Bond Access, Arbitrage, and Consumption Choices in a Two-Period Model
Summary
The question concerns a two-period consumption model in which an investor allocates wealth between a risky asset and a zero-coupon bond. The displayed equations appear to let the bond position transfer funds between periods, while the question asks why bond purchases or borrowing might not be represented symmetrically. The response emphasizes that the answer depends on the model’s discounting and market assumptions.
If prices are discounted at the risk-free rate, excluding an opportunity to invest in the bond can conflict with no-arbitrage reasoning: an investor should be able to carry funds forward at that rate. The response suggests that forcing an agent to avoid bond investment would require unusual preferences, such as extreme risk seeking, and questions whether that is a realistic behavioral assumption. It offers a qualitative argument rather than a derivation or calibrated example, and the original equations’ sign convention and asset availability are not fully clarified. The main lesson is to align the model’s trading opportunities and preferences with its pricing assumptions.
Key ideas
- A two-period consumption model should state whether investors can trade the risk-free bond.
- Using the bond rate for discounting while excluding bond investment can conflict with no-arbitrage assumptions.
- Preventing an investor from choosing bonds may require extreme risk-seeking preferences.
- The response is qualitative and does not fully resolve the equations’ sign convention.
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Full text
# convention in borrowing money in a multiperiod model # convention in borrowing money in a multiperiod model I have a question concerning the idea of consumption in multi period. The following is given $$C_1=W_0-xS_1+B$$ $$C_2=xS_2-BR$$ where $W_0$ is initial wealth $x$ is the weight on an asset with price / value $S_t$ $B$ is a zerobond at some riskless rate $1<R<2$ the question: how can it be justified (or can it at all be?) that the agent's only chance to transfer money into the second period is via buying the risky asset $S_1$ and not by buying zerobonds? would it not be much more realistic if one had $$C_1=W_0-xS_1\pm B$$ $$C_2=xS_2\pm BR$$? Thanks in advance for any suggestions. ## Answer by Phun (score 1) https://quant.stackexchange.com/a/16472 Your question depends on the discount factor you wish to use for pricing. If u use the risk-free rate (from the bond), it wouldn't be in line with the no-abitrage condition to assume an risk neutral agent can't/wouldn't invest in bonds to carry money into next period. To understand this: just assume a 1 period model with two outcomes for S, where both outcomes are more beneficial than investing in the bond. I believe, if you really want to force the desired behaviour into your economical framework with resonable parameters for your assets, you have to model agents with an really high amount of risk searching attitude. (CRRA/EZ Utility with risk aversion parameters way below 1, most likley even negative (I don't know if EZ is even defined or can be interpreted for s.th. like that)). This, however, wouldn't be a realistic setting. The best way to check wether your idea is reasonable or not is to ask yourself: Would I do it? My answer is a clear no.
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