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.

#QuantitativeFinance

Easy Math Editor

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Comments

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 - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$