Question
A charity invests $100, putting $50 in each of two assets, A and B. During the first period, A doubles and B loses half its value. At the midpoint, the charity must choose between two policies for the second period:
Rebalance: sell and buy assets without fees to restore a 50:50 split of the current dollar value.
Buy and hold: make no trades, keeping the same quantities of shares bought initially.
In the second period, there are just two possible return paths: with probability 2/3, A halves and B doubles; with probability 1/3, A doubles and B halves again. There are no other cash flows or costs, and the second-period path is the only uncertainty.
Find final wealth under each policy on both paths, as well as each policy's expected final wealth. Which policy has the higher expectation?
Get hint
Hint 1
First calculate the two asset positions' dollar values immediately after period one. Buy-and-hold does not put these positions back at 50:50 automatically.
Check your answer
Solution
Both policies start with $50 in each asset. After period one, A is worth \(50\times2=\$100\) and B is worth \(50\times(1/2)=\$25\), so midpoint wealth is $125. The actual buy-and-hold weights have drifted to \(100/125=80\%\) in A and 20% in B; a 50:50 policy requires a trade.
Rebalancing sells $37.50 of A and uses that exact amount to buy more B. It is self-financing and leaves $62.50 in each asset. On the reversal path (\(A\times1/2,\ B\times2\)), terminal wealth is \(62.50(1/2)+62.50(2)=31.25+125=\$156.25\). On the continuing-trend path (\(A\times2,\ B\times1/2\)), it is \(62.50(2)+62.50(1/2)=125+31.25=\$156.25\). Its expected terminal wealth is thus $156.25.
Buying and holding preserves the midpoint values $100 in A and $25 in B. On reversal, final wealth is \(100(1/2)+25(2)=\$100\). On continuing trend it is \(100(2)+25(1/2)=\$212.50\). Weighting the same two paths as above gives
\[E[W_{\mathrm{hold}}]=\tfrac23(100)+\tfrac13(212.50)=\$137.50.\]
The expected advantage of rebalancing is \(156.25-137.50=\boxed{\$18.75}\). Buy-and-hold wins on the continuing-trend path, but rebalancing wins on reversal. The comparison is not based on assuming that constant weights and unchanged share quantities mean the same thing: the midpoint trade changes the exposure to the second-period returns. Both policies use identical exogenous return paths and the rebalance requires no injected capital.
Assumptions
Both assets can be traded in fractions at the midpoint without fees, spread, tax or borrowing constraints.
No external cash flows occur between the initial investment and final valuation.
The first-period returns are known; the two second-period paths are mutually exclusive and exhaustive with probabilities 2/3 and 1/3.
Buy-and-hold means preserving initial share quantities, not target weight percentages.
Final Answer
Rebalance: $156.25 on either path, expectation $156.25. Buy and hold: $100 on the reversal path and $212.50 on the continuing-trend path, expectation $137.50. Rebalancing leads by $18.75 in expected wealth.