Varian (1987) pp 1 – 8 VERY IMPORTANT CONCEPT
Consider a market for assets that pay off in different states of nature –
This can be the outcomes of some random process and doesnt have to be full-fledged Arrow-Debreu states of the world ( which describe all possible circumstances)
Assume; individuals care about their wealth in different states of nature & prefer more wealth to less wealth in any state of nature.
𝑅=[■8(𝑅_11&⋯&𝑅_1𝐴@⋮&⋱&⋮@𝑅_𝑠1&⋯&𝑅_𝑆𝐴 )]

Payoff of asset 𝑎 in state 𝑠 given by 𝑅_𝑠𝑎
Number of assets = 𝐴
Number of states = 𝑆
An asset is described by a vector giving payoffs in each 𝑆 states of nature
This 𝑆 𝑏𝑦 𝐴 matrix gives the payoffs of each of the 𝐴 assets in each of the 𝑆 states.
Thus, the first security is described by the column vector (𝑅_11⋯𝑅_𝑠1)
(Payoff) R:Tesco  State 1 -Recession, Payoff 〖𝑎𝑠𝑠𝑒𝑡〗_𝑇𝑒𝑠𝑐𝑜
		⋮
(Payoff) R:Tesco  State 6 -Growth, Payoff 〖𝑎𝑠𝑠𝑒𝑡 〗_𝑇𝑒𝑠𝑐𝑜
Each column represents a security, each row gives the payoffs in a particular state of nature of each of the securities 
𝑅=a matrix which summarises the payoff characteristics offered by this collection of assets , imagine 10 assets ranging from stock 1, Tesco to stock 10 , Apple.
And states S ranging from worst (Recession) to best (Growth)


= [■8(𝑅_( 𝑃𝑎𝑦𝑜𝑓𝑓 𝑖𝑛 𝑆 𝑠𝑡𝑎𝑡𝑒 :𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛,   𝑇𝑒𝑠𝑐𝑜 1)&⋯&𝑅_( 𝑃𝑎𝑦𝑜𝑓𝑓 𝑠𝑡𝑎𝑡𝑒 𝑟𝑒𝑐𝑒𝑠𝑠𝑖𝑜𝑛 ,   𝐴𝑝𝑝𝑙𝑒 10)@⋮&⋱&⋮@〖𝑅 〗_(𝑃𝑎𝑦𝑜𝑓𝑓  𝑖𝑛 𝑆 𝑠𝑡𝑎𝑡𝑒: 𝑔𝑟𝑜𝑤𝑡ℎ, 𝑇𝑒𝑠𝑐𝑜 1)&⋯&𝑅_( 𝑃𝑎𝑦𝑜𝑓𝑓 𝑖𝑛 𝑆 𝑠𝑡𝑎𝑡𝑒 :𝑔𝑟𝑜𝑤𝑡ℎ , 𝐴𝑝𝑝𝑙𝑒 10) )]
Let 𝑥_𝑎 indicate the amount held of asset a

so then a portfolio (p/f) can be represented as a column vector 𝑥=(𝑥_1,⋯,𝑥_𝐴 )
Components of p/f 𝑥 can be + or – sign
Positive = a “long” position – holds the assets & thus entitled to receive the payoff  if state(s) materialise
Negative = a “short” position so one must pay-out the appropriate amount if the state occurs
The wealth in state s that one receives from holding a portfolio 𝑥=(𝑥_1,⋯,𝑥_𝐴 )is given by;
𝑤_𝑠=∑2_(𝑎=1)^𝐴▒〖𝑥_𝑎 𝑅_𝑠𝑎 〗
Or in matrix notation
		(■8(𝑤_1@⋮@𝑤_𝑠 ))=(■8(𝑅_11&⋯&𝑅_1𝐴@⋮&⋱&⋮@𝑅_𝑆1&⋯&𝑅_𝑆𝐴 ))(■8(𝑥_1@⋮@𝑥_𝐴 ))   or..

𝑤=𝑅𝑥
This equation illustrates the relationship between the ends and the means: the ends are the levels of wealth that the consumer can achieve in the different states of nature, which is what the consumer ultimately cares about. 
The means are the existing assets. 
By combining the existing assets into portfolios, the consumer can achieve different patterns of wealth across the states of nature. 
The patterns of wealth that can be achieved depend on the entire set of available assets. 
Thus, the market value of a single asset will typically depend on what other assets are available to combine with it. 
The central case is where the number of assets matches the number of states of nature. In this case, any pattern of wealth can be achieved by some portfolio of existing assets. 
To achieve a particular distribution of wealth
	𝑤=(𝑤_1,⋯,𝑤_𝑠)
 you must solve the system of equations for the p/f
	𝑥=(𝑥_1,⋯,𝑥_𝐴 ), that achieves that distribution of wealth & as the system of equations has exactly as many unknowns as equations, it will always be possible to solve for such a p/f. 	
If the assets outnumber the states of nature, there will be more unknowns than equations and several portfolios will exist that generate any particular distribution of wealth. 
If states of nature outnumber assets, then it will not be possible to solve the system of equations for all distributions of wealth—some patterns of wealth cannot be constructed using the existing set of assets. 
This latter case is presumed the most realistic, since one would generally think that people care about more outcomes than they have assets to trade. 
But the question is controversial. If people really care about achieving a certain distribution of wealth doesn't it seem likely that the market will offer an asset that will achieve such a pattern?
One important kind of assets are Arrow-Debreu securities. 
These are assets that pay off $1 ( 1 unit of numeraire)
….if and only if a particular state of nature occurs; otherwise they pay off zero. 
Thus the payoff pattern of an Arrow-Debreu security takes the form (0,..., 1,..., 0), where the 1 occurs in location s. 
Arrow-Debreu assets can be thought of as especially basic assets, in both the economic sense and the linear algebra sense. 
They are basic in the economic sense, since any pattern of payoffs can be constructed from portfolios of Arrow-Debreu assets. 
They are basic in the linear algebra sense in that they form a basis for the linear space of all payoffs. 
It is this fundamental nature of the Arrow-Debreu assets that makes them so important in analyzing the pricing of financial assets.
When investors choose portfolios they are in effect choosing a distribution of wealth across the states of nature. 
In making this choice they are constrained by their budget constraint: the constraint imposed by amount of wealth that they have to invest and the prices of the various assets that they face
To express the idea of a budget constraint, let the price of asset a be denoted by 𝑝_𝑎 and let 
𝑝=(𝑝_1,⋯,𝑝_𝐴)
…be the row of vector prices.
The value of a portfolio 𝑥=(𝑥_1,⋯,𝑥_𝐴 ) will then be given by  𝑝𝑥=∑2_𝑎^𝐴▒〖𝑝_𝑎 𝑥_𝑎 〗
This is like consumer theory where the value of a bundle of goods is expressed as  the sum of the expenditures on the various goods..
The difference is that the goods that are being chosen—the assets—are not the ultimate end of consumption. They are only the means to an end. What consumers really care about is the final distribution of wealth that different portfolios provide. Therefore, any two portfolios that provide the same pattern of wealth must be worth the same amount. 
Consider, for example, the case of the Arrow-Debreu securities. ……
If we know the price of each Arrow-Debreu security, we can value any asset. 
Since the payoff pattern of any asset can be achieved by some portfolio of Arrow-Debreu securities, the price of the asset must be equal to the price of the portfolio of Arrow-Debreu securities that realizes that same distribution of wealth across the states of nature. 
Let 𝜋_𝑠  be the price of the Arrow-Debreu security that pays $1 if state s occurs,
Let (𝑅_𝑠𝑎)be the payoff pattern of some asset a

Then the equilibrium price of asset a must be given by

 𝑝_𝑎=∑_(𝑠=1)^𝑆▒𝜋_𝑠  𝑅_𝑠𝑎

 Why?
2 Main Arguments. 
The intuition is that 𝜋_𝑠measures the 	value of a dollar in state s, and that the security pays off 𝑅_𝑠𝑎dollars in state s. Summing over all the possible states gives the value of security a. This argument is plausible, but rests on a notion of "value" that is somewhat slippery. A more compelling argument is based on arbitrage considerations.. 
If a complete set of markets for all Arrow-	Debreu securities exists, and the price of an asset 	ever deviated from the price of a portfolio of 	Arrow-Debreu securities that generates the 	same pattern of payoffs, then there would be a 	sure way of making money—just sell the 	security and buy the Arrow-Debreu portfolio, or 	vice-versa, depending on which was worth more. 
	If no arbitrage possibilities exist and a complete set 	of Arrow-Debreu securities are marketed, then any 	asset can be valued in terms of the prices of the 	Arrow-Debreu assets
This discussion shows that any asset can be valued in terms of a particular set of assets in the case where the number of assets matches the number of states of nature. 
However, it turns out that a similar result holds even if there are fewer assets than states of nature. But this demonstration requires a more substantive assumption about arbitrage possibilities, a point to which we now turn..
A Formalization of the No Arbitrage Condition
Lets consider the concept of arbitrage, or more precisely, the absence of arbitrage. 
What should this mean? Basically, the no arbitrage condition must rule out "free lunches"—configurations of prices such that an individual can get something for nothing. 
Any portfolio that pays off nonnegative amounts in every state of nature must be valuable to individuals, so if no free lunches exist, portfolios which are guaranteed to have nonnegative payoffs must have a nonnegative cost. 
Recall that a portfolio's pattern of returns can be represented by 𝑅_𝑥 and the cost of the portfolio by 𝑝_𝑥. 
Using this notation we can state the 
	No Arbitrage Condition

𝐼𝑓 𝑅_𝑥≥0, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑚𝑢𝑠𝑡 ℎ𝑎𝑣𝑒 𝑃_𝑥≥0
The no arbitrage condition loosely described above is now an explicit algebraic condition. 
This condition imposes some restrictions on the equilibrium prices 𝑝. 
Given a particular set of assets, as described by the payoff matrix 𝑅, only certain asset prices 𝑝 are consistent with the absence of arbitrage. 
What characterizes such prices? What restrictions does the assumption of no arbitrage impose on the asset prices? The No Arbitrage Condition implies that there must exist a vector of nonnegative "state prices" 
The No Arbitrage Condition implies that there must exist a vector of nonnegative "state prices 𝜋=(𝜋_1,…,𝜋_𝑠)such that the price of any existing asset is given by;
𝑝_𝑎=∑_(𝑠=1)^𝑆▒〖𝜋_𝑠 𝑅_𝑠𝑎 〗What’s the economic interpretation of the state prices 𝜋_𝑠? 
Consider an Arrow-Debreu asset that pays off $1 if state s occurs and zero otherwise. 
Then according to equation ppt25 the value of this asset must be given by 𝜋_𝑠. 
In other words, 𝜋_𝑠, measures the value of a dollar in state s. 
So equation on ppt25  says that to determine the value of any asset, examine how much it pays off in a given state, multiply that by the value of a dollar delivered in that state, and then sum over all the states. 
The existence of the state prices 𝜋 is a necessary and sufficient condition for the absence of arbitrage. 
If no arbitrage possibilities exist, then there must be state prices, and if the state prices exist, there can be no arbitrage possibilities. 
Hence the existence of the state prices is equivalent to the absence of arbitrage; any argument that follows from the absence of arbitrage must follow from the existence of the state prices and vice versa. 
The state prices 𝝅_𝒔  emerge as a consequence of the No Arbitrage Condition.
Whenever the market works efficiently enough to eliminate the possibilities of arbitrage, there must be state prices that can be used to value the existing assets. 
Of course the state price vector 𝜋 will not, in general, be unique. 
With 𝑆 states and 𝐴 assets, the equations in ppt 26 provide 𝐴 equations in 𝑆 unknowns. Thus there will in general be an 𝑆−𝐴 dimension set of solutions to this system. 
Only if the number of (independent) assets equals the number of states of nature will a unique set of state prices exist.
The No Arbitrage Condition is simple, and yet surprisingly powerful. 
It's linked to an important result in the theory of financial markets, the Value Additivity Theorem. 

Consider two securities 𝑎 and 𝑏, which have payoffs 𝑅_𝑠𝑎  and 𝑅_𝑠𝑏
Suppose the prices of the two securities are known 𝑝_𝑎  and 𝑝_𝑏
Consider a third security which has a random payoff given by some linear combination of the two securities. ( where A and B are arbitrary constants)
𝑅_𝑠𝑐=〖𝐴𝑅〗_𝑠𝑎+〖𝐵𝑅〗_𝑠𝑏
The No Arbitrage Condition implies that if no arbitrage opportunities exist in the market, then the price of an existing asset with a payoff vector 𝑅_𝑠𝑐
𝑝_𝑐=∑_(𝑠=1)^𝑆▒〖𝜋_𝑠 𝑅_𝑠𝑐 〗      using the equation for 𝑅_𝑠𝑐 above
𝑝_𝑐=∑_(𝑠=1)^𝑆▒〖𝜋_𝑠 𝑅_𝑠𝑐 〗
𝑝_𝑐=∑_(𝑠=1)^𝑆▒𝜋_𝑠  〖〖(𝐴𝑅〗_𝑠𝑎+〖𝐵𝑅〗_𝑠𝑏 ")" 〗_ 
𝑝_𝑐=∑_(𝑠=1)^𝑆▒〖𝐴𝜋_𝑠 〗 〖𝑅_𝑠𝑎+〖𝐵𝜋_𝑠 𝑅〗_𝑠𝑏〗_ 
𝑝_𝑐=〖𝐴𝑃〗_𝑎+〖𝐴𝑃〗_𝑏
Value Additivity Theorem
	Assume no arbitrage possibilities exist. Then the	price of a security whose payoffs are a linear 	combination of other assets must be given by the 	same linear 	combination of the prices of the 	other 	assets.

What can we reflect on here?
Note that the payoffs of the assets a and b are arbitrary; the payoffs could be highly correlated, either positively or negatively. 
It is well known that if two assets with negative correlation are combined, the riskiness of the resulting portfolio will be less than the riskiness of either asset held alone. 
Diversification has a natural value in a market with risky assets. Since a linear combination of two negatively correlated assets is less risky than either of the two assets held alone, it might seem that the combination portfolio would be worth more than the sum of the values of the two assets. 
Nevertheless, the Value Additivity Theorem states that the price of the portfolio consisting of both assets will just be the sum of the prices of the two individual portfolios. 

Why?
The answer is that the equilibrium prices of assets must already reflect the value of any kind of linear portfolio manipulation. 
If the value of the two assets in combination exceeded the sum of the values of the assets alone, for example, then arbitragers could just buy the two assets and sell the " mutual fund" consisting of the combination. 
Similarly, if the values of the individual assets were less than the combination, then arbitragers could "unbundle" the combination and make a pure profit by doing so. 
Since this repackaging offers a sure profit, it cannot exist in equilibrium. 
The Value Additivity Theorem is also the basic principle underlying the Modigliani-Miller theorem. (MM)
Modigliani-Miller theorem states that the value of a firm is independent of its financial structure; independent of the fraction of the firm financed by stock and the fraction financed by bonds. Again, the argument is based on the absence of arbitrage. 
If firms could change their value by changing the proportion of stocks and bonds they issue, then individual arbitrages could also repackage the existing stocks and bonds and make a sure profit. Hence, the value of the firm should depend only on the sum of the values of its stocks and bonds, not on whether the firm is weighted more heavily to debt or equity. 
Noting too, that the Modigliani-Miller example also illustrates the restrictions of Value Additivity Theorem. 
First, the Value Additivity Theorem and the No Arbitrage Condition        only apply to combinations of existing assets. If a firm issues new bonds with a different pattern of returns from the old bonds—perhaps because they have higher default risk—then the Value Additivity Theorem does not necessarily hold. 
If a financial operation creates assets that have a new payoff pattern not currently available through linear combinations of existing assets, then the Value Additivity Theorem, or the No Arbitrage Condition need not apply. 
If an economic shift changes the matrix of payoffs 𝑅_𝑠𝑎  so that new possibilities of consumption across states of nature are created,  the vector of state prices w will typically change.
Second, the No Arbitrage Condition only applies directly to linear operations. An asset whose payoffs are the sum of the payoffs of two different assets must have a price which is the sum of the prices of the two different assets. 
Similarly, an asset whose payoff is some multiple of some other asset must have a price which is a multiple of the other asset's price. 
If one asset is a nonlinear function of another asset's price, the No Arbitrage Condition cannot be directly applied, at least in the static framework examined here. Answer this Describe how the arbitrage principle, and the concept of “no arbitrage” in particular, can be applied systematically to uncover hidden relationships in asset prices. Your answer should refer to Varian (1987).                Answer this precise and to the point as well as bullet points  with simple words                                                  
from above data