Personal Investments • VBTLX Price in response to interest rate cuts

Nope. You use curve fitting for anything that has more than 1 cash inflow or 1 cash outflow.

The yield/rate of a bond/loan is the 1 interest rate where you can discount all cash outflows (e.g., coupon/interest payments and principle payments) back to the original value of the cash inflows.

With the emboldened text you just defined the current bond price without the need for curvefitting in the definition.

It is important to distinguish what to compute from the methods used to compute it. If you could derive a closed form expression analytically for what you defined, why would you need curvefitting to compute it?

OK, what is the closed deterministic formula that allows you to calculate the bolded part? To my knowledge, there is none.

Let me give you a trivial problem. A 10 year bond, purchased at a par value of $100, with a annual $5 coupon. The answer is obvious but what are the deterministic steps to calculate it?

It's actually an important mathematical result that there is no closed form. Suppose that the current value of the bond is Y, and let X be the interest rate you need to compute. Then you need to solve

Y = 5/(1+X) + 5/(1+X)^2 + ... + 5/(1+X)^9 + 105/(1+X)^10

for X. Multiplying through by (1+X)^10 gives a tenth-degree polynomial in X, or in (1+X) if you prefer. And it is a mathematical theorem that there is no closed-form solution for a polynomial equation of degree more than 4. (For degree 2, there is the quadratic formula, and there are formulas for degrees 3 and 4. But even with three terms, you only get a quadratic if the payments are equally spaced, such as a bond which makes payments exactly one and two years from today; to value such a bond on any future day gives a formula with non-integer powers which requires numerical methods.)

Statistics: Posted by rossington — Thu Jun 27, 2024 5:26 am — Replies 45 — Views 4780