Financial Advice for TDFs $($Target Date Funds$)$

We have seen how households save and consume over the life cycle and how the profile of wages may affect saving and investment decisions. You will replicate some of these graphs under different hypothetical scenarios of life$-$cycle profiles. The main result that you should have in your mind is the Merton$-$Markowitz result. That is$,$ if we have to make an asset allocation between stocks and bonds and the returns on stocks follow

rst $=\backslash$mu $+\backslash$epsi st

with the following assumptions:

The mean return $(\backslash$mu $)$ does not depend on time t $($and is constant at $6\%)$

The error term $\backslash$epsi st follows a normal process with a standard deviation $(\backslash$sigma $)$ equal to $20\%$ per annum

The investor earns in the bond market a fixed r$=2\%$ with certainty $(\backslash$sigma $=0)$ Then with $\backslash$gamma measuring aversion to risk we have

$\backslash$alpha $=(\backslash$mu $-$r$)/\backslash$gamma $*\backslash$sigma $2$

where $\backslash$alpha $=$S$/($S$+$B$),$ that is $\backslash$alpha is the share of the financial wealth invested in stocks $($total financial wealth$=$B$+$S$).$

Assume the investor has a wage increasing by $2\%$ per annum from age $20$ until $65$ and then retires at $65$ and earns a state pension $($not modelled here$)$ equal to $60\%$ of the last salary $(60\%$ is also known as the replacement rate$).$ The investor expects to live until $90.$

a$).$ Compute the present value of remaining wage income over the life cycle $($age $20-90)$ using r$=2\%,$ r$=4\%$ and r$=6\%.$

b$).$ Assume that the investor wants to follow the Merton$/$Markowitz advice. $\backslash$Mu erton and $\backslash$Mu arkowitz however ignore how wages affect our investment decision. Assume that an investor with $\backslash$gamma $=3$has decided to save $10\%$ of wages every period until retirement and nothing from his retirement income and assume they view the certain salary as a riskless bond. During retirement the investor draws down wealth by consuming the wealth divided by the remaining years of life $($if they expect to live for $10$ years and have $100000$ pounds, they consume $10000$ pounds which are deducted from financial wealth$).$ Please present in an excel spreadsheet your advice to this investor and discuss the economic intuition.

c$).$ Wages are not certain in reality but suffer from uncertainty. Following CAPM logic, we take the present value with r$=6\%$ rather than r$=2\%.$ How does the answer in $(2)$ change?

d$).$ Another investor with the same characteristics has a higher risk aversion coefficient $(\backslash$gamma $=5).$How does the investment decision change over the life cycle relative to $(2)?$

e$).$ Most consumers face high uncertainty about the promised pension payments they will receive due to government debt sustainability concerns and the ageing of the population. If$,$ in a rare event, pensions replace $0\%$ of the last wage $($some recent proposals in the political campaign in the US have this feature for rich households$),$ how does the investment choice get affected relative to the answer in $(2)?$

f$).$ If we stay with the assumptions in $(5)$ but the increase in wages is $0\%,$ how do the results change?