Asset Allocation
TipHow these exercises work
Each topic starts with a real scenario and a bar model, then three levels of exercises. Level 1 (Guided) fills in the blanks of a finished function. Level 2 (Practice) applies it to new numbers. Level 3 (Challenge) asks a new question with little scaffolding. The R and Python exercises run in your browser: click Run Code to try, Submit Answer to check, and use the hint or solution if you get stuck. Excel exercises have a solution you can reveal.
1 Book Topics
- Stocks, bonds, and cash: stocks grow most but can drop 30% to 50%; bonds grow slower and fall less; cash is the emergency fund and sits outside the investment split
- How much in stocks when you’re young: everyone agrees on mostly stocks; Bogle (roughly your age in bonds, later 80 to 85% stocks), Sethi (80 to 90%), Collins (100% while building), Felix on the Cederburg research (100% stocks, global), Housel (extra cash for peace of mind); 80 to 90% is a reasonable start
- US vs. international: 60/40 to 80/20 (US/international) is common; the Cederburg research goes to about one third US
- Target date funds: one fund that sets the split and follows a glide path toward bonds; check the fee and don’t mix it with other funds
- The three fund portfolio: total US stocks, total international stocks, total US bonds for about 0.05% (for example 60/30/10)
- Rebalancing: check once a year and act when any part drifts more than about 5 percentage points; send new money to whatever is below target
2 Good Years and Bad Years
A portfolio’s return is a weighted average of what each part earns.
Formula: Portfolio return = (Stock % × Stock return) + (Bond % × Bond return)
Example (stocks +10% average, −40% bad; bonds +4% average, 0% bad): an 80/20 split in a bad year is (0.80 × −40%) + (0.20 × 0%) = −32%, or −$3,200 on $10,000. In an average year it’s (0.80 × 10%) + (0.20 × 4%) = 8.8%, or +$880.
The second example adds more splits and two columns compared with 100/0: the average year gain you give up and the bad year loss you avoid. The third uses the real 2008 numbers from the book (stocks about −37%, bonds about +5%), where bonds actually went up.
Read the two columns together. Going from 100/0 to 60/40 gives up $240 in an average year but cuts the bad year loss from $4,000 to $2,400. Each step toward bonds trades a little growth for a softer fall.
2.1 Level 1: Guided
Fill in the blanks to finish portfolio_return(), then find the return of an 80/20 portfolio in a bad year (stocks −40%, bonds 0%). Give a decimal.
Put the stock share in A2 (0.8), the stock return in B2 (-0.40), and the bond return in C2 (0). Find the portfolio return in D2.
NoteSolution
=A2*B2 + (1-A2)*C2This returns -0.32, a loss of $3,200 on $10,000.
Whatever isn’t in stocks is in bonds, and earns the bond return.
def portfolio_return(stock_pct, stock_return, bond_return):
return stock_pct * stock_return + (1 - stock_pct) * bond_return
portfolio_return(0.80, -0.40, 0)Whatever isn’t in stocks is in bonds, and earns the bond return.
portfolio_return <- function(stock_pct, stock_return, bond_return) {
stock_pct * stock_return + (1 - stock_pct) * bond_return
}
portfolio_return(0.80, -0.40, 0)2.2 Level 2: Practice
A 60/40 portfolio of $10,000 has an average year (stocks +10%, bonds +4%). How many dollars does it gain?
Change A2 to 0.6, B2 to 0.10, and C2 to 0.04, then turn the return into dollars.
NoteSolution
=10000 * (A2*B2 + (1-A2)*C2)This returns $760, versus $1,000 for all stocks.
Stock share 0.60, stock return 0.10, bond return 0.04.
10000 * portfolio_return(0.60, 0.10, 0.04)Stock share 0.60, stock return 0.10, bond return 0.04.
10000 * portfolio_return(0.60, 0.10, 0.04)2.3 Level 3: Challenge
In 2008, stocks fell about 37% and bonds rose about 5%. Of the splits 100/0, 90/10, 80/20, 60/40, 40/60, and 0/100, what is the highest stock share that kept the 2008 loss to 15% or less? Give a decimal.
Put the stock shares in A2:A7, the 2008 return in B, and find the largest share with a return of -15% or better.
NoteSolution
In B2, filled down: =A2*-0.37 + (1-A2)*0.05. Then:
=MAXIFS(A2:A7, B2:B7, ">=-0.15")The answer is 0.4 (a 40/60 portfolio lost 11.8%).
Use max(s for s in splits if portfolio_return(s, -0.37, 0.05) >= -0.15).
splits = [1, 0.9, 0.8, 0.6, 0.4, 0]
max(s for s in splits if portfolio_return(s, -0.37, 0.05) >= -0.15)portfolio_return() is vectorized: pass all splits, then max(splits[...]).
splits <- c(1, 0.9, 0.8, 0.6, 0.4, 0)
returns <- portfolio_return(splits, -0.37, 0.05)
max(splits[returns >= -0.15])3 How Much to Move When You Rebalance
When one part grows or shrinks, move just enough to get back to the target split.
Formula: Amount to move = (Target % × Total) − Current amount
Example: $10,000 at 80/20, then stocks drop 30%. Stocks are $5,600, bonds $2,000, total $7,600 (stocks about 74%). (0.80 × $7,600) − $5,600 = $480 moved from bonds into stocks.
The second example runs several stock moves, up and down, and applies the book’s 5 percentage point rule to decide whether to act at all. A negative amount means sell stocks and buy bonds. The last example rebalances with new money only, so nothing gets sold.
Read a positive answer as “buy that much stock,” and a negative answer as “sell that much stock.” A 10% drop only moves the split about 2 points, so the 5 point rule says wait. With $1,000 of new money after the 30% drop, the full $1,000 goes to stocks and the gap closes from $480 to under $300 without selling anything.
3.1 Level 1: Guided
Fill in the blanks to finish rebalance_amount(). A $10,000 80/20 portfolio has $5,600 in stocks and $2,000 in bonds after a 30% stock drop. How much should move into stocks?
Put the target in A2 (0.8), stocks in B2 (5600), and bonds in C2 (2000). Find the amount to move in D2.
NoteSolution
=A2*(B2+C2) - B2This returns $480: sell $480 of bonds and buy stocks.
Where you should be (target × total) minus where you are.
def rebalance_amount(target_pct, total, current_amount):
return target_pct * total - current_amount
rebalance_amount(0.80, 5600 + 2000, 5600)Where you should be (target × total) minus where you are.
rebalance_amount <- function(target_pct, total, current_amount) {
target_pct * total - current_amount
}
rebalance_amount(0.80, 5600 + 2000, 5600)3.2 Level 2: Practice
Now stocks rise 30% instead: $10,400 in stocks and $2,000 in bonds. How much should move? (A negative answer means sell stocks.)
Change B2 to 10400.
NoteSolution
The same formula returns -$480: sell $480 of stocks and buy bonds, selling high.
The total is 10400 + 2000, and the current stock amount is 10400.
rebalance_amount(0.80, 10400 + 2000, 10400)The total is 10400 + 2000, and the current stock amount is 10400.
rebalance_amount(0.80, 10400 + 2000, 10400)3.3 Level 3: Challenge
The book’s rule: only rebalance when stocks drift more than 5 percentage points from the 80% target. Starting from $8,000 stocks and $2,000 bonds, stocks move −30%, −10%, −5%, +10%, or +30%. In how many of those cases should you rebalance?
Put the moves in A2:A6. Find stocks, the stock share, and the drift, then count drifts over 5 points.
NoteSolution
Stocks in B2: =8000*(1+A2). Share in C2: =B2/(B2+2000)*100. Drift in D2: =ABS(C2-80). Fill down, then =COUNTIF(D2:D6, ">5") returns 1. Only the 30% drop (73.7% stocks) crosses the band.
stocks = 8000 * (1 + stock_move), share is 100 * stocks / (stocks + 2000), drift is (share - 80).abs().
moves["stocks"] = 8000 * (1 + moves["stock_move"])
moves["share"] = 100 * moves["stocks"] / (moves["stocks"] + 2000)
moves["drift"] = (moves["share"] - 80).abs()
(moves["drift"] > 5).sum()mutate(stocks = 8000 * (1 + stock_move), share = 100 * stocks / (stocks + 2000), drift = abs(share - 80)), then filter(drift > 5) and nrow().
moves |>
mutate(
stocks = 8000 * (1 + stock_move),
share = 100 * stocks / (stocks + 2000),
drift = abs(share - 80)
) |>
filter(drift > 5) |>
nrow()