# Portfolio Theory

## Summary

If interest rates fall below a critical level there will be widespread switch to money from bonds. Resulting in the liquidity trap problem.

## Assumptions

Common features of modern portfolio analysis

1. Multiple real/financial assets to allow choice between them.
2. The possible assets differ in their risk (usually variance of the σ_returns) and return (expected yield) where the portfolio's  expected return is a weighted average of expected returns of each assets and variance of the portfolio is the weights^2 of the variances.
3. Investors tend to hold mixed portfolios (unless have strong preference for liquidity/high returns etc.) to take advantage of diversification (esp. if have negatively correlated returns).
4. Portfolio approach is a general eq approach where total financial wealth constrains choice and whether there are income and substitution effects between types of assets.
5. Transaction costs introduce frictions to lower desire to substitute assets.
6. Treatment of expectations is critical - generally it's assumed exp. are rational but with imperfect info.

## Model/Theory

1. ### Portfolio Approach to Keynes Transactions Demand

• Q is why individuals do individuals hold purchasing power in non-interest bearing form? Answer is transaction costs to switching between assets.
• Model optimise π_t=R_t-C_t
• assumes no risk attached to holding bonds (it's a world of certainty)
• where R_t = return = r_t(aveW_t-aveM_t)
• C_t=brokerage costs=c_t(aveW_t/aveM_t)
• subject to aveM_t where we find the optimal average money balances is aveM_t=(ct*aveW_t)^0.5/r_t
• Marginal benefit is constant, marginal cost increases where they intersect is equilibrium money balances (p136)
• Where a decrease in the interest rate lowers the marginal benefit of holding bonds, and increases the marginal benefit of holding money and thus the optimal average money balances rises.
2. ### Portfolio Approach to Keynes' Precautionary Demand

• Main assumption is risk-return correlation.
• Lower interest rate leads to same variance but less returns, which would usually lead to higher proportion of average money balances (but depending upon indifference curves could lead to higher if for example wants to maintain the same target returns)
• Higher risk leads to same expected returns but more variance (curve similarly rotates counter-clockwise).
• Crucially if the majority of investors are risk-averse diversifiers then the model is precautionary in character.If investors are primarily risk-averse plungers (opposite of diversifiers) or risk-loving gamblers then the model has instability associated with the speculative motive.
3. ### Portfolio Approach to Keynes' Speculative Demand

• Herding behaviour and the critical threshold can lead to large instability.
• In the model we assume certainty about future interest rates (and thus capital gain).
• If interest rates are less than the critical rate losses on bonds will outweigh the interest income where the best outcome is all money with zeroreturns.
• If interest rates are above the threshold the investor will hold all bonds and no money.
• Interest rate policy under these conditions both 1. rise in r not past threshold 2. fall in r not past threshold will lead to no change in behaviour. 3. rise in r over threshold --> switch from money to bonds 4. fall in r over threshold --> switch from bonds to money.

## Evaluation

• Main criticism of Baumol-Tobin transactions model is that it is conducted in a world in which bold holding is not risky.
• Main criticism of the speculative model is that it does not explain diversification.
• Can integrate the three approaches to have the qualities we want, but regardless the effect of an expected capital loss on bonds when interest rates are below the critical rate is the dominant feature of the models.
• If you change the assets in the portfolios, for example, to currency and capital you find the minimum risk portfolio includes both assets in inverse proportion to their variances. The crucial difference being that money is no longer assumed to be riskless.