On bitcoin

In the last month there has been a lot of discussion about cyber currency and specifically about the bitcoinm which has experienced a meteoric rise in value (relative to the dollar) over this winter season. This has prompted many to suggest that the bitcoin is the currency of the future. This may well be the case at some point.

However, it's important to curtail our expectations about the current possibilities that such a currency offers. In particular, some were quick to suggest that the bitcoin may offer an alternative to the euro for countries such as Cyprus. This is a provocative sound bite, but with no meaningful substance.

Proponents of such schemes may posit that even if not suitable for a whole a country to adopt, the bitcoin offers a way to earn a handsome profit because it is valued so highly. Such assertions neglect to mention that asset prices and currency rates reflect all publicly available information in that the market has already priced in all the publicly available information. In fact, since the internet fastened the speed of the distribution of news and expanded the market, adjustments in prices and rates is almost instantaneous. Hence, a small trader must outsmart the market in order to earn profit through such trades. A nearly impossible task.

But let me return to the claim that it is a good idea to switch your currency of exchange to the bitcoin for a second. Thanks to websites http://www.coindesk.com/price/ and http://epp.eurostat.ec.europa.eu/portal/page/portal/exchange_rates/data/main_tables I was able to find (close) rates of exchange for bitcoin and euro relative to US dollar (that is, how many US dollars a bitcoin and a euro buy at closing rates).

In Figure 1, we see the daily percentage change in value for bitcoin (blue line) and for euro (red line) for the period Aug 1 2013 - Dec 18 2013. The difference in variation is striking. The bitcoin varies greatly in value whereas the euro does not. Hence, holding all your assets in bitcoins might not be such a good idea as the value of your wealth would fluctuate enormously on daily basis. In contrast, the euro exhibited a certain stability which offers a certainty about the value of one's wealth.

Figure 1:

Yet the above comparison might not be fair as the euro has ECB backing it whereas the bitcoin does not. But, the proponents of the bitcoin suggest the absence of a central authority is a plus. Yet the data is clear that bitcoin is not the safest method of storing your wealth.

It could be of course that the bitcoin is fluctuating in such a manner due to the fact that it is new and people do not know much about it. In Figure 2, we see the daily percentage change in value for both the bitcoin (blue) and the euro (red) since their respective inception. It is immediately apparent the euro has never exhibited fluctuations of similar magnitude to those exhibited by the bitcoin.

Figure 2:

It is thus clear that for the time being, the bitcoin is not the safest storage of wealth. But more can be said. In particular, looking at the percentage change in value of the monthly average of each exchange rate we see a similar picture to the earlier. That is, the bitcoin (blue) displays wild fluctuations, while euro (red) does not. Therefore, the bitcoin may not be the safest storage even for longer horizons.

Figure 3:

Yet, in spite of the noted fluctuations in value, it might be possible that for an investor with mean-variance utility to use the bitcoin as part of a diversified portfolio. Specifically, if we look at the changes in daily value (note: traders gain by changes in value), since bitcoin's inception the mean change for bitcoin was .418 dollars with standard deviation 16.86, whereas for euro the mean change was .000279 with standard deviation .008. Hence, neither dominates the other. Moreover, as seen below the two changes seem uncorrelated.

 Figure 4:

 Hence, a portfolio divided between w percent bitcoin and 1-w percent euro would have variance

where DB is change in bitcoin value and DE is change in euro value. In fact, a portfolio which would have consisted a proportion 0<w<0.000000469069 of bitcoin would have had a smaller variance than a portfolio with just euros, and also would have had a higher expected return.

On the proposed first residence policy

Cyprus has been in the eye of the storm for a while now. An array of adverse shocks has hit the economy of the small island economy (a nice summary can be found in Zachariadis' analysis). Due to the severity and magnitude of both the liquidity shock and wealth effects, which ensued the "deposit-shares swap" (or as it has been wrongly dubbed "the haircut") and the resolution of Laiki took place, the government is worried about a surge in non-performing loans. What´s particular worrisome for the government is the possibility of home confiscations by banks as a response to mortgage defaults.

As a result, the government has put forward a policy that aims to address the problem of people ending up without a house. Specifically, if someone defaults on her mortgage (let`s call her the "defaulter"), then an can either buy or lease her mortgage contract/house from her bank (let`s call the owner of the contract, the "M.O."). The defaulter, however, has the right to live in the house (whose mortgage she defaulted on) as long as she is willing to pay a monthly rent to the M.O.. Moreover, the defaulter can, at a pre-decided point in the future, buy back her mortgage/house from the M.O. The buy-back price equals to the value of the mortgage (loan) when it was defaulted.

For example, suppose John had started off with a $100,000 mortgage and paid back 40% of it, but now decided to default. Then, an MO can buy his mortgage/house from John´s bank, but John can still live in his house as long as he pays rent. Moreover, at some point he can buy back his house so long he gives back the $60,000.

So, here is the key issue with the proposed policy that has also been a subject of some controversy among MPs; any amount John pays as rent is not intended to be used as mortgage repayment. In fact, a lot of people questioned the ability of the proposed policy to guarantee that no-one will end up homeless. At this point, I should also mention that the proposed policy plan is only concerned with the first home of each family. Hence, it does not protect homeowners from complete home confiscations of their second, or third etc., house.

In fact, this provision leads to some complexities that seems that have eluded most of our politicians. For instance, what defines a first home; if someone defaults on all his property, who decides which is one is his first home? And how is this decision being made?. These are some concerns that have not really received any attention, and unfortunately will not receive any here either.

My main focus is to address the concern about the payment of the rent mentioned above. Some have hinted at the immorality of making a house occupant pay rent for what used to be his house. Furthermore, people deem it unfair that the rent is not being used for mortgage repayment. Yet it is possible for the defaulters to benefit from this arrangement.

In order to illustrate this fact, let's introduce some notation so that we can do a bit of economics. Let "R" denote the monthly rent (assumed to be time invariant), while "Tmax" is the number of months an individual expects to live in a house, when she first moves in. It´s further assumed that an individual does not anticipate to ever move out, so Tmax essentially is the number of months an individual expects to live. "V0" is the value of the mortgage in period 0, and thus is the total amount that the individual has to pay back (for simplicity assume interest rate is 0). In such case, an individual who makes an average monthly payment of "P" towards her mortgage is expected to pay back the whole amount in "Y" years. That is, V0=P∙Y. Lastly, I assume no credit constraints in that an individual can borrow as much as she likes.

Given the above, in period 0 we cannot have Tmax∙R<P∙Y. To see this, suppose that we do. Then, an individual faces the following tradeoff. She could rent a house for all Tmax months, which will cost her Tmax∙R. Alternatively, she can buy a house worth V0 (assuming she borrow the whole value of the house). Then, she will have to pay back P∙Y. Clearly, the former dominates the latter, and hence no one would take up a mortgage. By a symmetric argument, it's obvious that we cannot have Tmax∙R>P∙Y, because if this was not the case, no one would be renting a house. So, if we want to derive an equilibrium in which we have both tenants and owner occupancy, we must have Tmax∙R=P∙Y.

If we solve for the average monthly mortgage instalment, we get

P = (R∙Tmax)/Y.

This formula enables the comparison between the average monthly mortgage payment (P) and rent (R). It's immediately apparent that P≥R when Tmax ≥Y. So, a natural question is whether Tmax ≥Y is a logical assumption to make. In fact, if we assume otherwise, then nobody would be a lender, because she would never receive her money back. This because Y represents the months it takes to repay the mortgage, while Tmax the maximum number of months an individual expects to live. Hence, lending money out would be mean that upon the death of the borrower, the lender would not have received all her money back. So, nobody would lend if Tmax <Y were not the case. Hence, we have Tmax ≥Y.

Consequently, we must have that P≥R; namely, that monthly rent is at most equal to the average monthly mortgage repayment. Hence, a defaulter will in fact observe a reduction in his monthly housing costs once his house is seized. Of course a defaulter will lose ownership, and will most likely have to move at some point to a new place. This will impose further costs, which are ignored in this simple analysis. Furthermore, one could argue that instead of forcing rent upon defaulters is suboptimal to an extension of Y.

These arguments, however, do not negate the fact that a defaulter will experience a decrease in her monthly housing fee. Moreover, observe that maximum value of Y is Tmax, and that if Y=Tmax, then the borrower  will simply die right after she gets full ownership of the house. In such a case, renting and "owning" a house are ultimately equivalent. If, instead,Y is increased but maintained below Tmax, then an individual can earn house ownership before she dies. However, given the equilibrium condition stipulates P = (R∙Tmax)/Y and Tmax ≥Y, then R will still be lower than P, and thus for a sufficiently impatient individual (i.e., sufficiently small discount factor), renting will still be superior.

All in all, should it be left to the market forces, then we would expect that the government`s proposed policy will enable the defaulters to swap the combination (renting, low housing fee) for (ownership, high housing fee). However, it conceivable to anticipate that the determination of the rent will not be decided by market forces, but will be instead capped by the state. Of course a rent fee cap only make senses, from the point of view of the state, if the cap is placed below the equilibrium rent fee. In other words, it is quite conceivable to expect that the rent paid by the defaulters will be well below the equilibrium rent, which in turn is less than the mortgage payment, as established above. Hence, the government´s proposed policy can greatly benefit defaulters.

Is It Time To Have A Child?

During the last month I found myself entrapped in a series of discussions concerning the best time to have a child. While not my favourite topic in the world, deciding when to have a child is one of the biggest decisions a household has to make. This initiated my interest to examine this decision by using simple economic modelling. (Note: I omit the mathematical characterisation of the model in this post, but hopefully I will post it in the near future.)

Of course my simple analysis omits several factors that come into play in the decision making process, but I find that key features of such a decision are still illustrated. Before moving on to the results, I should identify my simplifying assumptions:

A1: Each household can only have one child.
A2: Each household has two "parents".
A3: Before a child is born, both parents earn wages and receive the same wage in each period.
A4: After the child is born, only the highest paid parent continues to work.
A5: Parents live for T years.
A6: Parents derive a constant pleasure each period from having a child and exert a constant effort each period by raising a child. But total effort increases the longer it takes to have a child (i.e., younger parents are more “efficient”).

The key assumptions are A2 & A4, which collectively boil down to the requirement that parents decrease their time at work in order to raise their child. So, A2 & A4 can be replaced by requiring parent(s) to shift some of their "working time" towards "raising time". Assumption A1 is taken so as to simplify the analysis, while A3 is made in order to generate a difference in income after a child is born. Finally, A5 can be considered as the life expectancy.

In essence, the problem of deciding when to have a child is an optimal stopping problem, where the household stops to be childless and switches to parenthood. So, given that households have finite lives A5 is our constraint.

In the simplest model, households cannot accumulate savings and yet the results seem intuitive. On the one hand, I find that the higher the wages the household has to forfeit from when it has a child, the longer it takes to have a child. Similarly, the longer the household expects to live, the longer it will postpone parenthood. On the other hand, the higher the pleasure the household derives from having a child, the sooner it will decide it have a child. Effort has exactly the same effect as pleasure. That is, the higher the per period effort that needs to be exerted, the sooner the child will come. It's also interesting to note that pleasure and effort have the most profound effect, followed by forfeited wages and life expectancy.

Then I proceed to extend the model in order to allow for savings, child and general costs (note: costs are the same each period). Moreover, I impose the requirement that in equilibrium the household must either spend or leave as bequest all of its earned income. This is a neat condition as it gives us the optimal time to have a child (since it must hold in equilibrium).

In the extended model, the opportunity cost of having a child comprises of lost wages and child costs. The effect of the latter is ambiguous, while the effect of lost wages is clear-cut. In particular, the higher the wage (that has to be given up) the sooner the household will choose to have a child. This might seem counterintuitive at first as it contradicts the results of the simple model above. However, recall that this wage is earned prior to the child birth, so the higher it is, the faster the household will accumulate sufficient savings in order to have a child.

This channel allows the wage earned after the child is born to also affect the optimal time to have a child. As in the case of forfeited wages, I derive a negative association. That is, the higher the wages after the child is born are, the sooner the child will come. In this case, the higher wage will enable the household to avoid depleting their savings too early, and thus will be able to support its child. Finally, as in the simple model, I find that effort and pleasure decrease the optimal time to have a child.

In conclusion, despite the model's simplicity it's interesting to find the varying effects wages can have. My initial hunch was that of a positive relationship, which was confirmed in the simplest model. However, it was intriguing to unravel the effects of savings and how they can reverse the effects of wages.