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Unique positive solution

Suppose that \( y_i, C_i, F \) are non-negative reals, then prove that the following equation has a unique positive real solution for \( y \):

\[ \frac{ F } { ( 1 + y_T ) ^ T} + \sum_{ i = 1 } ^ T \frac{ C _ i } { ( 1 + y_i) ^ i} = \frac{ F } { ( 1 + y ) ^ T} + \sum_{ i = 1 } ^ T \frac{ C _ i } { ( 1 + y) ^ i} \]


Hence, conclude for bonds (with non-negative integers rates and coupon rates), there is a unique Yield to Maturity.

Note by Calvin Lin
2 years ago

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Let us write it as \(K=f(y)\). Now \(f(y)\) is continuous and monotone decreasing function of \(y\) and \(0=f(\infty)\leq K\leq f(0)\). By intermediate value theorem there exists one and by monotonicity, there exists only one solution of \(K=f(y)\) Abhishek Sinha · 2 years ago

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