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Yield to Maturity as an Implicit Discount Rate

Article Quant Q&A · Author: Phibert
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Summary

The document introduces yield to maturity as the single continuously compounded rate that equates a bond’s market price with the present value of its fixed payments. It clarifies that each payment time marks when a coupon or other cash flow is received, while the time difference in the pricing equation measures how far that payment lies from the valuation or issue date. An example uses monthly coupon dates to illustrate the elapsed-time convention.

The response explains that the rate generally cannot be isolated by rearranging the equation and compares the calculation to finding an internal rate of return. It suggests solving numerically with a spreadsheet goal-seeking tool. The explanation is brief: it gives neither a numerical solution nor details about root-finding, compounding conventions, accrued interest, or how multiple possible roots should be handled.

Key ideas

  • Yield to maturity is the rate that discounts a bond’s scheduled payments to its market price.
  • Each payment date identifies when a cash flow is received.
  • The exponent uses the elapsed time from valuation or issue to each payment date.
  • The yield is generally found numerically, similarly to an internal rate of return.

Tags

Fixed incomeDerivatives pricingStatistics
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# Yield to Maturity


# Yield to Maturity












For a bond with market price $P_t$ and fixed payments $c_n$, I'm told the yield to maturity is given by the solution $Y$ to the equation

$P_t=\sum_{n=1}^N c_n e^{-Y(t_n-t)}$.

Firstly, I'm not great a rearranging such equations to not sure how to find an expression for $Y$ from here.

Also could someone explain what each $t_n$ is? As in, in the equation what's the difference between the fixed $t$ and the $t_n$?

## Answer by BCLC (score 0, accepted)

https://quant.stackexchange.com/a/11233

I think $t_n$ is the time the nth coupon is paid and $t_n$-t is the time difference between the time the coupon is paid at the time the bond is issued.

So if a bond is issued on May 10 and coupons are paid on June 10, July 10 and August 10, then the $t_n$ - t's are 1/12, 2/12 and 3/12.

Y cannot be solved directly. It's like Internal Rate of Return. Use Goal Seek in Excel.

https://ph.answers.yahoo.com/question/index?qid=20081123061034AA419b4 YTM: http://www.youtube.com/watch?v=NV0S8pKvje8 Goal Seek: http://www.youtube.com/watch?v=7QIBvMPC9vM

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.