# A ten year bond pays 11% interest on a \$1000 face value annually. If it currently sells for \$1195, what is i?

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The yield is essentially the IRR of the cash flow hence the rate of return in the form of 1 + rate / 100 is such that the following equation will be satisfied:

1195 = 0.11 * 1000 / R + 0.11 * 1000 / R^2 + 0.11 * 1000 / R^3 + ... + 0.11 * 1000 / R^10 + 1000 / R^10

As it's awkward to work with so many coupons being discounted to a present value, we can use the summation of a geometric sequence equation to rewrite this as:

1195 = 0.11 * 1000 * ( ( 1 - 1 / R^11 ) / ( 1 - 1 / R ) - 1 ) + 1000 / R^10

As this is a 11th order polynomial, it has to be solved by numerical methods such as Newton's, Secant or Binary, that is it has to be solved by trial and error. There are explicit solutions for lower order polynomials such as the quadratic equation for a 2nd order polynomial but it's generally accepted that there are only numerical method solutions for 5th order or higher polynomials. By binary (high and low guessing) we have:

R = 1.0808 which is 8.08%

The yield of a ten year bond with 11% per annum coupons paid annually but selling at a market price of \$1,195 is 8.08%

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