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  • astral_tsar astral_tsar Jan 20, 2004 9:26 AM Flag

    Value Estimate -- Comments?

    Fair enough, though perhaps the approach I used is more "unusual" in academic circles than in practice?

    What got me "riled up" is a very real content issue here. My objection w/r/t double-counting book value isn't nitpicking, it's the whole ball of wax. Growth comes from reinvestment. Here, I'll show you.

    The problem shows up in examples as simple as a savings account. Lemme contrast the "B" approach to a savings account ("A").

    B. Assume that I really do receive checks in the mail, with absolute certainty, beginning 1/1/04 and continuing on the first of the year forever. The payments are:
    1/1/04 $1
    1/1/05 $1.05
    1/1/06 $1.05^2 = $1.1025
    1/1/07 $1.05^3,
    1/1/08 ...and so on.

    Assume a risk-free discount rate r = 6%. What's the present value of those checks?

    Well, net present value turns out to be

    $1/(r-g) = $1/(6%-5%) = $100.

    Wow. Great deal.

    A. Now look deeper. Where might such checks come from? Consider a bank account that contains $20 and pays g=5% interest on the first day of every year. I leave the interest IN the savings account and let it compound.

    Here are the interest payments deposited in the account:
    1/1/04: $1
    1/1/05: $1.05
    1/1/06: $1.1025
    1/1/07: ...and so on

    Growth "g" is 5%. Exactly the same as the checks. I'm rich! But there's a catch: now I've assumed some real-world mechanism for *producing* the checks. So what's the net present value of leaving my money in that savings account?

    Less than $20.

    Right? The account loses the race against the 6% discount rate by 1% per year. I'm losing value. If I leave it there forever under these conditions, the value is zero.

    Ok, what if I want checks mailed to me? Well, the values of those checks are determined by the 5% interest rate:
    1/1/04: $1
    1/1/05: $1
    1/1/06: $1
    1/1/07: ...and so on.

    Present value:

    $1 / 6% = $16.67.

    Still less than $20. I never take the principal out, remember. All I have is the checks. But now they don't grow!

    So if I have total control over the savings account, how close can I get to the $100 value of case A? $20. I can withdraw all the money now.

    C. How come?
    Very simply from the ROI picture. What is the approximate present value of a $1000 10-year treasury bond that pays 4%? The present value is $1000! That's how they price these things!

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