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Math Help - Interest Rate Risk Help...

  1. #1
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    Cool Interest Rate Risk Help...

    Both Bond Sam and Bond Dave have 9 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 3 years to maturity, whereas Bond Dave has 19 years to maturity...

    If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Sam? Dave?

    How the heck do I do this? I'm so confused....


    I thought all I had to do was find the interest rate at 9% and then at 11% and just note the difference, but I came up with a 1% difference which was incorrect...
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  2. #2
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    Is this a problem given out by your math teacher?
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  3. #3
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    Quote Originally Posted by Ranger View Post
    I thought all I had to do was find the interest rate at 9% and then at 11% and just note the difference, but I came up with a 1% difference which was incorrect...
    What do you mean 'find interest rate' - which interest rate did you find exactly?
    I think you need to play with the relationship between the implied interest rate and the price of the bond (via yield to maturity, or "ytm"). If the implied interest rate were to rise by 2% in both cases, how will this impact the ytm ->price of each bond? Is that what you calculated?
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  4. #4
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    Yes, this is an extra credit assignment from my professor. We haven't learned how to do interest rates yet so yea =(


    I have no clue what I was doing, I read my chapter on this and they give no examples or anything..

    I thought:

    N= 3*2 = 6
    PMT= 45 (9%*1000=90/2)
    PV= -1000
    FV= 1000
    YTM= 4.5%

    So an increase of 2% would be 5.5%? Diffence = 1%...But that is 100% wrong =(
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  5. #5
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    Oh, I think I got it. No guarantee though )))

    At 9%, you know the bonds are sold at their face value and the coupon payment match exactly the current interest rate -> so PV of the bond (or their market value) equals their face value

    If interest rate rise to 11% (9+2), then the bonds will no longer be compensating fully for the time value of money with their 9% coupon payments -> their market value will go down. I think you know this already. How do we calculate their new market value? We discount the expected cash flows from the investment into the bond at the new discount rate, 11%, to arrive at new fair value (market value) - because 11% is the new market price of borrowing.

    I did this for Sam, see attached file. I got new market value for Sam of about 950, which is 5% down on the original price of 1000. I did it just by adding the cash flows from Sam and discounting them at 11% (or 11/2=5.5% in each half year period).

    You can of course apply annuity formula to calculate 19 year bond cashflows, I just did it from first principles because I don't remember the formula. Again, this is what I think, no guarantee that this is the right answer.

    same n dave.xls
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  6. #6
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    Assume $1,000 bonds; both have currently a PV of $1,000.

    Since I'm not sure what you're trying to do, I'll give you the PV's at 11% cpd semi-annually:
    SAM: ~950.04
    DAVE: ~841.95

    Enjoy !

    To Volga: formula for PV of payment flow:
    PV = p[1 - 1/(1 + i)^n] / i
    p = payment amount (45)
    i = interest rate (.055)
    n = number of payments (6 and 38)
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  7. #7
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    Oh cheers, I am happy to see that I got the same answer )))
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  8. #8
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    Sam & Dave Bond Price Calculation

    Quote Originally Posted by Ranger View Post
    Both Bond Sam and Bond Dave have 9 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 3 years to maturity, whereas Bond Dave has 19 years to maturity...

    If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Sam? Dave?

    How the heck do I do this? I'm so confused....


    I thought all I had to do was find the interest rate at 9% and then at 11% and just note the difference, but I came up with a 1% difference which was incorrect...
    I used an online Bond Price Calculation tool to provide the price calculation as listed below, the tool is found here Bond Price calculation - ThinkandDone.com

    BOND SAM PRICE CALCULATION WHEN YTM is 11%

    Compounding = semi annually
    Par Value = 1000
    Coupon Rate = 0.045
    Market Rate = 0.055
    N = 6

    Non Zero Bond Price Formula
    Coupon Rate x Par Value x PVIFA(ytm%, n) + Par Value x PVIF(ytm%, n)

    PVIFA Formula
    PVIFA(ytm%, n) = [1 - v] / ytm%
    v = 1 / (1 + ytm%)^n
    PVIFA(ytm%, n) = [1 - { 1 / (1 + ytm%)^n }] / ytm%

    PVIFA Calculation
    v = 1 / (1+0.055)^6
    v = 0.7252458330246
    PVIFA(0.055, 6) = [1 - 0.7252458330246] / 0.055
    PVIFA(0.055, 6) = 0.2747541669754 / 0.055
    PVIFA(0.055, 6) = 4.9955303086437

    PVIF Formula
    PVIF(ytm%, n) = 1 / (1 + ytm%)^n

    PVIF Calculation
    PVIF(0.055, 6) = 1 / (1+0.055)^6
    PVIF(0.055, 6) = 1 / 1.3788428067619
    PVIF(0.055, 6) = 0.7252458330246

    Non Zero Bond Price Calculation
    Price = 0.045 x 1000 x 4.9955303086437 + 1000 x 0.7252458330246
    Price = 224.79886388897 + 725.2458330246
    Price = 950.04

    BOND DAVE PRICE CALCULATION WHEN YTM is 11%

    Compounding = semi annually
    Par Value = 1000
    Coupon Rate = 0.045
    Market Rate = 0.055
    N = 38

    Non Zero Bond Price Formula
    Coupon Rate x Par Value x PVIFA(ytm%, n) + Par Value x PVIF(ytm%, n)

    PVIFA Formula
    PVIFA(ytm%, n) = [1 - v] / ytm%
    v = 1 / (1 + ytm%)^n
    PVIFA(ytm%, n) = [1 - { 1 / (1 + ytm%)^n }] / ytm%

    PVIFA Calculation
    v = 1 / (1+0.055)^38
    v = 0.1307394139635
    PVIFA(0.055, 38) = [1 - 0.1307394139635] / 0.055
    PVIFA(0.055, 38) = 0.8692605860365 / 0.055
    PVIFA(0.055, 38) = 15.804737927936

    PVIF Formula
    PVIF(ytm%, n) = 1 / (1 + ytm%)^n

    PVIF Calculation
    PVIF(0.055, 38) = 1 / (1+0.055)^38
    PVIF(0.055, 38) = 1 / 7.6488028336976
    PVIF(0.055, 38) = 0.1307394139635

    Non Zero Bond Price Calculation
    Price = 0.045 x 1000 x 15.804737927936 + 1000 x 0.1307394139635
    Price = 711.21320675713 + 130.7394139635
    Price = 841.95
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  9. #9
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    Quote Originally Posted by dexteronline View Post
    IBOND DAVE PRICE CALCULATION WHEN YTM is 11%

    Compounding = semi annually
    Par Value = 1000
    Coupon Rate = 0.045
    Market Rate = 0.055
    N = 38

    Non Zero Bond Price Formula
    Coupon Rate x Par Value x PVIFA(ytm%, n) + Par Value x PVIF(ytm%, n)

    PVIFA Formula
    PVIFA(ytm%, n) = [1 - v] / ytm%
    v = 1 / (1 + ytm%)^n
    PVIFA(ytm%, n) = [1 - { 1 / (1 + ytm%)^n }] / ytm%

    PVIFA Calculation
    v = 1 / (1+0.055)^38
    v = 0.1307394139635
    PVIFA(0.055, 38) = [1 - 0.1307394139635] / 0.055
    PVIFA(0.055, 38) = 0.8692605860365 / 0.055
    PVIFA(0.055, 38) = 15.804737927936

    PVIF Formula
    PVIF(ytm%, n) = 1 / (1 + ytm%)^n

    PVIF Calculation
    PVIF(0.055, 38) = 1 / (1+0.055)^38
    PVIF(0.055, 38) = 1 / 7.6488028336976
    PVIF(0.055, 38) = 0.1307394139635

    Non Zero Bond Price Calculation
    Price = 0.045 x 1000 x 15.804737927936 + 1000 x 0.1307394139635
    Price = 711.21320675713 + 130.7394139635
    Price = 841.95
    1000 / 1.055^38 + 45(1 - 1/1.055^38) / .055 = 841.95262...
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