Question

Consider a two-asset model with money paying the positive given interest rate Rm and the bonds paying a return R which has the expected value μb and standard deviation σb. Show diagrammatically the effects of the following for the proportions held of the two assets:

a) The government imposes a tax on the excess return (R – Rm) on bonds, with a corresponding refund if the return is negative.

b) The government imposes a tax on a positive excess return (R – Rm) on bonds, but without any refund if the return is negative.

c) Show the effects for the above cases if Rm= 0

Answer 1

Bonds generate income which may be taxable. Interest on corporate bonds is taxable. In case of tax-saving bonds, tax is imposed on end returns, but exemptions are also provided while paying income tax. An investment portfolio's expected return, μb in this case, indicates the return amount which is anticipated to be generated by the portfolio. The standard deviation σb is a measurement on how much the return is deviating from μb, that is, the mean.

Let us consider that in the two-asset model, the safe return rate is given by i. It is given that the return on bonds is R. Therefore, the risky asset return is given by -

R = i+ (R-i)

i) In case of a tax system that is proportional, whether the income from bonds is positive or it is negative, the same rate of tax is applied. The investors are burdened but only on R. To explain this, let tax rate be t* on the bond's excess return, then after paying the tax, the investor is left with -

i(1-t) + (R-i)(1-t*). Hence, the investor becomes indifferent to the t* value. This is because tax on excess returns can be undone if the investor holds a higher amount of the risky asset compared to the safe asset. So, the burden is of little amount. Also, if excess returns are negative, on the risky asset, then the revenues from the corporate tax should also be negative, even if on average, they are positive. Also, in case of negative excess return, tax should be negative. This increases the investor's burden.

ii) If the safe return rate is 0, the investors have to bear a net burden of corporate tax. This discourages corporate investment and the tax incidence analysis is relevant in this case.

The CAPM depicts the association between the expected return and the systematic risk for the asset. As shown in the diagram below, expected returns is plotted on the y-axis and expected risk is plotted on the x-axis. As per our explanation, it is also seen on the graph that higher expected risks yields higher expected returns. According to the Modern Portfolio Theory, the expected return increases with an increase in the risk, beginning with a rate that is risk-free. Any portfolio that fits on the Capital Market Line (CML) gives the best optimal portfolio with the best return possible depending on the risk to manage their risk. For perfect optimization of an investor's portfolio, the efficient frontier curve is used as shown in the graph.

iii)