Dollar Cost Averaging
versus
Value Averagingan appendix to DCA versus Value |

Suppose that we buy stock each month at prices P - We invest $
**A**each month. - At the start of month
**k**we invest $**A**, hence buy**U**_{k}= A/P_{k}units. - At the start of month
**N**we have U_{1}+ U_{2}+ ... + U_{N}= A(1/P_{1}+ 1/P_{1}+ ... + 1/P_{N}) Units Held. - Our Total Cost, after N investments at $A per month, is $NA.
- The average price paid is:
(Total Cost)/(Units Held) = NA/{A(1/P _{1}+ 1/P_{1}+ ... + 1/P_{N})}= 1/{(1/N)(1/P _{1}+ 1/P_{1}+ ... + 1/P_{N})}= 1/{Average of the Reciprocal Prices}
H(P_{1}, P_{2}, ...P_{N})
= N/[1/P_{1}+1/P_{2}+...+1/P_{N}], the
Harmonic Mean of the prices.On the other hand, the average stock price (the Arithmetic Mean), over these N months, is (1/N)(P the :Big QuestionIs the average DCA cost per unit less than the Average stock price, or, to put it differently:
Is the
or, to put it differently:
Is:
or, to put it differently:
Is: 1/{(1/N)(P ???
_{1} + P_{2} + ... + P_{N})}
or, to put it differently:
First, notice that, for equal values, say P Further, suppose we replace two P-values by their average, say: - replace each of P
_{1}and P_{2}by (1/2)(P_{1}+ P_{2})
x = (1/2)(P_{2}-P_{1}) to P_{1}
and subtracting x from P_{2}
... thereby changing each to P = (1/2)(P_{1}+P_{2}).
The effect of this is to leave the
Average, hence the Reciprocal of the Average, unchanged.
However, the Average Reciprocal will be
That's because
initial Average Reciprocal was greater than the Reciprocal of the Average.
- We insist that our portfolio increase by $
**B**at the start of each month. - At the start of month
**n**our Portfolio Value must be**nB**(n increases, at $B each). - If
**P**is the current stock price, then the number of units we hold is (Portfolio Value)/(Stock Price) =**nB/P** - Now we must buy (or sell) units so our Portfolio Value increases by exactly $
**B**. - The sequence of portfolio values (before and after buying or selling),
*etc.*is like so:
Table 1
_{1} + 2 g_{2} + ... + Ng_{N})/{N(N+1)}
is a ... of monthly returns, with recent returns weighted more heavily.
(See
Weighted AverageWeighted Averages.)
Our Cost per Unit involves such a weighted average (with N replaced by N-1):
The Average Cost per Unit will become negative if this
We can now compare the Cost per Unit for DCA and VA, like so: - Let G
_{1}= 1 + g_{1}, G_{2}= 1 + g_{2}, ...*etc.*be the monthly Gain Factors (meaning that, if the unit price has increased by a factor 1.23 in month 7, then G_{7}= 1.23) - The unit prices, at the start of each month, are P
_{1}, P_{2}= P_{1}G_{1}, P_{3}= P_{1}G_{1}G_{2}, P_{4}= P_{1}G_{1}G_{2}G_{3}...*etc.*
Of course, normally (with increasing unit prices), one can also achieve a negative cost per unit by investing $1K, waiting for your portfolio to double, withdrawing $1K, waiting for your portfolio to double, withdrawing $1K, waiting ... Of course, who's interested in the Average Cost per Unit?
, we invest $A each year.DCATo determine the Annualized Return for DCA, we must solve for R from the equation: A(1+R)Suppose, for , we require that our portfolio grow by $B each year.VATo determine the Annualized Return for VA, we must solve for R from the equation: B(1+R) If we set x = 1+R, the two equations become:
Here's a
>zzzZZZ Here are more examples.
Occasionally, the Annualized Gain for DCA is greater than for VA ... >zzzZZZ |