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Risk of Stocks in the Long Run Barnstable College Endowment Case Solution
The under discussion piece of literature takes into account a very basic and simple assumption that, in the long run, the ‘risky assets’ will outperform the safer investment types (such as bonds and treasury bills). Therefore, the concerned authority is provided with two distinct options to invest in. The purpose of this report is:

To analyze that whether the instrument used to measure risk is suitable

To see whether any other method can be used to measure risk

To measure the costs and benefits of using the risk assessment technique

To analyze what the best course of action can be taken at the moment (among the two proposed options)
Following questions are answered in this case study solution:

Background
i. Comparison of Shortfall Method with VAR Model
ii. Empirical Analysis on Shortfall Method

What are the side effects of using Shortfall?
i. Table 1: Cost of Benefit guarantee as a function of Maturity
ii. Net cost to the Sponsor of fund
Risk of Stocks in the Long Run Barnstable College Endowment Case Analysis
1. Background
Before moving forward to the discussion of shortfall, it is vital to define this term and subsequently interpret its meaning. Also, it is crucial to explore as to why this method is used widely to calculate risk and why this method is operating at the first place.
i. Comparison of Shortfall Method with VAR Model
The only method used by major financial institutions all over the world is the value at risk or VAR model. VAR measures the potential of maximum loss in a portfolio of assets. Mathematically, it can be defined as:
“The loss level that will not be exceeded with a certain confidence level during a certain period of time is known as VAR”.
Many financial analysts criticize this method by pointing out that the VAR does not hold subadditivity condition, which means that diversification does not reduce VAR. This statement is in contrast with the modern portfolio theory. Also, as VAR ignores statistical properties of a significant loss beyond the quantile point of interest, it is not a preferred option for analysts. Therefore, expected shortfall has been referred as a substitute for VAR model. Expected shortfall is a coherent risk measure and calculates the conditional expected loss beyond VAR. It is a risk that the rate of return earned on an investment portfolio will be less than the risk free rate of interest over some time period. In general, this is a good measure of risk as it can take into consideration the subadditively condition. The expected shortfall theory suggests that if the investment horizon is long, than any entity should hold more stock than bond. Various similar prepositions are also proposed by this theory.
ii. Empirical Analysis on Shortfall Method
However, empirically speaking, Samuelson and Merton proved that most of these prepositions are wrong. They proved that, under standard assumptions about the probability distribution of stock returns and investor preferences, the length of the investment period should not affect optimal asset mix (Bodie, 1991). Therefore, according to the authors, shortfall misguides many investors about the optimal mix of the assets in the long run. However, this is not a weakness of the shortfall in itself as these propositions are projected from the theory itself without any strong proof. Koji Inuia and Masaaki Kijimab also studied shortfall as a measure of risk. In their study, both of them carried out empirical analysis of ES, taking it to be a risk measure. The results show that any coherent spectral risk measure is given by a convex combination of expected shortfalls, and an expected shortfall is optimal in the sense that it gives the minimum value among the class of plausible coherent risk measures.
In consideration with the current situation at hand, ES appears to be mildly effective as the preposition that stocks outperform the bonds and other instruments in the long run are empirically not proved yet. At the same time, reliance on merely an assumption or observation from the market can be unwise.
2. What are the side effects of using Shortfall?
Regardless of the fact that expected shortfall is a suitable measure of risk or not, it is vital to study the side effects of taking into consideration this technique. If the future outlook is gloomy and you want to beat the average interest rates in the long run, you would make sure that you invest in such vehicles or instruments that pay you a predetermined amount in relation to your fixed initial payments. In this way, you can make sure that you evade from expected shortfall and achieve your goal in the long run. The best pertinent example in this case could be the ‘defined contribution plan’ which is a type of mutual fund that pays you a predetermined amount in the future. Any sponsor, who has to pay a defined benefit, would immunize himself by either investing in bonds or equities. But at this point it is vital to define in theoretical terms what the cost of the shortfall is. The cost of the shortfall is the additional amount of money one has to add at the starting date, in order to, ensure that at the ending date, the value of the investment will yield more than the risk free rate. Insuring against a shortfall is similar to a put option with the required amount of the exercise price after the desired period. But how could the mathematical cost (value) of such option vary with time? The following table answers this question by assuming a portfolio which is invested entirely in stocks. The table is derived using blackScholes model of option pricing assuming volatility of 20% and a return of 8%.
i. Table 1: Cost of Benefit guarantee as a function of Maturity
Cost of Benefit guarantee as a function of Maturity 

N (Years) 
Value of Put as %age of present value of benefits 
1 
7.98 
5 
17.72 
10 
24.84 
20 
35.54 
30 
41.63 
50 
52.08 
The benefit/return on the option increases with time and this increase are less pronounced, the smaller percentage of stocks.
ii. Net cost to the Sponsor of fund
This table does not represent the net cost to the sponsor providing insurance against shortfall risk. Any surplus in the fund will reduce sponsor’s cost because sponsor owns some fraction, ∅ of the surplus. The net cost to the sponsor will be:
Cost to the Sponsor=(1∅)* Value of Put option
If the sponsor owns 100 of entire pension surplus, its net cost of providing the insurance would be zero.
There are two distinct options available in the current scenario. First of all, the selling of put options on S&P 500 (with dividends reinvested) seems to be a good strategy. The 2nd option related to the creation of a trust or an entity that will sell shares and will also manage S&P 500 shares on the asset side. According to Heinz Weisshaupt, the best action one can take to insure against this risk is to buy a binary option on the traded asset (Weisshaupt, 2003). Hence, according to Weisshaupt, the first option is the best course of action at the moment. Lets analyse the 2nd proposed option in detail. The 2nd entails the creation of a legal trust or entity which will manage the portfolio. At the same time, the entity will offer shares. If the emphasis is put on the long run gains, then the first option is still viable as it will comprise put options on an ‘all stock portfolio’ (S&P 500). As described above, if the portfolio comprises only of stocks, than the long term return will be tremendous in comparison to beating the meagre 6% Tbill rate.
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