# Johan de Witt - The first calculation on the valuation of life annuities

### The Waerdye

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The "Waerdye van Lijf-renten naer proportie van Los-renten", is a treatise that attempts to price life annuities respective of their true value. Life annuities were awarded against an upfront payment - commonly 12mio guilders - to the buyer. The buyer, or a person appointed by him or her, afterwards received a periodical payment of usually 1mio guilders a year. The payments ceased when the "life" on which the annuity was written died. Since buyer, annuitant and life could be a single entity, or divided into parties of one or two, an annuity contract could involve multiple people. Generally, the life was chosen as a young child that was expected to have a longer life span. The annuity's price was not age-dependent on the choice of life (Stamhuis, 1999). Redeemable annuities were a similar instrument, but the liable party had the option to redeem the annuity for a certain sum.

Despite its mathematical added value, the "Waerdye" must be seen and understood in the context of its publication. De Witt presented it as a piece of supportive reasoning for a political argument, which explains the unique format of the paper. The treatise was written for an audience of mathematically inexperienced politicians and hence worded in a very accessible way. De Witt's objective was to convince the ministers of the favourable nature of life annuities over other forms of state financing - hence, the paper is decidedly positive and does not mention potential disadvantages of this instrument. Structurally, de Witt combined an explanatory approach based on calculated examples with a structure that is inspired by Christiaan Huygen's "van rekeningh in spelen van geluck" (1660). De Witt used presuppositions, propositions, and corollaries to structure his paper; Huygen used "voorstellen", or propositions (Stamhuis, 1999). With regards to content, he structured the paper as follows.

In the preface, the intention and motivation for the treatise are stated, namely a proof for the "well-known fact" (de Witt, 1671) that life annuities are a better and more lucrative means of financing than redeemable annuities. He presents several advantages of life annuities over redeemable annuities: on the one hand, annuities are beneficial to the government since they terminate unfailingly, whereas redeemable annuities and bonds have a more perpetual and insecure character. On the other hand, they are also advantageous to the buyer, since the rate of life annuities trading at the time rendered a higher pay-out than a comparable redeemable annuity at 4%. These arguments he presents in his introductory paragraph:

Noble and Mighty Lords,

In so extensive an administration as that of the united country of Holland and West Friesland, it is better, as I have several times stated to your Lordships, for several reasons perfectly well known to you, to negotiate funds by life annuities, which from their nature are infallibly terminable, than to obtain them at interest, which is perpetual, or by redeemable annuities; and that it is likewise more useful for private families, who understand the economy well, and know how to make a good employment of their surplus in augmenting their capital, to improve their money by life annuities, than to invest it in redeemable annuities or, at interest at the rate of 4 per cent per annum; because the above mentioned life annuities, which are sold even at the present time at 14 years' purchase, pay, in fact, much more in proportion than redeemable annuities at 25 years' purchase. I have consequently respectfully to submit too your Lordships the unchallengeable proof of my assertion. [...]. (...) in presupposing that the redeemable annuity is and will be current at 25 years' purchase, or at the rate of 4 per cent per annum, we must find at how many years' purchase the life annuity should be sold, to be in proportion to the aforesaid redeemable annuity, in such manner that the life annuity may, if not with mathematical precision, at least in its discovered value, be more advantageous to the purchaser than an annuity with the same capital."(Hendricks, 1852)

The three presuppositions that follow elaborate on the two components he used for his calculation: expectation value and mortality rates. The "Waerdye" is considered the first manuscript that explores probability calculations on a public finance problem, which emphasises the significance of the treatise (van Brakel, 1976).

In the first presupposition, de Witt explains what he means by an equal contract - a notion he may have obtained from his law background -, namely a contract with equal probabilities for each outcome. The second and third presuppositions specify his assumptions about mortality rates. He forms four groups of people that he assigns absolute survival probabilities to - a group from age (3, 53), a group aged (53, 63), a group aged (63, 73), and finally a group aged (73, 80). Hence, he assumes that the average person that is written on an annuity lives up to age 80. To each of these groups, de Witt assigns an absolute probability of survival, in relative

proportions1: 3/2 : 2 : 3. These values seem to be arbitrary, but in a supplement to the treatise de Witt declares that "I have very carefully extracted from the registers of your Lordships some thousands of cases of persons upon whose lives annuities have been purchased" (Hendricks, 1852), which suggests that he might have approximated the mortality rates from real data. It should be noted that these probabilities do not specify chance of survival at year n given that the person has survived up until year n - 1; rather, de Witt assigns equal probabilities to any two events in the same group. De Witt is making the simplified assumption that a person has the exact same probability to survive any one year in the specified period; for instance, an individual is as likely to survive another year at age 4 as he is at age 40. Since life annuities were paid out in half-yearly time intervals, he proceeds to use the half-years (1,99), (100, 119), (120,139), and (140, 153) for his calculations. The relative proportions 1 : 3/2 : 2 : 3 translate in terms of mortality chances. For example, a person in age category (63,73) has a chance of dying the coming half-year twice as high as a person in age category (3,53), a person in category (73, 80) three times as high.

To calculate the value of the annuity, de Witt assumes an annuity payment of 1mio guilders, or equivalently 20mio stuivers yearly. He sums up the present values of the annuity, provided the life survives half-year i. For half-year 1 for instance, the value of the annuity at present is calculated as

present value of the annuity at half-year 1 = $10,000,000/((1+0.4) ^{0.5}) = 9,805,806.76$

and for half-year 2 we have $10,000,000 / 1.04 = 9,615,384.62$, giving a value of
$9,805,806.76 + 9,615,384.62 = 19421191.38$

for the annuity if the life survives the first year. It should be observed that de Witt is calculating for a half-yearly payoff of 10,000,000 stuivers, hence amounting to a yearly income of 20,000,000 stuivers. This results in the denominator being taken to the power of n/2, since n denotes a unit of a whole year. Present values can be calculated for each of the 153 half- years, using the general formula for calculating present values:
present value of the annuity at half-year $i = C/(1+i)^{n/2},$

where C is the sum paid out to the annuitant half-yearly (here 10 million stuiver), i the rate of return (4% yearly) and n the number of years (divided by 2 to indicate half-years). He uses i = 4% since de Witt is comparing to redeemable annuities, which are trading at that rate.

For a life living up to age 80, a value of 497,679.26 stuivers is calculated. This is the amount that one ought to set aside at 4% today to be able to pay 10,000,000 when the life survives its 791/2 th year.

De Witt then combines the present values with the aforementioned survival chances. He gives a table of the 153 half-yearly present values, calculated in the above manner, and undersigned by the two bookkeepers T. Bellechiere and J. Lense as being error-free. The sum of all 153 present values gives the amount that needs to be invested at 4% today to be able to pay out 10 million stuivers each half-year until a person is 80 years old, which is the age de Witt assumes that life terminates at. However, de Witt realises correctly that not every single annuity holds this value, since not every life has the certainty of living up until 80. Ingenuously, he takes the novel step of weighting the present value calculation with his suggested mortality rates, accounting for the fact that people die before the age of 80.

Each half-yearly present value has the chances 1/128, 2/3*128, 1/2*128 or 1/3*128 according to de Witt's assumption. The divisor 128 is the number of chances 100 + 2/3* 20 + 1/2* 20 + 1/3* 14. De Witt sums the weighted present values to arrive at a final value of 320,032,130 stuivers, which divided by 20 gives about 16 mio guilders. Given in a formula, de Witt calculates

value of annuity = $^{1}/_{128}$ where

Most curiously, one notices that de Witt exactly inverts the probabilities he presupposed as 1: 3/2 : 2: 3, and uses instead 1: 2/3 : 1/2 : 1/3. This was first observed by Orchard (1852) and stated as a tremendous miscalculation.

Hald (2003) recalculates with the initially assigned probabilities and arrives at a final value of the annuity of 18.9mio guilders. Speculations as to why the mistake occurred vary. Hald (2003) presumes deliberate manipulation of the numbers to make the result politically convincing. De Witt knew that a result of nearly 19mio guilders - an increase from the common 12mio guilders of more than 50% - would never have been accepted by the ministers or the public. Kupers (2014) claims that this is in fact how mortality data behave, since even though mortality rates are higher at old age, due to decreased absolute numbers in that age group, in absolute terms more people die at young age than at old. He also notes that de Witt was evidently working under time pressure since the last sections of the treatise are less detailed and shorter than the introductory parts, and he also did not have time to check some of the tables and calculations until afterwards. It remains unclear whether de Witt became aware of his miscalculation or not. Personally, I tend to agree with the argument that he deliberately manipulates the calculation to arrive at a favourable outcome. It was foremost an effort to convince the ministers of using annuities as a state financing tool, and it was important to arrive at a "nice" result. The assumed probabilities seem to provide an outcome that fulfils this criterion, and it is hard to believe that the diligent and careful mathematician de Witt would have made this mistake. The paper was also checked by numerous people, among them Johannes Hudde. Why he did not correct his assumptions or elaborate on why he changes them in his calculation, remains an unanswered question.

De Witt's main intention with the paper was to present a compelling argument for increasing the number and price of annuities. However, as an experienced politician, he was aware that an increase to almost 19 mio guilders was improbable to be accepted. Yet, after arriving at a value of 16 mio guilders, de Witt proceeds to conclude the treatise with four arguments that show that this is likely to be an understated valuation.

First, he states that the selection of the life may be done with select criteria and hence a buyer normally chooses a life in full health, which will likely prove to have high survival chances in the first while. The subgroup of the population that is selected to be signed under annuities may have better survival chances than other parts of the population, but it might also happen that a seemingly healthy person develops an illness and proves short-lived. De Witt points at a selection bias here.

Secondly, he writes that

he advantage resulting from the aforesaid selection is so much more considerable, as one half year of life, at the commencement of and shortly after the purchase of the life annuity, is of greater value to the annuitant, with respect to the price of such a purchase, than (...) half-years during which the person upon whom the annuity is purchased might live after the said purchase." (Hendricks, 1852)

What his writing seems to suggest is that the annuity is more valuable to the annuitant at an early stage and hence the selection of a "good" life is crucial. He might hint at the fact that due to inflation, the payouts of the annuity will be of greater value in the first years than later on. For example, 10 million stuivers may decrease in monetary value over time. This makes an annuity beneficial to the government since the lump sum of money paid for the annuity consists of monetary units that are worth more at the time of the purchase than at later payout dates.

His third reason acknowledges that his mortality estimates may be wrong and hence give an inaccurate result. He states that there are phases in life that are more dangerous to health than others, and that younger individualy may not be as exposed as later in life. He also points at gender-related difference, for instance due to female risks at childbirth and pregnancy.

Lastly, de Witt is giving room for the possibility of a person reaching an age above 80 years old, which would alter his calculation.

Concluding the "Waerdye", de Witt reiterates his strong recommendation to rely on life annuities as a main instrument for state financing.

What stands out throughout the paper is that even though structurally there is mathematical terminology, it is first and foremost a political statement. The use of examples and careful explanation of the calculations show that de Witt's primary aim was to convince the meeting attendants of a policy initiative. He was very aware that he was not addressing a mathematical audience, and even if he knew of the mathematical relevance of his calculations, he stressed the real-life implications of the results far more. The politician shines through far more than the mathematicus, and de Witt arguably uses his calculations to substantiate his favoured hypothesis. It is curious how his chosen mortality rates arrive at a result that strategically makes a very good attest of his opinion, and it must be assumed that he was aware of that. Additionally, he does not mention any opposing views or arguments against life annuities, such as inflation, the applied transfer tax, or the fact that a poof of life was necessary to receive the payment. These are all considerations substantiating the argument that his miscalculation was deliberate.

De Witt's Waerdye did not have a big impact on the policy of annuity issuance at the time. Annuities were raised from 12 guilders to the already previously proposed 14 guilders - ignoring or at least not fully following de Witt's suggestion (Stamhuis, 1999). De Witt's work on mortality rates suggested that age-dependent schemes for annuities would be sensible, also making annuities more attractive to older people. This was first done on a larger scale in 1672 in Amsterdam under Hudde as Burgomaster, but would not become common practice until the end of the 1700s (Kupers, 2014).

It is nevertheless unchallengeable that de Witt delivered a calculation and mathematically correct formula that prove his profound wit and understanding that are ahead of his time.

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