## Third party life insurance

An alumnus of IIT Madras has made a contribution to the institute in a very interesting manner. He has assigned IITM as a beneficiary of his life insurance policy. The amount isn’t large – \$100,000, which I would think is lower than the median amount insured in pure life insurance (as opposed to insurance combined with investment) policies in India.

The important thing is if there are any regulatory implications of this. Typically, insurance companies don’t allow you to assign your policies to random third party beneficiaries, since it can result in adverse incentives – the random third party can murder you, for example (it’s common, however, for employers to be beneficiaries of key employees’ life insurance). However, things might be different here since the receiving entity is an institution.

If insurance firms are willing to write such policies, I wonder if this could be a scalable and sustainable method to donate to institutions of one’s choice. Or if it is simply better to will the same amount of your life’s savings to go to the institution after your death.

PS: I found the original article on LinkedIn and found it incredibly difficult to link to the text and picture. Hence just put the picture here. Another reason why LinkedIn sucks.

## Continuous and barrier regulation

One of the most important pieces of financial regulation in the US and Europe following the 2008 financial crisis is the designation of certain large institutions as “systemically important”, or in other words “too big to fail”. Institutions thus designated have greater regulatory and capital requirements, thus rendering them at a disadvantage compared to smaller competitors.

This is by design – one of the intentions of the “SiFi” (systemically important financial regulations) is to provide incentives to companies to become smaller so that the systemic risk is reduced. American insurer Metlife, for example, decided to hive off certain divisions so that it’s not a SiFi any more.

AIG, another major American insurer (which had to be bailed out during the 2008 financial crisis), is under pressure from its activist investors led by Carl Icahn to similarly break up so that it can avoid being a SiFi. The FT reports that there were celebrations in Italy when insurer Generali managed to get itself off the global SiFi list. Based on all this, the SiFi regulation seems to be working in spirit.

The problem, however, is with the method in which companies are designated SiFis, or rather, with that SiFi is a binary definition. A company is either a SiFi or it isn’t –  there is no continuum. This can lead to perverse incentives for companies to escape the SiFi tag, which might undermine the regulation.

Let’s say that the minimum market capitalisation for a company to be defined a SiFi is \$10 billion (pulling this number out of thin air, and assuming that market cap is the only consideration for an entity to be classified as a SiFi). Does this mean that a company that is worth \$10 Bn is “systemically important” but one that is worth \$9.9 Bn is not? This might lead to regulatory arbitrage that might lead to a revision of the benchmark, but it still remains a binary thing.

A better method for regulation would be for the definition of SiFi to be continuous, or fuzzy, so that as the company’s size increases, its “SiFiness” also increases proportionally, and the amount of additional regulations it has to face goes up “continuously” rather than being hit by a “barrier”. This way, the chances of regulatory arbitrage remain small, and the regulation will indeed serve its purpose.

SiFi is just one example – there are several other cases which are much better served by regulating companies (or individuals) as a continuum and not classifying them into discrete buckets. When you regulate companies as parts of discrete buckets, there is always the temptation to change just enough to move from one bucket to the other, and that might result in gaming. Continuous regulation, on the other hand, leaves no room for such marginal gaming – marginal changes aer only giong to have a marginal impact.

Perhaps for something like SiFi, where the requirements of being a SiFi are binary (compliance, etc.) there may not be a choice but to keep the definition discrete (if there are 10 different compliance measures, they can kick in at 10 different points, to simulate a continuous definition).

However, when the classification results in monetary benefits or costs (let’s say something like SiFis paying additional regulatory costs) it can be managed via non-linear funding. Let’s say that you pay 10% fees (for whatever) in category A and 12% in category B (which you get to once you cross a benchmark). A simply way to regulate would be to have the fees as a superlinear function of your market cap (if that’s what the benchmark is based on).

## Optimal risk sharing

The wife moved to Ann Arbor over the weekend, where she will be spending three months. She took an Air France flight (AF191) in the wee hours of Sunday morning, and then switched to a Delta flight at the legendary Charles de Gaulle. I must mention upfront that she seems to have had a peaceful journey.

Except that people following the same schedule exactly twenty four hours earlier would not have. AF191 that departed from Bangalore i n the wee hours of Saturday morning returned to Bangalore after a bomb scare. The flight was subsequently cancelled.

There are many risks to flying. Schedules nowadays are packed so closely that your flight might be delayed. Occasionally it might be cancelled even, sometimes without a good reason. A delay might sometimes mean that you miss your connecting flight.

The question is who bears the risk on this one. If I’m booked on a flight that gets cancelled or delayed (because of which I miss my connection), whose responsibility is it that I’m transported to my destination? There are three possibilities – the passenger himself, the airline and an external insurer. The question is which of these is most optimal.

The traditional model in aviation as I understand it is that it is the airline’s responsibility. While this makes sense because a large number of delays/cancellations are on account of faults on account of the airline, even when the delay is not due to the airline’s fault, the airline is best placed in terms of mitigating the risk.

Leaving the risk on the passenger has the advantage that he can choose his own risk profile. If you are flexible about your trip, you might choose to go without insurance, and take the hit yourself. If you’re a frequent flyer, then the “insurance cost” thus saved will compensate for the occasional delay. Yet, the problem with this kind of a model is that people tend to underestimate the risks, and will more often than not not insure, and get hit badly when the delay happens.

Which brings us to the final absorber of risk – the insurance company. I’d purchased “travel insurance” for a recent trip, and there was a component on account of delayed or missed flights. If my flight was delayed by a certain amount of time, my insurer would pay me a fixed amount of money.

While this financial hedging is good, it may not adequately represent the costs of making a new booking (including the hassles) when my flight is delayed or cancelled. So this is not a workable solution at scale.

Another solution is for the insurer to guarantee that you will reach your destination by a certain time in case your flight gets delayed or cancelled. This might work out to be more expensive than a fixed cash payout but this removes the cost and hassle of figuring out the next best alternative on the part of the customer. The problem, however, is correlation. Insurance works when people’s risks are uncorrelated or negatively correlated. Here they are positively correlated – all passengers on Saturday’s AF191 to Paris were affected similarly, and this pushes up the cost for the insurer to rebook people.

Unless they tie up with the airline itself! If they reach an agreement with the airline such that the airline commits to transport the stranded passengers, then this “positive correlation” I mentioned earlier will be taken care of. Seems workable, right? Except that what is being insured here is the risk that the airline abandoned in favour of the passenger, who insured against it from an insurer, who reinsured it with the airliner! Can we just cut out the middle men?

From this rather unscientific argument above, it looks like airlines are best placed to insure passengers against disrupted flight schedules. Back in the days of regulated air fares where competition had to be “on service”, airlines would take responsibility. This might have disappeared with the move towards unbundling over the last 2-3 decades. For good reason – insuring a schedule results in an additional (albeit hidden) cost, and getting rid of it can result in cheaper (base) fares.

Yet, given that airlines are best placed to insure schedules, we need a solution. Maybe they can charge a premium for insuring schedules apart from the base fares? Or would they argue that the current “unrestricted fares” are such insured fares (implying the premium is rather high)?

Short of  government mandated regulation, what is the best way for allocating the risk of disrupted flight schedules, and pricing it appropriately?

Tailpiece: A decade ago, our valuation professor (at IIM Bangalore) had told us that “risk cannot be eliminated. It can only be mitigaged by selling it to someone who can handle it better”.

## Means, medians and power laws

Following the disbursement of Rs. 10 lakh by the Andhra Pradesh government for the family of each victim killed in the stampede on the Godavari last week, we did a small exercise to put a value on the life of an average Indian.

The exercise itself is rather simple – you divide India’s GDP by its population to get the average productivity (this comes out to Rs. 1 lakh). The average Indian is now 29 and expected to live to 66 (another 37 years). Assume a nominal GDP growth rate of 12%, annual population increase of 2%  and a cost of capital of 8% (long term bond yield) and you value the average Indian life at 52 lakhs.

People thought that the amount the AP government disbursed itself was on the higher side, yet we have come up with a higher number. The question is if our calculation is accurate.

We came up with the Rs. 1 lakh per head figure by taking the arithmetic mean of the productivity of all Indians. The question is if that is the right estimate.

Now, it is a well established fact that income and wealth follow a power law distribution. In fact, Vilfredo Pareto came up with his “Pareto distribution” (the “80-20 rule” as some people term it) precisely to describe the distribution of wealth. In other words, some people earn (let’s stick to income here) amounts that are several orders of magnitude higher than what the average earns.

A couple of years someone did an analysis (I don’t know where they got the data) and concluded that a household earning Rs. 12 lakh a year is in the “top 1%” of the Indian population by income. Yet, if you talk to a family earning Rs. 12 lakh per year, they will most definitely describe themselves as “middle class”.

The reason for this description is that though these people earn a fair amount, among people who earn more than them there are people who earn a lot more.

Coming back, if income follows a power law distribution, are we still correct in using the mean income to calculate the value of a life? It depends on how we frame the question. If we ask “what is the average value of an Indian life” we need to use mean. If we ask “what is the value of an average Indian life” we use median.

And for the purpose of awarding compensation after a tragedy, the compensation amount should be based on the value of the average Indian life. Since incomes follow a Power Law distribution, so does the value of lives, and it is not hard to see that average of a power law can be skewed by numbers in one extreme.

For that reason, a more “true” average is one that is more representative of the population, and there is no better metric for this than the median (other alternatives are “mean after knocking off top X%” types, and they are arbitrary). In other words, compensation needs to be paid based on the “value of the average life”.

The problem with median income is that it is tricky to calculate, unlike the mean which is straightforward. No good estimates of the median exist, for we need to rely on surveys for this. Yet, if we look around with a cursory google search, the numbers that are thrown up are in the Rs. 30000 to Rs. 50000 range (and these are numbers from different time periods in the past). Bringing forward older numbers, we can assume that the median per capita income is about Rs. 50000, or half the mean per capita income.

Considering that the average Indian earns Rs. 50000 per year, how do we value his life? There are various ways to do this. The first is to do a discounted cash flow of all future earnings. Assuming nominal GDP growth of about 12% per year, population growth 2% per year and long-term bond yield of 8%, and that the average Indian has another 37 years to live (66 – 29), we value the life at Rs. 26 lakh.

The other way to value the life is based on “comparables”. The Nifty (India’s premier stock index) has a Price to Earnings ratio of about 24. We could apply that on the Indian life, and that values the average life at Rs. 12 lakh.

Then, there are insurance guidelines. It is normally assumed that one should insure oneself up to about 7 times one’s annual income. And that means we should insure the average Indian at Rs. 3.5 lakh (the Pradhan Mantri Suraksha Bima Yojana provides insurance of Rs. 2 lakhs).

When I did a course on valuations a decade ago, the first thing the professor taught us was that “valuation is always wrong”. Based on the numbers above, you can decide for yourself if the Rs. 10 lakh amount offered by the AP government is appropriate.

## Probability of accidental death

So I’ve received two separate SMSs from my bankers over the last few days. One of them asks me to sign up for the Pradhan Mantri Suraksha Bima Yojana at Rs. 12 per annum for an insurance against accidental worth Rs. 2 lakhs. The other SMS asks me to sign up for a more general life insurance scheme (the Pradhan Mantri Jeevan Jyoti Yojana) by paying a premium of Rs. 330 per annum. Here is a poster that describes the two schemes:

Considering that you can insure yourself against all kinds of death for a premium of Rs. 330 per annum, and you can insure yourself against accidental death alone for a premium of Rs. 12 per annum, what this implies is that the probability of death by accident given death is 12/330 or 3.6%. Which seems rather low considering that it’s mostly the younger population that is covered by these insurance schemes!

That aside, it’s a good move by the government to increase insurance penetration. I don’t know about accidental death, but the rate on the life insurance is pretty good, and there is a reasonable cut for the banks too for distributing this instrument. And going by the principle that you should be insured for about 5-7 years’ of annual income, Rs. 2 lakhs is a decent amount (India’s mean income is USD 1500 (~INR 90,000) per head. But the median income is likely to be much lower ).

Moreover, the implementation of these schemes is rather simple, since the premium directly goes from your bank account and you can sign up with a SMS, and there are no medical tests. Hopefully this scheme will take off and the insurance penetration in India will increase significantly.

As an aside, I wonder what impact this will have on the life insurance industry which thrives in selling plans that are a combination of insurance and investment. Now that this scheme shows off the real cost of insurance (Rs. 330 for a Rs. 2 lakh insurance), customers might become more discerning about these combo plans and see through the margins the insurers are making, and this may not be all that good for the insurers. Though this might be offset by these insurers themselves becoming underwriters to the government plan itself.

Disclosure: I’ve worked as a consultant with a leading Indian life insurance firm.

## The problem with Indian agriculture, and government

The problem with the Indian agriculture sector is that the government takes a very “cash view” of the sector while what is required is a “derivative view”.

So Congress VP Rahul Gandhi railed on in a rally about how the current Narendra Modi government is anti-farmer, and pointed out at the land acquisition amendment bill and the lack of raising of “minimum support price” as key points of failure. Gandhi was joined at the rally by a large number of farmers, who reports say were primarily very pissed off about the failure of their rabi crops thanks to unseasonal rains in the last month and a bit.

If the government were to take Gandhi’s criticism seriously, what are they expected to do? Not amend the land acquisition act, or amend it in a different way? Perhaps, and we will not address that in this post, since it is “out of syllabus”. Increase the Minimum Support Price (MSP)? They might do that, but it will do nothing to solve the problem.

As I had pointed out in this post written after a field trip to a farm, what policymakers need understand is that farming is fundamentally a business, and like any other business, there is risk. In fact, given the number of sources of uncertainty that exist, it can be argued that farming is a much riskier business than a lot of other “conventional” businesses.

So there is the risk of high prices of inputs, there is risk of bad weather, there is risk of a glut in supply that leads to low prices, there is a risk that the crop wasn’t harvested at the right time, there is a risk that elephants trampled the field, or there is a risk that there might be a new strain of bugs that might destroy the crops. And so forth. And given that most farmers in India are “small”, with limited land holdings, it needs to be kept in mind that they don’t have diversification as a (otherwise rather straightforward) tool to mitigate their risks.

And when the farmers face so many risks, what does the government do? Help them mitigate at max one or two of it. One of them is the “minimum support price” which is basically a put option written by the government, for free, in favour of the farmers. All it entails is that the farmer  is assured of a minimum price for his wares if market prices are too low at the time of harvest. In other words, it helps the farmer hedge against price risk.

What other interventions do Indian governments do in farming? There are straightforward subsidies, all of the input variety. So farmers get subsidised seeds, subsidised fertilisers, subsidised (or in several cases, free) electricity, occasional subsidies in irrigation, subsidised loans (“priority sector lending” rules), and occasionally, when shit hits the fan, a loan waiver.

Barring the last one, it is easy to see that the rest are all essentially input subsidies, making it cheaper for the farmer to produce his produce (I’m proud of that figure of speech here, and I don’t know what it’s called in English). Even loan waivers, while they happen when market conditions are really bad, are usually arbitrary political decisions, and never targeted, meaning that there are always significant errors, of both omission and commission.

So if you ask the question of whether the government, through all these interventions, make the business of farming easier, it should be clear that an answer is no, for while it makes inputs cheaper and helps farmers hedge against price risk, it doesn’t help at all in mitigation of any other risks. Instead, what the government is essentially doing is by paying the farmers a premium (subsidised inputs, free options) and expecting them to take care of the risks by themselves. In other words, small “poor” farmers, who are least capable of handling and managing risk, are the ones who are handling the risk, and at best the government is just providing them a premium!

The current government has done well so far in terms of recognising risk management as a tool for overall wellbeing. For example, the Jan Dhan Yojana accounts (low-cost bank accounts for the hitherto unbanked) come inbuilt with a (albeit small) life insurance cover. In his budget speech earlier this year, the Finance Minister mentioned a plan to introduce universal insurance against accidental death. Now it is time the government recognises the merits of this policy, and extends it to other sectors, notably agriculture.

What we need is a move away from “one delta” cash subsidies and a move towards better risk management. The current agricultural policies of successive governments basically ensure that the farmer makes more when times are good (lower inputs costs, free put options (MSP) with high strike price), and makes nothing when times are bad. Rudimentary utility theory teaches us that the value of a rupee when times are good is much lower than the value of a rupee when times are bad. And for the government, it doesn’t really matter as to when it spends this money, since its economic cycle is largely uncorrelated with farmers’ economic cycles. So why waste money by spending it at a time of low marginal utility as opposed to spending it at a time of high marginal utility?

In other words, the government should move towards an institutionalised system of comprehensive crop insurance. Given the small landholdings, transaction costs of such insurance is going to be high, and the government should help develop this market by providing subsidies. And this subsidy can be easily funded – remember that the government is already paying some sort of a premium to farmers so that they manage their own risk, and part of this can go towards helping farmers manage their risk better.

It is not going to be politically simple, for the opposition (like Rahul Gandhi) will rail that the government is taking money away from farmers. But with the right kind of messaging, and subsidies for insurance, it can be done.

## Pricing railway safety

Yet another railway accident has happened. As someone on twitter pointed out,

The problem with the Indian Railways is that there is no real measure of safety. How do we know how much safer the trains and tracks are compared to last year? Given the way the Railway finances are put out currently, there is no way to figure this out. Without the railways putting out more disclosures, is there a way to put a number on how safe the Indian Railways are? In other words, is there a way to “price” railway safety?

As you are well aware, and as the above tweet points out, it is standard practice in Indian Railway accidents for the Railway Minister to announce an ex-gratia payment to the families of the dead and the injured in case of any accident. I’m not sure if there is a formula to this but one cannot rule out the arbitrariness of this amount. As I had pointed out in an earlier post on RQ, accident compensation needs to be predictable and automatic. Can we use this to price railway safety?

First of all, we need to point out that the railways follows a cash accounting system, and thus doesn’t need to account for any contingent liabilities such as ex-gratia payment (last weekend I sat through an awesome lecture by Prof. Mukul Asher (councillor to Takshashila) on public finances, and he pointed this out). Hence, it would be prudent on behalf of the Indian Railways to hedge out this contingent liability.

How do you hedge a contingent liability? By buying insurance! What the Indian Railways needs to do is to buy group accident insurance – all the ex-gratia payments will then by paid out by the insurance company, and the railways will only pay a premium to these companies, thus hedging out the risk! And this process will help put a price on railway safety!

How is that? Let us say that given the railways’ bad record in safety, and its continued promises that safety will be improved each year, the railways decides to take up group accident insurance on an annual basis. Let us say that there is a competitive bidding process among general insurers in India (both public and private sector) to provide this insurance (railways is a large organization, and insuring them will be a matter of prestige, so companies will bid for it). The premium as determined by this competitive bidding process is the price of railway safety!

We can do better – instead of buying one overall policy, the Railways can think of insuring different routes separately, or perhaps zones. This will help put a price on the safety of each route or zone! There will be some transaction cost, of course, but price discovery will happen, and we will be able to put a price on risk!

But then, this is all wishful thinking. It is unlikely this will happen because:

1. Given the cash accounting system followed by the railways, there is no incentive to hedge contingent liabilities
2. Buying insurance means increasing scrutiny. The railways will not want to be scrutinized too hard. It is currently an opaque organization and it will want to be that way.
3. Given the railways are wholly government owned and there are government owned general insurers, there might be some collusion which might  result in underpricing the risk.
And so forth…

Nevertheless, the point of this post is that it is possible to put a price on safety!

Nikhil Service Station is one of the more popular petrol bunks in South Bangalore. If you wonder why you have never heard of it, however, it is because nobody refers to the petrol bunk by its real name. The bunk is owned by Anitha Kumaraswamy, wife of former Karnataka Chief Minister HD Kumaraswamy and daughter-in-law of former Prime Minister HD Deve Gowda. The service station came into service in the early 2000s. It had been allotted by Indian Oil in the “Kargil martyrs” quota. It is now a landmark in South Bangalore, and popularly known as “Deve Gowda Petrol Bunk”.

The reason I’m bringing up the issue of the petrol bunk is to draw a parallel with the scam-ridden Adarsh Cooperative Housing Society in Mumbai, which was again ostensibly built for “serving and retired army personnel”. The Adarsh scam is in the news once again due to the rejection of the report by the Maharashtra cabinet and the connection with arrested diplomat Devyani Khobragade.

The question about Adarsh that nobody is asking is this – why is it the government’s business to construct housing for “serving and retired army personnel”? Why is it that the government should compensate families of martyrs with petrol bunks and LPG dealerships and not cash? Aren’t these structures designed to be scammed?

Nobody argues that the army must be paid well. Nobody argues, either, that army personnel should be generously insured and compensated, given the hazardous nature of their jobs. My argument, however, is that this insurance and compensation should be universal and standardized. Allotments such as housing and LPG dealerships are discretionary by nature, and that makes them prone to abuse.

Consider for example, a housing society the government constructs for “serving and retired army personnel”. Let us say that the society has 500 apartments. How does the government choose who gets these apartments? And in what way are the 500 such chosen personnel different from those that did not get the allotment? Does this discretionary allotment not leave the system to abuse? Does this not lead to unhealthy competition among the “serving and retired army personnel”? Do we want that in our armed forces?

On a similar note, after each railway accident, we have the railway minister announcing a discretionary compensation for the dead and injured. The question is why this should be discretionary. Cannot the railways simply buy group insurance for all its passengers, which is automatically paid out upon an accident?

The argument I’m making is that some of  the processes we follow are designed to be scammed. In the time of tragedy, either in an accident or in battle, what we need is a standardized and predictable response on behalf of the government agencies. By not putting that in place, the system is prone to abuse.

## Pricing fines for ticketless travel

In large mass transit systems such as those in Mumbai (or even Chennai), ticket checking turnstiles can significantly slow the flow of human traffic. The sheer number of passengers that use these transit systems daily makes it impossible to check the ticket of each and every traveler. Hence, the Railways, rather than checking the tickets of every passenger, instead relies on random checks. During these random ticket checking efforts, people traveling without a ticket are asked to pay a fine. This, the Railways hope, will be deterrent enough for people to purchase tickets before travel.

The very existence of this market, however, implies that fines for ticket less travel are not being priced properly. The math is fairly simple: if the price of the ticket is $p$ and the probability of your ticket getting checked is $frac {1}{N}$, then the fine for ticketless travel should be strictly greater than $Np$. If not, it works out cheaper for your to pay the fine each time you are caught rather than buying the ticket.

So what role is being played by these “informal insurance companies”? Risk management! People don’t like risk. While on an average your ticket might be checked only once in 30 days (number pulled out of thin air), there is no reason that you will not be pulled up for ticket less travel multiple times in a month. By outsourcing the risk to a central party who pools the risks (from several commuters), you have a steady cash out flow and are hedged against getting caught multiple times (you might get caught but your insurer pays the fine). In fact, this is how insurance works in other sectors also.

What should the Indian Railways do to drive these “informal insurance companies” out of business? Currently, if the fine is $S$, $S le Np$. From this equation, you can see that the Railways can do one of three things so that this inequality gets reversed – the price of a ticket can be reduced – but that would be equivalent to cutting off the nose to spite the face, for it would have significant adverse impact on the railways’ revenues. Next, N can be reduced, or in other words the frequency of surveillance be increased. This, too, is not easily implementable since the Railways will have to invest in additional resources to check tickets. The last option is to increase $S$, and there is nothing that prevents the railways from doing this!

How will this work, though? By raising the cost of fines for ticketless travel while keeping the frequency of ticket checking constant, the “premium” a commuter will have to pay to these insurance companies will increase. If the fine amount is increased to a certain level, the premium a commuter will have to pay to buy ticketless travel insurance will exceed the price of buying tickets! And the insurance market will implode.

While this seems like a simple solution in theory, I’m not confident of it being implemented any time soon. Who knows – one might have to go to the Union cabinet to increase the level of fines in local trains. That’s how our railways is structured.

## Why the rate of return on insurance is low

I’m currently doing this course on Asset Pricing at Coursera, offered by John Cochrane of the University of Chicago Booth School of Business. I’m about a fourth of the way into the course and the beauty of the course so far has been the integration of seemingly unrelated concepts. When I went to business school (IIM Bangalore) about a decade ago, I was separately taught concepts on utility functions, discount rates, CAPM, time series analysis and financial derivatives, but these were taught as independent concepts without anybody bothering to make the connections. The beauty of this course is that it introduces us to all these concepts, and then shows how they are all related.

The part that I want to dwell upon in this post is the relationship between discount factors and utility functions. According to one of the basic asset pricing formulae introduced and discussed as part of this course, the returns from an asset is a positive function of the correlation between the price of the asset and your expected consumption growth. Let me explain that further.

The basic concept is that one’s utility function is concave. If you were to plot consumption on the X axis and utility from consumption on the Y-axis, the curve would look like this:

In other words, let us say I give you a rupee. How much additional happiness would that give you? It depends on what you already have! If you started off with nothing, the additional happiness out of the rupee that I gave you would be large. However, if you already have a lot of money, then the happiness you would derive out of this additional rupee would be much lower. This is known in basic economics as the law of diminishing marginal utility, and is also sometimes called the “law of diminishing returns”.

So, let us say that tomorrow you will either have Rs. 80 or Rs. 120 (the reason for this difference in payoff doesn’t matter). Let us call these as states “A” and “B ” respectively. Now, suppose I’m a salesman and I offer you two products. Product X  pays you Rs. 20 if you are in state A but nothing if you are in state B. Product Y pays you Rs. 20 if you are in state B and nothing if you are in state A. Assuming that you can end up in states A or B with equal probability, which product would you pay a higher price for?

The naive answer would be that you would be indifferent between the two products and would thus pay the same amount for both. However, rather than looking at just the payoffs, you should also look at the utility of the payoffs. Given the concave utility function, you would derive significantly higher happiness from the additional Rs. 20 when you are in State A rather than in State B (refer to appendix below). Hence, you would pay a premium for product X relative to product Y.

Now, from a purely monetary perspective, the payoffs from X and Y are equal. However, you are willing to pay more for product X than for product Y. Consequently, the expected returns from product X will be much lower than the expected returns from Y (define returns as $frac {payoff}{price} - 1$. Hence, for the same payoff, the higher the price the lower the returns). Keep this in mind.

Now let us come to insurance. Let us take the example of car insurance. Most of  the time this doesn’t pay off. However, when your car gets smashed, you are compensated for the amount you spend in getting it fixed. What should be your expected return from this product?

Notice that when your car gets smashed, you will need to spend money to get it repaired. So at the time of your car getting smashed, the amount of money (and consequently consumption) is going to be lower than usual. Hence, the marginal utility of the insurance payout is likely to be higher than the marginal utility of a similar payout at a point in time when your consumption is “normal”. This is like product X above – which gives you a payoff at a time when your consumption level is low! And remember that you were willing to expect lower returns from X. Similarly, you should be willing to expect a lower rate of return from the insurance product!

Technical Appendix

A standard utility function used in finance textbooks is parabolic. Let us assume that for a consumption of $C$, the utility is $- (200-C)^2$. The following table shows the utility at various levels of consumption:

Consumption          Utility
80  (A)                  -14400
100                        -10000
120  (B)                 -6400
140                        -3600

Notice from the above table that getting the payoff of 20 when you are at A increases your utility by 4400, whereas when you are at B, the payoff of 20 increases your utility by only 2800. Hence, your utility from the payoff is much higher when you are at A than at B. Hence, you would pay a higher price for product X (which pays you when your consumption is low) than product Y (which pays you when your consumption is already high)