Abstract
In the previous two chapters, we have restricted ourselves to the case of two time periods, one for investing and one for receiving payoffs. For many applications it is, however, necessary to allow for models with more than two time periods. In particular one can then study re-trading on the arrival of new information. Nevertheless we will see that many of the insights we have won for the two-period model will be useful also for multi-period models.
“It will fluctuate.” John P. Morgan’s reply, when asked what the stock market will do.
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Notes
- 1.
The mathematical term for an event tree is a filtration, \(\mathcal{F}_{0},\mathcal{F}_{1},\mathop{\ldots },\mathcal{F}_{T}\), i.e. a sequence of partitions of a set \(\{1,\mathop{\ldots },s\}\) such that \(\mathcal{F}_{0} =\{\{ 1,\mathop{\ldots },s\}, \varnothing \}\), \(\mathcal{F}_{T} =\{\{ 1\},\{ 2\},\mathop{\ldots },\{s\}\}\) and for all \(e_{t} \in \mathcal{F}_{t}\) there exists \(e_{t-1} \in \mathcal{F}_{t-1}\) such that \(e_{t} \subseteq e_{t-1}\).
- 2.
Note that investors’ preferences are defined over consumption and not over the depot value. The utility function representing the investors’ time preferences and risk attitude determines the consumption, which is smoothed over the realized states.
- 3.
Note that in contrast to Chap. 4, k = 0 does not denote the risk-free asset but consumption. This is because long-lived assets that are re-traded are rarely risk-free since their prices might fluctuate.
- 4.
In principle, all quantities will be dependent on the entire history/path ω t – or at least on the realized state ω t – and we should write, e.g., \(q_{t}^{k}(\omega ^{t})\), as there might be a different path ω′t for which \(q_{t}^{k}(\omega '^{t})\) is a different value. Thus, in writing \(q_{t}^{k}\) above we not only name a function where instead its value is meant, but we are not precise on which value we actually mean, either. Doing so is therefore – if not stated differently – understood just as an abbreviation for ease of reading. Please observe that we cannot write q k(t), because there is not “one” function q or q k which gets evaluated at two different points in time: q t and q t+1 are two different functions, as they are defined on two different domains (Ω t: = Ω 0 × … ×Ω t and \(\varOmega ^{t+1} =\varOmega ^{t} \times \varOmega _{t+1}\) respectively).
- 5.
The budget constraint is defined over the wealth in period t and t − 1, where the sum \(\sum _{k}\left (\frac{D_{t}^{k}+q_{t}^{k}} {q_{t-1}^{k}} \right )\lambda _{t-1}^{i,k}\) is the compound interest rate.
- 6.
This may arise if there are more states than insurance possibilities.
- 7.
Re-arrange the budget constraints for c, insert the result in the utility function and differentiate with respect to the bond holdings s. Finally, evaluate the marginal utilities at c = w.
- 8.
Note that in reality, the growth rate is unknown while the spot rate curve can be observed at all times. Thus, a falling term structure is typically an indicator for a recession (see [CS09]).
- 9.
Note that s 12 does not depend on the state u or d. This is because the forward contract is written in t = 0 without conditioning on the state realized in t = 1. The amount borrowed/lent will be effective for the budget constraint in t = 1.
- 10.
We give only the strict monotonic variant of arbitrage (compare Chap. 4 for other definitions).
- 11.
In the insurance context, “fair” means that the insurance premium must be equal to the expected damages.
- 12.
We use the same notation as for the similar example given in Chap. 4.
- 13.
To simplify expressions we have assumed a constant interest rate.
- 14.
See [HK78] for more details.
- 15.
Considering different speeds of adjustment in economic dynamics was first suggested by Samuelson [Sam64].
- 16.
In fact, trading is so fast nowadays, that even the physical distance from the stock exchange plays a role, as buying and selling orders are transmitted via cables and since the speed of light and particularly the processing times of signals at switching points are limited, a trading computer on Wall Street can react so much faster than its colleague a few blocks away that the latter does not stand a chance in competing when trying to exploit new information on the short-run.
- 17.
To simplify matters we first suppress all time and uncertainty dependence in this generic one step ahead optimization problem.
- 18.
R s i, k is the return agent i expects to get from asset k if state s occurs.
- 19.
Recall that in Chap. 4 k = 0 was the risk-free asset.
- 20.
I.e. the wealth not consumed, but spent on financial assets.
- 21.
To be mathematically correct one should introduce a different symbol for a utility function if it depends on different variables than before. However, adding more notation will be confusing to many readers.
- 22.
Again, the notation is used: μ(R i) is a vector in \(\mathbb{R}^{k}\) but R f is a scalar. A more correct notation would be \(\mu (\tilde{R}^{i}) - R_{f}\nVdash \), where \(\tilde{R}^{i}\) denotes the matrix of risk assets and \(\nVdash \in \mathbb{R}^{k}\) is a vector with 1 in each entry.
- 23.
The first assumption is not restrictive, since in case two agents were to disagree on the dividends in one of the states, one might introduce more states and let the agents disagree over the occurrence of the states. The second assumption is strong. The only excuse we have is that it is sufficient to generate interesting dynamics – and that it is convenient in the Brock-Homes-Model.
- 24.
For a proof, we show that the iteration \(X_{n+1}:= aX_{n} + b\) always converges if | a | < 1. To see this, we compute the fixed point of the iteration as \(X = b/(1 - a)\), i.e. if \(X_{n} = b/(1 - a)\) then \(X_{n+1} = X_{n}\). Then we consider the squared difference of X n to the fixed point and show with a small computation that it is decreasing. From this we can deduce that X n indeed converges. We can apply this auxiliary result to the dynamic system above.
- 25.
This result follows from the decreasing marginal utility. The more wealth an investor receives the lower is his marginal utility.
- 26.
Note that up to now we did not make any assumption on how the portfolio strategy \(\lambda _{t}^{i,k}(\omega ^{t})\) executed at ω t is determined!
- 27.
Formally, this identity follows from aggregating \(W_{t}^{i} =\sum _{k}(D_{t}^{k} + q_{t}^{k})\theta _{t-1}^{i,k}\) over all agents noting that \(\sum _{k}q_{t}^{k} = (1 -\lambda ^{c})\sum _{k}W_{t}^{i}\).
- 28.
That is to say on all paths except for those that are highly unlikely, i.e. those that have measure zero according to the probability measure P. For example, if P is i.i.d., every infinite sequence in which some state is not visited infinitely often has measure zero.
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Hens, T., Rieger, M.O. (2016). Multiple-Periods Model. In: Financial Economics. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49688-6_5
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