Dollar Cost Averaging versus Value Averaging an appendix to DCA versus Value
 Dollar Cost Averaging ... the Average Cost per Unit

Suppose that we buy stock each month at prices P1, P2, etc. and we're buying units each month: U1, U1, etc.

• We invest \$A each month.
• At the start of month k we invest \$A, hence buy Uk = A/Pk units.
• At the start of month N we have
U1 + U2 + ... + UN = A(1/P1 + 1/P1 + ... + 1/PN) 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/P1 + 1/P1 + ... + 1/PN)} = 1/{(1/N)(1/P1 + 1/P1 + ... + 1/PN)} = 1/{Average of the Reciprocal Prices}
We recognize the Cost/Unit as H(P1, P2, ...PN) = N/[1/P1+1/P2+...+1/PN], the Harmonic Mean of the prices.

On the other hand, the average stock price (the Arithmetic Mean), over these N months, is

(1/N)(P1 + P2 + ... + PN)
the Big Question:

Is the average DCA cost per unit less than the Average stock price, or, to put it differently:

Is the Harmonic Mean less than the Arithmetic Mean ???

or, to put it differently:     Is:
1/{(1/N)(1/P1 + 1/P1 + ... + 1/PN)}     less than     (1/N)(P1 + P2 + ... + PN)     ???

or, to put it differently:     Is:
(1)     (1/N)(1/P1 + 1/P1 + ... + 1/PN)     greater than     1/{(1/N)(P1 + P2 + ... + PN)}     ???

or, to put it differently:
Is:
(2)     the Average Reciprocal     greater than     the Reciprocal of the Average     ???

First, notice that, for equal values, say P1 = P2 = ... = PN = V, then each side of (1) equals 1/V, so the Average Reciprocal is equal to the Reciprocal of the Average.

Further, suppose we replace two P-values by their average, say:

• replace each of P1 and P2 by (1/2)(P1 + P2)
That means adding x = (1/2)(P2-P1) to P1 and subtracting x from P2 ... thereby changing each to P = (1/2)(P1+P2).

The effect of this is to leave the Average, hence the Reciprocal of the Average, unchanged.
(The Average, namely (1/N)(P1 + P2 + ... + PN), is unchanged by adding and subtracting the same amount, x.)

However, the Average Reciprocal will be reduced since:
(1/2){1/(P1+x) + 1/(P2-x)} = (1/2){1/P + 1/P} = 1/P = 2/(P1+P2) which is less than (1/2){1/P1 + 1/P2}.

That's because
2/(u+v) < (1/2)(1/u + 1/v)
provided   4 u v < (u+v)2 ... where we've multiplied by 2uv(u+v)
provided   0 < u2 - 2 uv + v2
provided   0 < (u - v)2     which is true, unless u = v.
 So, we continue to replace pairs of prices - the largest and smallest (or the Maximum and Minimum) - by their average, leaving the Reciprocal of the Average unchanged ... but continually reducing the Average Reciprocal. Each time we do this the Maximum gets smaller and the Minimum gets larger: This ritual continues until all prices are equal to the Average Price:     (1/N)( P1+P2+ ... +PN) ... and that provides the minimum Average Reciprocal.
Hence (unless the prices were equal) the initial Average Reciprocal was greater than the Reciprocal of the Average.

Conclusion?
The average cost per unit, using DCA, is less than the average of the unit prices.

Note:
If, instead of investing a fixed number of dollars each month, you bought a fixed number of units, then your Average Cost per Unit would be the Average Unit Price: (1/N)( P1+P2+ ... +PN).

 Value Averaging ... the Average Cost per Unit

• 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
 Start ofMonth StockPrice Portfolio(before purchase) Portfolio(after purchase) Cost TotalUnits 1 P1 0 B B B/P1 2 P2 (B/P1) P2 2B 2B - (B/P1) P2 2B/P2 3 P3 (2B/P2) P3 3B 3B - (2B/P2) P3 3B/P3 ... ... ... ... ... ... N PN ((N-1)B/PN-1) PN NB NB - ((N-1)B/PN-1) PN NB/PN
Note that the monthly "Cost" may be positive or negative.

Conclusion?
After N investments, we have:
 Total Cost = B + {2B - (B/P1) P2} + {3B - (2B/P2) P3} + ... + {NB - ((N-1)B/PN-1) PN} = B + {B - B (P2/ P1-1)} + {B - 2B (P3/ P2-1)} +... + {B - (N-1)B (PN/ PN-1-1)} = B + (B - B g1)+ (B - 2B g2) +...+ (B - (N-1)B gN-1)       where g1 = P2 / P1 - 1 is the gain for month 1 etc. etc. = NB - B (g1 + 2 g2 + ... + (N-1)gN-1)       which may be negative ... if the gains are large enough!
Note that, although we've made N investments (by the start of month N), only N-1 months have passed.
 Total Units = N B/PN
hence
 Cost per Unit = {Total Cost} / {Total Units} ={NB - B [g1 + 2 g2 + ... + (N-1)gN-1]} / {NB/PN} = PN {1 - [g1 + 2 g2 + ... + (N-1)gN-1]/N}
Now a thing which looks like     2(g1 + 2 g2 + ... + NgN)/{N(N+1)}     is a Weighted Average ... of monthly returns, with recent returns weighted more heavily.     (See Weighted Averages.)

Our Cost per Unit involves such a weighted average (with N replaced by N-1):
g = 2[g1 + 2 g2 + ... + (N-1)gN-1]/{(N-1)N}

Conclusion?
Whether the Cost per Unit is negative after N investments depends upon the value of g, the Weighted Average monthly return:

VA Cost per Unit (after N investments) = PN [ 1 - {(N-1)/2}g]
where g is the Weighted Average return:
 g = 2/{(N-1)N} [g1 + 2 g2 + ... + (N-1)gN-1] = 2/{(N-1)N} [(P2/P1-1) + 2 (P3/P2-1) + ... + (N-1)(PN/PN-1-1)] = 2/{(N-1)N} [G1 + 2 G2 + ... + (N-1) GN-1 - {1+2+3+...+(N-1)}] = 2/{(N-1)N} [G1 + 2 G2 + ... + (N-1) GN-1] - 1
where 1+2+3+...+(N-1) = (N-1)N/2   and   G1 = P2 / P1   is the Gain Factor for month 1 etc. etc.

The Average Cost per Unit will become negative if this Weighted Average return satisfies: g > 2/(N-1)
... which is almost inevitable ... eventually ... for rising markets.

 DCA versus VA ... the Average Cost per Unit

We can now compare the Cost per Unit for DCA and VA, like so:

• Let G1 = 1 + g1, G2 = 1 + g2, ... etc. be the monthly Gain Factors
(meaning that, if the unit price has increased by a factor 1.23 in month 7, then G7 = 1.23)
• The unit prices, at the start of each month, are P1, P2 = P1G1, P3 = P1 G1G2, P4 = P1 G1G2G3 ... etc.

If CD is the DCA Cost per Unit, then:
 CD = H (P1, P2, ... PN)     ... the Harmonic Mean so 1/CD = (1/N)(1/P1 + 1/P2 + ... +1/PN) = (1/NP1)[1 + 1/G1 + 1/G1G2 +... +1/G1G2...GN-1] = (1/NP1)[1 + 1/K1 + 1/K2 +... +1/KN-1]
where K1, K2 ... are the Gain Factors over 1, 2 etc. months.
Hence CD = P1 H (1,K1, K2, ... KN-1)
where H is the Harmonic Mean
of the cumulative Gain Factors.

If CV is the VA Cost per Unit, then:

 CV = PN [ 1 - {(N-1)/2}g] = P1 KN-1 [ 1 - {(N-1)/2}g]
where g = 2/{(N-1)N} [g1 + 2 g2 + ... + (N-1) gN-1] is the Weighted Average of monthly Gains.

Question:

Is CV < CD ???

Yes, provided:

g > {2/(N-1)} [1 - H / KN-1]
which is often true, especially for increasing unit prices (see sample charts at the right where we assume the unit price gains are Normally distributed with Mean = 10%, Standard Deviation = 30% and investments are made once per year).

Note that (see the charts) the DCA Cost per Unit is always lower than the Average Unit Price.

The VA Cost per Unit may ... or may not ... be lower than the Avg Price.

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?
Investors, presumably, are more interested in their returns, and the above charts show that DCA may be better, or worse, than VA ... when it comes to Annualized Returns.

 DCA versus VA ... Annualized Returns
Suppose, for DCA, we invest \$A each year.
To determine the Annualized Return for DCA, we must solve for R from the equation:
A(1+R)M + A(1+R)M-1 + ... + A

= current portfolio (after M years) = A[1/P1+1/P2+...+1/PM]PM

where M is the number of years and the first \$A investment grows to A(1+R)M after M years
and the second \$A investment grows to A(1+R)M-1 after M-1 years etc. etc.
... and, after M years, we have a portfolio worth A[1/P1+1/P2+...+1/PM]PM = M A PM / H(P1, P2, ...PM)

Suppose, for VA, we require that our portfolio grow by \$B each year.
To determine the Annualized Return for VA, we must solve for R from the equation:
B(1+R)M + {B-B(P2/P1-1)}(1+R)M-1 + {B-2B(P3/P2-1)}(1+R)M-2 + ... + {B-(M-1)B(PM/PM-1-1)}
= current portfolio (after M years) = MB

where the first \$B investment grows to B(1+R)M after M years and the second \${B-B(P2/P1-1)} investment grows to {B-B(P2/P1-1)} (1+R)M-1 after M-1 years etc. etc. (See Table 1 and following, for the periodic investments.)

If we set x = 1+R, the two equations become:

DCA:       D(x) = xM + xM-1 + ... + x + 1 = M PM / H   (the solution is 1 + Annualized Return)

where H = M/[1/P1+1/P2+...+1/PM] is the Harmonic Mean.

VA:       V(x) = xM + (1-g1)xM-1 + (1-2g2)xM-2 + (1-3g3)xM-3 + ... + (1-(M-1)gM-1) = M
or         V(x) = D(x) - { g1xM-1 +2g2xM-2 +3g3xM-3 + ... +(M-1)gM-1 } = M
or         V(x) = D(x) - G(x) = M   (the solution is 1 + Annualized Return)

where G(x) = g1xM-1 +2g2xM-2 +3g3xM-3 + ... +(M-1)gM-1
and g1 = P2/P1 - 1, g2 = P3/P2 - 1, ... are the Gains for month 1, month 2, etc.
(which may be positive or negative)

Here's a typical example, where the annual gains (over 20 years) are randomly selected
... from a normal distribution with Mean = 10%, Standard Deviation = 30%:

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Here are more examples.
Where the blue curves cross, that gives the Annualized Gain for DCA.
Where the green curves cross, that gives the Annualized Gain for VA.

Occasionally, the Annualized Gain for DCA is greater than for VA ... but not often!

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