A 2-year bond has an annual coupon which is an integer percentage between 1% and 99% (e.g. 3%, 17%, but not 7 1/2%). The bond’s price is quoted as a percent of par (e.g. 93, 108, but not 98.5). The bond’s yield is an exact positive rational number.

If the bond is trading at a premium (price above 100), what is the least the price could be?

If the bond is trading at a discount (price below 100), what is the least the price could NOT be?

This will require a spreadsheet unless you thought of something cleverer than I did.

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For those who don’t know how bonds work: you pay a price P for the bond, then after one year you get the first coupon interest payment of C, then after two years you get the second coupon interest payment of C plus the principal payment of 100. The interest rate which makes the present value of those payments equal the price is the yield Y, which is usually an irrational number; this problem investigates when it can be an exact rational number.

This post has gotten a few hits but no one has tackled it. If Y is the yield, then we have the equation P = C/(1+Y) + (100+C)/((1+Y)^2).

Rewriting X = 1/(1+Y) we have the quadratic equation (100+C)X^2 +CX -P=0. This has a rational root when the discriminant (c^2 + 4*P*(100+C)) is a perfect square.

Putting this formula into a spreadsheet and comparing square roots with their truncations, we get that the smallest premium price with a rational yield is 108 when the coupon is 75%, and the yield is then 66 2/3%. (Paying 75 discounted by a factor of 3/5 followed by 175 discounted by a factor of 9/25 has a present value of 45 + 63 = 108.)

The smallest discount price which does not have a rational yield solution for any coupon 1%-99% is 27. For a price of 27, the first coupon which gives a rational yield is 156% corresponding to payments of 156 discounted by a factor of 9/64 and 256 discounted by a factor of 81/4096, giving 351/16 + 81/16 = 432/16 = 27. This corresponds to a yield of 611 1/9%.

BONUS QUESTION: Are there any values of the price (expressed as an integer percent) for which no coupon (expressed as an integer percent) gives a rational positive yield?

Answer to Bonus Question: I can prove that 2-year bonds priced at 101 and 102 never have a rational yield for any coupon, but for larger values of P, the coupon C=P(P-102)+1 gives a rational yield. This question wasn’t interesting enough, but I think I can come up with a better one about bond math.