Speaking of Leibniz

弁護士　永　島　賢　也
Kenya Nagashima, Attorney at law
2008/05/08 (first published: 2001/8/14)

Leibniz coefficient

When people hear the name Leibniz, they are likely to think of the mathematician (1646–1716) who discovered calculus.

When Leibniz is mentioned in the context of litigation for damage claims (particularly in litigations relating to automobile accidents), however, it nearly always means the Leibniz coefficient.

For example, one of the methods used to calculate the amount of damages in a case in which the plaintiff has partially lost the ability to work because of the aftereffects of a traffic accident is to calculate the present value of lost income by multiplying the income by a fixed proportion of working capacity lost, and then again by a Leibniz coefficient that corresponds to the period of disability.

For example, if a male salaried worker who was earning 5 million yen per year lost 35% of his working capacity due to aftereffects of his injury when his disability became permanent at age 50, the present value of lost income is calculated as:

5,000,000×0.35× 11.2740 = 19,729,500

The number 11.2740 in the above expression is the Leibniz coefficient, which corresponds to the duration of disability (17 years from age 50 to 67 in this case).

What current amount will be worth 5 million yen in a year’s time?

A Leibniz coefficient is used to calculate the interim interest to be deducted. Let us look at some examples.

How much money will one need now if the amount were to be worth exactly 5 million yen one year from now? Given that a lawsuit is assumed, the interest rate used in this calculation will be the statutory interest rate for civil affairs (5%), which is prescribed by Article 404 of the Civil Code, rather than the current rate on your bank deposit.

According to this calculation, if you have 4,761,904 yen right now, you will have about 5 million yen after one year at 5% interest. In other words, the present value of the 5 million yen one year from now is 4,761,904 yen.

4,761,904×1.05 ≒ 5,000,000

Likewise, the present value of a position that will earn 5 million yen in two years’ time is 4,535,147 yen.

4,535,147×1.05×1.05 ≒ 5,000,000

Likewise, the present value of a position that will earn 5 million yen in three years’ time is 4,319,187 yen.

4,319,187×1.05×1.05×1.05 ≒5,000,000

Then, the present value of a position that will earn 5 million yen per year for three years is 13,616,238 yen.

4,761,904 + 4,535,147 + 4,319,187 = 13,616,238

Where does the Leibniz coefficient come into play?

Where does the Leibniz coefficient come into play in these calculations?

Take the example of the first year, and look at the ratio of 4,761,904 yen, which will be worth 5 million yen in a year’s time, to 5 million yen. The question is: by what number do you need to multiply 5 million yen to get 4,761,904 yen?

The answer is 0.95238095.

(5,000,000×0.95238095 ≒ 4,761,904)

This 0.958095 is the Leibniz coefficient (one year, present value).

What, then, will the Leibniz coefficient be for the second year? Let’s look at the ratio of 4,535,147 yen, which will be worth 5 million yen in two years’ time, to 5 million yen. Again, the question is: by what number do you need to multiply 5 million yen to get 4,535,147 yen? The answer is 0.90702948.

5,000,000×0.90702948 ≒ 4,535,147

In the same manner, the Leibniz coefficient for the third year is calculated as 0.86383760.

5,000,000×0.86383760 ≒ 4,319,187

Then the present value of a position that earns 5 million yen per year for three years is calculated by simply summing the values for these years:

(5,000,000 0.95238095) +
(5,000,000 0.90702948) +
(5,000,000 0.86383760) ≒ 13,616,238

By taking the 5 million yen in this equation out of the parentheses, the expression becomes:

5,000,000 ×
(0.95238095 + 0.90702948 + 0.86383760)

If you sum all of the Leibniz coefficients in the parentheses (present values), you get 2.72324803.

That is, the present value of a position that will earn 5 million yen per year for three years can be calculated by multiplying 5 million yen by 2.72324803.

Present value and present value of annuity

The Leibniz coefficients are organized into a table of the present values of annuity in a format that sums the present values for two years, three years, four years, and so on. The coefficients in this table of present values of annuity are often used as the so-called “Leibniz coefficients.” In short, successive additions of the present Leibniz values produce the Leibniz table of the present values of annuity.

The Leibniz table of the present value of annuity shows 3 years = 2.72732. So you simply multiply the annual income (for example, 5 million yen) by this number. This saves you considerable calculation time.

That is, the present value of a position that will earn 5 million yen per year for three years can be calculated by multiplying 5 million yen by the Leibniz coefficient for 3 years (i.e., present value of annuity):

5,000,000×2.7232 = 13,616,000

Date when disability becomes permanent

Now, at what point in time should a calculation of present value be based?

It seems reasonable to use the time at which the aftereffects have become permanent disability (which will be indicated by a physician in his certificate) as the present time. In this case, it is reasonable to use the date on which the disability became permanent as the starting point for calculating the delay damages for aftereffects.

On the other hand, if delay damages are to be claimed from the date of the traffic accident, the present time would be the date of the accident. Of course, as the aftereffects will not be confirmed during the period from the date of the accident to the date the disability becomes permanent, it will be unreasonable to include an amount for this period.

In other words, if the date on which the disability becomes permanent is to be used as the basis for the calculation of the present value, the delay damages are incurred from that date, whereas to be consistent, the date to calculate the present value must be the date of the accident if delay damages from the date of the accident are to be claimed. To consider delay damages to be incurred before a permanent disability, which can only be recognized once the physical impediments become permanent, is absurd because they would earn interest retroactively.

If disability becomes permanent five years after a traffic accident

The actual calculation is simple. Returning to the case of the male salaried worker in the first example, he was 50 years old and earning 5 million yen a year when his disability became permanent. He lost 35% of his working capacity, and it took five years for the disability to become permanent.

5,000,000 0.35
{(Leibniz coefficient for 22 years) - (Leibniz coefficient for 5 years)}

In other words:

5,000,000×0.35 (13.1630 - 4.3294)
5,000,000×0.35×8.8336 = 15,458,800

Since the Leibniz coefficient (present value of an annuity) is a sum of the individual present values, the amount over the 17 years from six to 22 years after the accident can be calculated by subtracting the Leibniz coefficient for 5 years of the present values of annuity from that for 22 years of the present values of annuity.

If so, it will help clarify the arguments by having the court exercise the authority to ask for an explanation and ask the plaintiff (victim), who is claiming delay damages from the date of the wrongful act, whether the date of the present value of damages for physical impediments should be based on the date of the traffic accident or the date on which his/her disability became permanent.

Calculation of loss of income due to aftereffects

Incidentally, the calculation of income lost due to aftereffects varies according to specific facts. In an abstract sense, assuming the actual income prior to the accident, the proportion of working capacity lost is assessed (e.g., 100% for vision loss in both eyes, or 79% for loss of all toes on both feet) and lost income is calculated from the date on which the disability became permanent as the start date of the period of lost working capacity. The Leibniz method is used to calculate a deduction of interim interest during this period.

* Interest rate of 5% per year: There is criticism of calculating interest according to the 5% yearly rate prescribed in the Civil Code. This is because calculations assuming an interest of 5% a year are somewhat unrealistic in the present age of ultra-low interest rates.

* Fluctuation of annual income: The calculations assume that annual income remains constant. If annual income fluctuates from year to year, use of the Leibniz coefficient (present value of annuity) will produce errors. So calculations are more accurate if the annual amounts after deducting the annual interim interest are summed.

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