David
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David asked in Education & ReferenceHomework Help · 1 decade ago

Bond Redemption is the name of this question. Help, please!?

The board of directors of Maven Co. agrees to redeem some of its bonds in two years. At that time, $1,102,500 will be required. Suppose the firm presently sets aside $1,000,000. At what annual rate of interest, compounded annually, will this money have to be invested in order that its future Value be sufficient to redeem the bonds?

Solution: Let r be the required annual rate of interest. At the end of the first year the accumulated amount will be $1,000,000 plus the interest, 1,000,000r, for a total of

1,000,000 + 1,000,000r = 1,000,000(1+r)

Under the compound interest, at the end of the second year the accumulated amount will be 1,000,000(1+r) plus the interest of this, which is 1,000,000(1+r)r. Thus, the total value at the end of the second year will be

1,000,000(1+r) + 1,000,000(1+r)r

This must equal $1,102,500:

1,000,000(1+r) + 1,000,000(1+r)r = 1,102,500

Since 1,000,000(1+r) is a common factor of both terms on the left side, we have

1,000,000(1+r)(1+r) = 1,102,500 <--how did it factor like that?

1,000,000(1+r)^2 = 1,102500 <-- shouldn't (1+r)(1+r) equal 1+r+r+r^2 or 1+2r+r^2 instead of just

(1+r)(1+r)

sorry if that's a lot of reading for an answer, but this is it copied straight out of the book. not clicking for me and financial algebra is the WORST for me. I'd much rather work with parabolas.

1 Answer

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  • Anonymous
    1 decade ago
    Favorite Answer

    Your initial problem required finding the rate at which $1 Million dollar will have a future value of $1,102,500 in 2 years

    Finding future value with this formula

    FV = PMT. (1+r)^n

    FV/PMT = (1+r)^n

    (FV/PMT)^1/n = 1 + r

    r = [(FV/PMT)^1/n] - 1

    r = [1,102,500 / 1,000,000]^1/2 -1

    r = [1.1025]^0.5 -1

    r = 1.05 - 1

    r = 0.05

    r = 5%

    FV = 1,000,000 (1.05)^2

    FV = 1,000,000 (1.1025)

    FV = 1,102,500

    Now back to your algebra question

    1,000,000(1+r) + 1,000,000(1+r)r = 1,102,500

    1,000,000(1+r) is common in the first term and the second terms

    think of dividing both sides of the equation by 1,000,000(1+r)

    [1,000,000(1+r)+1,000,000(1+r)r]/1,000,000(1+r) = 1,102,500/1,000,000(1+r)

    1,000,000(1+r)/1,000,000(1+r) + 1,000,000(1+r)r/1,000,000(1+r) = 1,102,500/1,000,000(1+r)

    1 + r = 1,102,500 / 1,000,000(1+r)

    (1 + r) = 1,102,500 / 1,000,000(1+r)

    Now multiplying both side with (1+r)

    (1 + r)(1 + r) = 1,102,500 / 1,000,000

    No Multiplying both side with 1,000,000

    1,000,000(1 + r)(1 + r) = 1,102,500

    We can rewrite this as

    1,000,000(1 + r)(1 + r) = 1,102,500

    1,000,000(1 + r)^1 (1 + r)^1 = 1,102,500

    1,000,000(1 + r)^1+1 = 1,102,500

    1,000,000(1 + r)^2 = 1,102,500

    (1 + r)^2 is the same as what you said r^2+2r+1

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