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Thread: Averaging down

  1. #1
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    Averaging down

    I need help figuring out how to find the number of share to purchase to bring down an average to a certain number.

    I have 1,200 shares of xyz at \$9.34 and need to know how many to purchase at \$8.5 to bring down the average of all shares to \$9.00.

    How would I write a formula to solve for this and be able to interchange the variables as well.

    Thanks in advance.
    Last edited by skeeter; Aug 22nd 2017 at 01:49 PM.
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  2. #2
    SGS
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    Re: Averaging down

    Quote Originally Posted by NumbersDontLie View Post
    I need help figuring out how to find the number of share to purchase to bring down an average to a certain number.

    I have 1,200 shares of xyz at \$9.34 and need to know how many to purchase at \$8.5 to bring down the average of all shares to \$9.00.

    How would I write a formula to solve for this and be able to interchange the variables as well.

    Thanks in advance.

    Let X = # of shares to buy at $8.5. Total shares after purchase will then be 1200 + X.

    1200 * 9.34 + 8.5 * X - (1200 + X) * 9

    Solve for X!

    Steve
    Last edited by skeeter; Aug 22nd 2017 at 01:50 PM.
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    Re: Averaging down

    Quote Originally Posted by SGS View Post
    Let X = # of shares to buy at $\$$8.5. Total shares after purchase will then be 1200 + X.

    1200 * 9.34 + 8.5 * X - (1200 + X) * 9

    Solve for X!

    Steve
    That minus sign should be an equals sign. To find the average, you take the sum of the price of all of the shares (you have 1200 at $\$$9.34 and x at $\$$8.50) and divide by the number of shares, which is 1200+x (as Steve suggested), and that average is $\$$9:

    $\dfrac{9.34\cdot 1200 + 8.5x}{1200+x} = 9$

    Multiply both sides by 1200+x to get Steve's equation (but change the minus sign to an equals sign).
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  4. #4
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    Re: Averaging down

    Quote Originally Posted by SGS View Post
    Let X = # of shares to buy at $8.5. Total shares after purchase will then be 1200 + X.

    1200 * 9.34 + 8.5 * X - (1200 + X) * 9

    Solve for X!

    Steve
    Thanks, I solved by simplifying (?) the equation (?) :

    11208 + 8.5x - (10799 + 9x)
    11208 + 8.5x - 10799 - 9x
    11208 - 10799 + 8.5x - 9x
    408 - 0.5x

    0 = 408 - 0.5x

    I added the " 0 = " but was it a given that it was always there when solving for one variable?

    -408 = -0.5x

    dividing both sides by -.05 to isolate x

    -408/-0.5 = x

    dividing two negative numbers for some reason seems weird to me to get the final answer, did I do something wrong or not optimal with my thinking process?

    816 = x

    .:. I need to buy 816 shares at $8.5 to get a net average price of $9 given that I already have 1,200 shares at $9.34.

    ---

    Is there a way to explain in English how I would come along this
    Quote Originally Posted by SGS View Post
    1200 * 9.34 + 8.5 * X - (1200 + X) * 9
    to solve for problems like mine in the future?

    Also how do I place dollar signs on this forum without it formatting everything together $123 word space word $456
    Last edited by NumbersDontLie; Aug 22nd 2017 at 03:03 PM. Reason: i posted this before seeing slipeternal's reply
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  5. #5
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    Re: Averaging down

    The formula is

    $p = h * \dfrac{c - t}{t - m}, \text {where}$

    $\text p = \text { number of shares to PURCHASE;}$

    $\text h = \text { number of shares currently HELD;}$

    $\text c = \text { average COST per share of shares currently held;}$

    $\text t = \text { TARGET average cost per share after purchase; and}$

    $\text m = \text { current MARKET price per share.}$

    Proof

    $t = \dfrac{ch + mp}{h + p} \implies ht + pt = ch + mp \implies$

    $pt - mp = ch - ht \implies p(t - m) = h * (c - t) \implies p = h * \dfrac{c - t}{t - m}.$

    Let's try it in your example.

    $p = 1200 * \dfrac{9.34 - 9.00}{9.00 - 8.50} = 1200 * \dfrac{0.34}{0.50} = 816.$

    Let's check.

    Your total cost after purchase = $1200 * 9.34 + 816 * 8.50 = 11208 + 6936 = 18144.$

    Your number of shares after purchase = $1200 + 816 = 2016.$

    The new average cost per share = $\dfrac{18144}{2016} = 9.00.$

    And that was your target. You may not always get an exact answer because you cannot buy fractional shares.
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  6. #6
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    Re: Averaging down

    Quote Originally Posted by JeffM View Post
    The formula is

    $p = h * \dfrac{c - t}{t - m}, \text {where}$

    $\text p = \text { number of shares to PURCHASE;}$

    $\text h = \text { number of shares currently HELD;}$

    $\text c = \text { average COST per share of shares currently held;}$

    $\text t = \text { TARGET average cost per share after purchase; and}$

    $\text m = \text { current MARKET price per share.}$

    Proof

    $t = \dfrac{ch + mp}{h + p} \implies ht + pt = ch + mp \implies$

    $pt - mp = ch - ht \implies p(t - m) = h * (c - t) \implies p = h * \dfrac{c - t}{t - m}.$

    Let's try it in your example.

    $p = 1200 * \dfrac{9.34 - 9.00}{9.00 - 8.50} = 1200 * \dfrac{0.34}{0.50} = 816.$

    Let's check.

    Your total cost after purchase = $1200 * 9.34 + 816 * 8.50 = 11208 + 6936 = 18144.$

    Your number of shares after purchase = $1200 + 816 = 2016.$

    The new average cost per share = $\dfrac{18144}{2016} = 9.00.$

    And that was your target. You may not always get an exact answer because you cannot buy fractional shares.
    First, thanks for your knowledge and time.

    Secondly, trying to commit this to memory might be beyond the scope of this forum but...
    Is there a way to summarize or better explain why we divide (c - t) by (t - m) or just a generalized statement in layman/english terms about the right side of the equation?

    I get that it works for my example and that it also works for different values but committing it to memory without a further explanation is a bit difficult for me.

    Quote Originally Posted by SlipEternal View Post
    To find the average, you take the sum of the price of all of the shares (you have 1200 at $\$$9.34 and x at $\$$8.50) and divide by the number of shares, which is 1200+x, and that average is $\$$9:

    $\dfrac{9.34\cdot 1200 + 8.5x}{1200+x} = 9$

    Multiply both sides by 1200+x to get Steve's equation (but change the minus sign to an equals sign).
    I think the above best explains things for me currently in english and I could reproduce it if I came to the situation again, but now I am wondering about the other explanation

    ------

    Also what is the purpose of proofing? Please excuse my ignorance/lack of knowledge on the subject. (I did see/get that breaking down and rearranging the formula did work based what I learned in algebra class though)
    Last edited by NumbersDontLie; Aug 22nd 2017 at 04:23 PM. Reason: answering my own questions
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  7. #7
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    Re: Averaging down

    Can I explain the formula intuitively. Sort of. (c - t) is how far you want to average down. In your example, we want to average down by 9.34 - 9.00 or 0.34.
    (t - m) is how close to current market you want to get. In your example, it is 9.00 - 8.50 or 0.50. The ratio of 0.34 over 0.50 is about 2/3. So we have to buy about 2/3 of what we already have. And 816 is close to 2/3 of 1200. I do not know whether that helps you feel that the formula matches up with your intuition.

    I do not recommend memorizing many formulas. It is a waste of energy, and frequently you remember the wrong formula. What is more important is to remember how I got the formula. I started with defining the relevant variables and showed a common sense but general computation for calculating a new average cost namely

    The historical cost per share of the old shares times the number of old shares PLUS the market price per share times the number of new shares purchased. That gave me the total cost of old and new shares together. Obvious, no? And how many total shares do we have after buying the new shares? Obviously the number of old shares plus the number of new shares. Again, obvious. So the average cost per share is the total cost divided by the total number of shares. Nothing esoteric.

    Now we play with algebra to solve for the number of shares to be bought. That is totally mechanical. No real thinking involved.
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