# 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
3 years, 3 months 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 $$K=f(y)$$

- 3 years, 3 months ago