Modeling Reinvested Dividends in a Five-Year Stock Portfolio

Summary

The document asks how to express the terminal value of a stock portfolio over five years when annual price returns and dividends follow separate lognormal models. It proposes adding a price return and a dividend amount for one period, then asks whether that gives the correct result.

The key modeling issue is that dividends are reinvested, so each year's price growth and dividend payment affect the shares held in later years. A suitable recursion updates portfolio value using the price growth factor and the dividend yield relative to the starting stock price, then applies that update across all five years. The post does not provide a derivation or answer, so it offers no evidence or worked result. The stated models also leave details such as dividend timing and whether the dividend amount is paid per share or per initial portfolio value to be interpreted carefully.

Key ideas

• Reinvested dividends change the number of shares held in subsequent years.
• The one-period portfolio update must account for both price growth and the dividend yield.
• The proposed expression adds a return factor and a dividend amount without a clear consistent unit basis.
• A five-year terminal value requires applying the annual update recursively across all five periods.

Tags

# Determining the investment strategy


# Determining the investment strategy












I have the following problem:

Consider the 5 year investment strategy and given the yearly portfolio returns $S_{t+1}/S_t$ and dividends $D_{t+1}$ paid at $t+1$ which are modeled as:

$\frac{S_{t+1}}{S_t}=e^{\mu+0.2X_{t+1}}$ and $\frac{D_{t+1}}{S_t}=0.05e^{-0.05^2/2+0.05Y_{t+1}}$ where both $X_i$'s and $Y_i$'s are independent and standard normally distributed.

The task is to determina a function $f$ for a investion of 1000000 dollars in a portfolio of stocks and reinsvesting the dividends in the portfolio of stocks such that the value of the portfolio in 5 years can be expressed as some function $V_5=f(\mu,X_i,Y_i)$ for $i=1,2,3,4,5$

I basically think this is just by using the given models for returns and dividends: $V_5 = 1000000e^{\mu+0.2X_{t+1}}+0.05e^{-0.05^2/2+0.05Y_{t+1}}$

However I am not sure if I have done this right? Any suggestions?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.