Dynamic Strategies for Asset
Allocation
Four Dynamic Strategies:
• buy-and-hold;
• constant mix;
• constant-proportion portfolio insurance; and
• option-based portfolio insurance
Payoff and Exposure diagrams
• Payoff diagram of a given strategy relates to the
portfolio performance over a certain period of
time to the performance of the stock over the same
period.
• Exposure diagram relates to the
decision of the strategy.
Buy-and-hold strategy - an initial
strategy (say, 60/40 stocks/bills) that is
bought and then held.
Payoff Diagram
Buy-and-Hold ($100)
Value of assets ($)
100
100% stocks
100% Bills
Value of stock mkt
Payoff Example of 60/40 stock/bill
buy-and-hold strategy
Value of assets ($)
slope=0.6
40
value of stock mkt
• Exposure diagram relates the dollars
invested in stocks to total assets;
it shows the decision rule.
Buy-and-Hold
Desired stock position ($)
slope=1 (100%
in stocks)
100% in bills (slope=0)
Value of assets ($)
Exposure Diagram: 60/40 Stock/Bill
Buy-and-Hold Strategy
Desired stock position ($)
slope=1
60
40
100
Value of Assets ($)
constant-mix Strategies
• It maintains an exposure to stocks that is constant
proportion of wealth
• Dynamic approaches -when the relative values of
assets change, purchases and sales are required to
return to the desired mix.
• Consider a 60/40 stock/bills constant mix strategy
(or $60 in stocks and $40 in bills for a total
investment of $100).
Exposure Diagram for 60/40
constant-mix strategy
Desired stock position
60
slope=0.6
100
Value of Assets
Suppose the rule is to set 10% threshold, i.e, rebalance
after 10% increase or decrease in stock price.
For Example:
Initial Change Rebalance
St. Mkt
100
90
90
Stocks
Bills
total assets
60
40
100
54
40
94
56.4
37.6
94
Due to “change”
stock/tot. asset =54/94=57.4%
After rebalance (i.e. buy more $2.4 stocks), i.e.,
stock/total asset =56.4/94=60%
The general rule of constant-mix
strategy is to buy stocks when their
prices are falling and to sell stocks
when they are rising.
Payoff Diagram of 60/40
constant mix strategy
Value of Assets
Buy-and-Hold
constant-mix
40
Value of stocks
When will Constant-mix outperform Buyand-Hold Strategy?
• Consider a case in which stocks fall from 100 to
90, the recover to 100. The market is flat, but it
oscillates back and forth.
• Buy-and-hold strategy - same
• Constant-mix strategy will do better than the buyand-hold because it buys more stocks as they falls.
When shares later increases in prices, the more
share purchased will enhance the return for the
Constant-Mix Strategy
• Other cases include: large volatility and price
reversals.
Constant-Proportion Strategies
• Constant-proportion strategy takes the form:
Dollars in stocks = m(Assets - Floor)
where m is a fixed multiplier.
• Three special cases:
(1) If m >1, the strategy is called the constantproportion portfolio insurance strategy (CPPI).
(2) If m=1, floor= value of bills, this strategy is
the buy-and-hold strategy.
(3) If 0<m<1, floor= 0, the strategy is the
constant-mix strategy.
Exposure diagram for CPPI
Desired position in stocks
50
slope=m=2
75=floor
100
Dollars in stocks= 2(100-75)
=$50
Value of assets
Thus, the initial investment for CPPI is
50/50 stock/bills mix.
Under the CPPI, when a stocks fall in price, say from
$50 to $45, the total asset value will be $95 (=45+50).
The new appropriate stock position = 2(95-75) = $40,
implying sale of $5 of stocks and investment of the
proceeds in bills. If stock prices rise in value, stocks
should be bought.
CPPI strategy sells stocks as they fall and buy stocks
as they rise in value.
In a bull/bear market, CPPI will do well as it calls for
buying/selling stocks as price rises/falls.
Price reversals hurt CPPI investors
because they sell on weakness only to see the market rebound
and buy on strength only to see the market weaken.
Payoff Diagram for CPPI
value of assets
25/75 buy-and-hold
50/50 buy-and-hold
Value of stock market
Concave and Convex Strategies
• Strategies that “buy stocks as they fall, and sell
stocks when they rise,” giving rise to concave
payoff curves are called concave strategies.
• Concave strategies do very poorly in flat in down
or up market, but tend to do well in oscillating
market. Eg: Constant-mix strategies.
• Convex strategies are those “buy stocks when they
rise or sell stocks when they falls”, e.g.. CPPI
strategies
• Convex strategies do well down or up market
Convex strategies represent the
purchase of portfolio insurance
because it has a floor value;
Concave strategies represent the sale of
portfolio insurance.
Convex and concave strategies are
mirror images of each other.
When a portfolio combines a convex concave
strategies, it results in a buy-and-hold strategy
with a linear payoff diagram.
Option-based Portfolio Insurance
• Option-based portfolio insurance (OBPI)
strategies begin by specifying an investment
horizon and a desired floor value at that horizon.
• The value of the the floor is the present value
value of the specified number discounted using
the riskless rate.
• Strategies involve buying of Tbills and call
option. At maturity, the tbills ensure the floor
value and option will have upside potential
Payoff Diagram for OBPI
value of assets
Value of stock market