Yield to Maturity (YTM) is the total return anticipated on a bond when it is held until its maturity date. It is essentially the internal rate of return (IRR) of the bond based on its cash flows, which include periodic coupon payments and the face value paid at the end of the bond’s term.
To compute the Yield to Maturity (YTM) for a fixed income security, we need to find the discount rate that makes the present value (PV) of its future cash flows equal to its current market price. Mathematically, this can be expressed as:
$$P = \sum_{i=1}^{n} \frac{C_i}{(1 + YTM)^{i}} + \frac{M}{(1 + YTM)^{n}}$$
Where:
- P is the current market price of the bond,
- n is the number of periods until maturity,
- Ci is the coupon payment received in period i,
- M is the face value of the bond, and
- YTM is the Yield to Maturity
The equation above is non-linear, and finding an analytical expression for the YTM is often difficult. Instead, you can use iterative numerical methods to solve for it, such as the Newton-Raphson method or a simple trial-and-error approach.
To better illustrate the process, let’s work with an example:
Suppose we have a bond with a face value of 1, 000, three years to maturity, an annual coupon rate of 5%, and a current market price of 960. We want to find its YTM. Using the equation above, we have:
$$960 = \frac{50}{(1 + YTM)^1} + \frac{50}{(1 + YTM)^2} + \frac{1050}{(1 + YTM)^3}$$
To solve for YTM, we can use the Newton-Raphson method, which iteratively refines the estimate through the following equation:
$$YTM_{k+1} = YTM_k - \frac{f(YTM_k)}{f'(YTM_k)}$$
Here,
- YTMk is the current estimate of YTM,
- YTMk + 1 is the next estimate of YTM, and
- f(YTMk) and f′(YTMk) are the function and its derivative evaluated at the current estimate.
In our case, the function f(YTM) is:
$$f(YTM) = \sum_{i=1}^{3} \frac{C_i}{(1 + YTM)^{i}} + \frac{M}{(1 + YTM)^{3}} - P$$
To apply the Newton-Raphson method, we also need the first derivative of f(YTM) with respect to YTM, which is:
$$f'(YTM) = -\sum_{i=1}^{3} \frac{iC_i}{(1 + YTM)^{i+1}} - \frac{3M}{(1 + YTM)^{4}}$$
By iteratively updating our estimate using the Newton-Raphson equation and starting with an initial guess for YTM, we’ll eventually converge to a solution.
In this example, starting with an initial guess of YTM0 = 0.05 (or 5%), after a few iterations, we’ll find that the YTM is approximately 6.34%.
Keep in mind that in practice, the Yield to Maturity may be subject to various assumptions, such as reinvestment of coupon payments at the YTM rate, or that the bond is held to maturity. These assumptions may not always hold true, which could lead to a deviation between estimated and realized returns.