM return e? How useful is the KBM result for the investor’s investment decision?

A logical investor will ask, if the KBM interest rate was constructed for securities and is also applied to fund investments, why should it not be suitable for comparison with fixed-interest securities? It is used in both cases, isn’t it? Why is there a need to backpedal on the settlement? Can this really be justified by the different course of the �invoice� capital commitment? May the explicitly listed IRR method be calculated with the IRR determination equation, but explained with the vocabulary of KBM? Is such a transplantation of KBM statements into the classical IRR method permitted and what liability issues does it trigger?

The all-rounders among the financial service providers have been selling fund participations of different providers and different types of funds for years, such as real estate funds from Germany and abroad, ship participations, energy funds, film production funds, leasing funds, games production funds, etc. and recently also funds with second-hand life insurance policies in different currencies, as well as private equity funds with the active support of rating agencies, which almost without exception use the IRR method.

Again and again, one encounters prospectus offers and matching ratings of well-known analysis houses that weight the yield statements differently and visibly and verbally mix IRR and KBM statements. From a liability point of view, not only should rating agencies and banks be aware of the differences, but they should also take note of the fact that incorrect statements on returns trigger liability claims.

In many cases, brochures on yield forecasts state, among other things, that the total return on the investment can be easily calculated with the help of the capital commitment method. All that is needed is to multiply the �calculated� average capital commitment by the KBM interest rate e (effective interest rate) and multiply this value by the term of the investment to obtain the total return. �If an investor wants to answer the simple question of the absolute total return from his capital investment, then he only has to interest the average imputed tied-up capital with the internal rate of return and then multiply it by the term.�2

Above all, one wonders why this information is necessary, since one can read off the total return almost directly if one adds up all returns or distributions and subtracts the capital investment?3 In principle, one does not need complicated formulas for this.

How do you arrive at the interest rate e, the average �calculative� capital commitment (DDGK) and what does the KBM method say?

In the securities example from Table 1, fixed-interest securities with a nominal interest rate of 4 per cent per annum are purchased for EUR 100,000 on 1.1.2005, the interest on which is paid on 31.12. each year. subsequently paid out to the buyer or investor. At the end of the investment, in addition to the interest for the past year, he also receives 100 percent of the entire capital investment back. In total, the recoveries amount to 140,000 euros. If the investor deducts his invested 100,000 euros, then he has achieved total earnings of 40,000 euros in a state without taxes. This total income is also called payment surplus or liquidity surplus.

But what does the total income or liquidity surplus or payment surplus have to do with the interest rate, or what interest rate at what basis is this based on?

The capital commitment should explain this. The simple capital commitment shows in Table 1 that the investor’s capital investment is reduced annually by the nominal interest of 4,000 euros flowing back and that the simple capital commitment falls steadily. In the penultimate year, there are still – 64,000 euros tied up, but due to the last interest payment and the repayment at 100 percent of the capital invested, the total of 104,000 euros jumps to the payment surplus of 40,000 euros.

The simple capital commitment decreases, but not the mathematical capital commitment in the 4th column of Table 1, which remains the same until the end of the investment. To prove this, the respective �integer� capital commitment is determined by compounding the capital input (as the first capital commitment) of -100,000 euros with the �search interest rate� e = 4 percent per annum and deducting the nominal interest of 4 percent per annum at the end of this interest period. The respective capital commitment gained in this way (= respective account balance of the fictitious account) is again compounded at 4 percent per annum (interest is added to the arithmetical KB of the previous year) and 4 percent nominal interest is deducted as inflow. Thus, �Step by Step� one step forward and one step back is rowed and the respective �calculated� capital commitment does not change in this fixed-interest security.

The fictitious account balance (respective capital commitment) in the year before last is also compounded with the KBM interest rate e and balanced with the sum of the last nominal interest and the 100 percent repayment. The account balance or the calculated, fictitious capital commitment is always zero at the end of the investment period in KBM.

Thus, the effective interest rate e is the interest rate of the respective mathematical capital commitment. Since the respective imputed capital commitment has not changed in the fixed-interest example, it is also explained that the average imputed capital commitment does not shift. So e can also be interpreted as the rate of return on the average notional capital commitment (DDGK) over the life of the loan.

Finding this result is quite cumbersome, when there is a simpler way to arrive at the liquidity surplus. This applies to the simple case in which 100,000 euros are invested and also repaid and the returns remain absolutely constant.

The payment or liquidity surplus or �total return� can be easily determined by adding up all investment values. What message does this value send to the investor or intermediary or other user of KBM? At first, it just shows that the investment seems to be worth it because more is flowing out than is being put in. This is because the total return is 140,000 euros over the investment period and -100,000 euros is the capital investment. This makes a total of 40,000 euros in income or liquidity or payment surplus.

If one asks what interest rate is behind this and what the interest rate refers to (does it refer to 100,000 euros of capital investment?), one arrives at the following results? If it refers to the 100,000 euro capital investment, then the investor, with the calculated interest of 4 per cent per annum in 10 years investment period would have to expect approximately 148,024.42 euro (without taxes), but the calculation in table 1 in column 1 speaks only of returns in the order of 140,000 euro?

KBM’s calculations actually look different. The simple capital commitment in Table 1 column 2 already shows a decrease in the actual capital commitment without interest. If the effective interest rate e (compounding of the respective capital commitment with e) comes into play, then the picture changes not inconsiderably, because the interest rate e is sought which brings the respective mathematical capital commitment at the end of the investment to the value zero (the respective KBM account balance corresponds to the respective mathematical capital commitment). In the case of Table 1, the interest rate is e = 4 per cent per annum, which again corresponds exactly to the internal rate of return = 4.0 per cent per annum. It is further noticeable that in Table 1, the 2nd column and the 4th column differ only by the KBM effective interest volume. With e, the interest rate is found that corresponds to the notional KBM (effective) interest volume at which the sum of the liquidity surplus and the cumulated KBM effective interest volume becomes exactly zero. This also means that the last mathematical capital commitment (last KBM account balance) in each case is zero, as shown in column 4.

The respective arithmetical KB – sometimes also called imputed, pagatoric or dynamic capital commitment because of the compounding with (1+e) – accrues interest at the interest rate e.

In fact, the iteration procedure4 looks for the interest rate in the KBM that generates a (negative) KBM (effective) interest volume, which makes the balance of the calculated KBM interest volume and the liquidity surplus end up at zero. The effective interest rate e of KBM corresponds exactly to the internal rate of return r of the IRR method in the security example.

We can verify these relationships in the sample in Table 2.

If you now think that the capital investment of 105,000 euros earns interest at e = 10.14 percent, you are mistaken. Instead, KBM’s average mathematical capital commitment (sometimes also called average dynamic total capital = DDGK) of 103,589 euros earns interest at an average rate of 10.14 percent per annum over the term of 10 years; however, the interest is not calculated according to the Leibniz equation, which would be:

Kn = 103.589 �*(1+0.101362)10 = 272.029 �

Unfortunately, an actual capital investment of -105,000 euros and an interest rate of 10.1362 percent per annum would have resulted in 275,734 euros as the final value, while an equally high �interest� on the fictitious DDGK of(-)103,589 euros would only result in �272,029 euros�.

Kn = 105,000 �*(1+0.101362)10 = 275,734 �

How is one supposed to explain to an investor with KBM how his capital investment of -105,000 euros earns interest, when the fictitious DDGK of (-)103,589 euros does not correspond at all to the actual capital investment? The DDGK is ultimately a notional value that is multiplied by the interest rate e to obtain the total return in the formula:

Total yield = DDGK * e * term

and reflects the effective interest volume on the right-hand side of the equation and the calculated KBM account balance in the following relationship

achieved:

Total yield – [ DDGK * e * term ] = 0

Or rather

Simple KB +[-Effective interest volume]=0

Interest rate and duration are only linked linearly (average interest rate times duration), whereas in compound interest the relationships between interest and time factors are always linked non-linearly in the Leibniz formula:

Kn = K0*(1+i)n.

How should investors now be shown the return on their capital investment with the help of KBM? Should we tell the investor that the fictitious DDGK of – 103,589 euros earns interest at 10.14 percent per annum, while the investor has actually invested exactly -105,000 euros? Conversely, wouldn’t the investor be right if he only paid in 103,589 euros because he only receives the interest rate of 10.14 percent on this value? After all, the investor does not want to know how a fictitious mathematical capital commitment �verzins�, but which interest rate his original capital investment is subject to? This is not identical with the DDGK. That is absolutely certain! It is unlikely to be mere coincidence if there is a match between the capital invested and the fictitious DDGK.

KBM is looking for the effective interest volume that compensates 100 percent of the payment surplus and leads the notional mathematical capital commitment or the notional KBM account balance at the end of the investment to zero.

Liquidity surplus +[negatives Effektivzinsvolumen] = 0

For this to succeed, the effective interest rate e must generate a (negative) interest volume5 on the respective mathematical capital commitment. In effect, the investor does not receive the notional �KBM reinvestment interest� (WAP) from the investment series. The KBM approach merely simulates a fictitious reinvestment that results in the same interest rate as in the IRR. Accordingly, KBM’s inherent reinvestment assumption [Verzinsung der Vorjahres-KB mit e)] makes it equally unsuitable for the investor who only wants to consume the returns. The strategic reinvestor would also have to meet the tough reinvestment conditions, as in IRR. This can also be illustrated with the fictitious, calculated KBM account history.

The KBM effective interest rate e demonstrably does not indicate the interest on the capital input, but the �Effective �interest on a fictitious value (=DDGK). Our adequate answer to the investor’s question about the return on his capital investment in the sample example would be that he can expect a return on his current capital investment of, for example, only 7.18 percent per annum. KBM does not have this answer. It defers to interest on the notional DDGK.

With the help of the formula for the compound interest calculation of the universal scholar Gottfried W. Leibniz and a pocket calculator, the investor could calculate that, with a capital investment of -105,000 euros, after 10 years he would have a credit balance of 210,000 euros at 7.18 percent interest, in a state without taxes on the interest and without costs for reinvestment. He’s a long way from that. But this fact is not apparent from the KBM. There, only the average calculated interest on a notional value (DDGK) is mentioned. This DDGK varies from participation to participation in different amounts for different forms of participation. Thus, KBM is ultimately useless for measuring returns in theory and practice, because it does not measure the return on capital employed, but the average return (interest per period) of a fictitious quantity.

Another question remains. If IRR and KBM lead to the same return result of 10.14 percent per annum, surely KBM should also logically generate �additional interest � as is the case with IRR?

To find the answer, we look in Table 12 in the last column. All interest from KBt-1*e is cumulated here. It generates �KBM additional interest� of – 105,000 euros6. The negative sign does not bother us because we have applied e to the respective (negative) capital commitment7, according to convention. KB0 starts with a negative value from the investor’s point of view. This is logical because we set the first capital commitment at the outflow of the first equity installment. It is conclusive according to the convention according to which everything that flows out at the investor is printed with a negative sign and everything that flows in (returns) is printed with a positive sign.

The interest on the previous year’s account balance [KBt-1*e], not otherwise the respective KBt = [ KBt-1*(1+e)] is determined, establishes the �reinvestment� beyond the inflow of distributions and leads to the same result e as the IRRreturn rater in the IRR method via the step-by-step interest calculation.

The review of the KBM effective interest e for a domestic real estate fund, ship fund, film leasing fund, (cash contribution = 54,900 euros) life insurance fund with US policies and a private equity fund of funds investment with 105,000 equity sums in each case (distributed over time, among other things) including a 5 percent premium and commencing on 1.6.2005 yields the following unsurprising result: In all investment examples (with the exception of the FWP), the DDGK deviates to a greater or lesser extent from the capital contribution.

The product from

DDGK x e x Duration

represents the effective interest volume (cumulative effective interest per period) over the entire term or investment period. Therefore, Table 4 illustrates the notional KBM final account balance:

At this point, at the latest, every user must notice that a fictitious effective interest (reinvestment) volume is searched for and found in KBM, the fictitious interest of which the investor does not receive. He receives only the simple liquidity surplus without notional �effective� interest. In the IRR method, he also does not receive the reinvestment interest (additional interest from r), which is automatically allocated to him for methodological reasons. While the KBM target determination concludes with the KBM final account balance (final value concept) = 0, it succeeds in the IRR, in the initial value concept, in which the balance of all present values achieves the value zero at the start of the investment. The method analysis presented shows that both methods calculate with notional reinvestment interest rates, although the original investment series does not provide for this, but only show the same �return ratios� through the WAP. But there is one more surprise. As a variant of the KBM method (positive capital commitments are compounded with the MISF interest rate and simultaneously negative capital commitments are compounded with the exogenous MISF reinvestment interest rate), the MISF return8 shows the same return result as in the IRR and KBM methods if the MISF reinvestment takes place at the IRR interest rate. The MISF interest rate then equals the IRR interest rate and the KBM result. Knowing these facts while maintaining the two respectively three methods with calculation and interpretation mix in the prospectuses, the use is either fraud or at best incorrect advantageous statements according to § 264a StGB. Whether intent, gross negligence or slight negligence is involved is a matter for the judiciary to decide after examining the facts of the case.9

Short content of KBM: KBM does not solve the yield representation and interpretation either, because it accrues interest on negative capital commitments with only one unit interest rate e in the same way as �positive capital commitments�. It says nothing about the return on the investor’s capital investment, but only about the return on a fictitious mathematical variable, the DDGK. Often, simple capital commitment and calculated capital commitment (average or respective KB) are mistakenly considered to be identical. This creates hopeless confusion of language in the prospectus statements. For investment structures with changing signs, the KBM is just as unsuitable as the IRR because it treats positive and negative investment values with the same KBM interest rate (compounding). It remains completely incomprehensible that in the offer prospectuses and WP audit reports and rating statements the IRR yield is defined mathematically as the interest rate that compresses the balance of the present values of all investment values to the value zero and is interpreted with the vocabulary of the KBM statement e = r as the interest rate that shows the return on the average notional capital commitment or the respective notional capital commitment as a notional measure of the interest rate. This is where the inadmissible mixing (to be precise: the transplantation of the KBM interest rate statement into the financial mathematical IRR instruments) of two yield method results or their economic interpretations takes place. In both (with MISF in all three) return statements, the rate of return does not refer to the investor’s capital investment. However, the investor is interested in the return on his capital (investment) until the end of the investment, comparable to the return on a savings account. The IRR and KBM do not provide a clear answer to this question, accurate information. In the IRR, the rate of return r is the rate of return on the IRR investment expenditure (= present value of all negative investment values, constructed from the endogenous discount rate r), if all returns are reinvested at the IRR interest rate = r or additional payments are financed. In the KBM, the DDGK is supposed to interest with e. But the DDGK is not identical to the capital employed at the beginning of the investment and the simple capital commitment during the investment period and constantly changes its �dress� as the individual investment values change. A fictitious yardstick (=DDGK) that is constantly changing is useless as a yardstick for measuring returns, as is the IRR investment expenditure, which is constantly changing due to endogenous influence, as a fictitious capital investment.

Dr. Johannes Fiala

Edmund J. Ranosch

(Credit and Rating Practice 01/2006, 1-6)

Courtesy ofwww.krp.ch.

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## About the author

Dr. Johannes Fiala has been working for more than 25 years as a lawyer and attorney with his own law firm in Munich. He is intensively involved in real estate, financial law, tax and insurance law. The numerous stages of his professional career enable him to provide his clients with comprehensive advice and to act as a lawyer in the event of disputes.

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