Why Expected Yield to Maturity Differs from Yield on Expected Price
Summary
The document considers how to summarize a callable bond valued across stochastic interest-rate paths. It compares averaging the yield to maturity calculated separately for each scenario with first averaging scenario prices and then finding the yield implied by that expected price. These operations generally produce different results because yield to maturity is a nonlinear transformation of price and scenario cash flows.
The answer recommends treating expected price as the more direct economic quantity, then deriving a yield from it if a single yield is required. This approach is also described as less computationally demanding than calculating and averaging a yield for every path. The distinction matters when interpreting reported metrics: an average of scenario yields answers a different question from the yield implied by the average price. The document gives a conceptual argument and a convexity illustration, but no general formula for the size or direction of the difference across all bonds and scenarios.
Key ideas
- Yield to maturity is nonlinear in price, so the yield of an average price need not equal the average of scenario yields.
- Scenario-based valuation can produce a separate bond price and yield for each interest-rate path.
- Expected price is a direct monetary measure and can be calculated before deriving an implied yield.
- A yield implied by expected price and an average of pathwise yields represent different summaries.
- The recommended summary depends on the economic question being asked.
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# Aggregate Yield to Maturity with Stochastic Interest Rate Paths
# Aggregate Yield to Maturity with Stochastic Interest Rate Paths
Suppose I am valuing a callable bond using stochastic interest rate paths (LMM generated for example) and I wish to express yield to maturity as a single value.
Would it be appropriate to average the cash flows at each time point and determine the yield to maturity for those cash flows (less computationally expensive) or would it be more appropriate to find a yield to maturity for each string of cash flows and average the yield to maturity (more computationally expensive)?
## Answer by Attack68 (score 2, accepted)
https://quant.stackexchange.com/a/77968
Suppose you had two curve scenarios, $C_1$ and $C_2$ with probabilities, $p_1$ and $p_2$, then the two ytms under each scenario are:
$$ y_1 = y_1(P_1(C_1)), \; y_2 = y_2(P_2(C_2)) $$
Where the prices of the bond ($P$) are the discounted cashflows under the curves.
Your expected ytm is:
$$E[y]= p_1y_1 + p_2y_2$$
Your question asks if this is equal to evaluating the ytm of the expected prices, i.e.,
$$ E[y] = y(p_1P_1(C_1) + p_2P_2(C_2)) $$
It is not.
The ytm function is not homogeneous and it is non-linear, so that,
$$py(P) \ne y(pP)$$
It is quite easy to find a combination of yields and prices such that:
$$ p_1y_1(P_1) + p_2y_2(P_2) \ne y(p_1P_1 + p_2P_2) $$
proving it is not true in general, and more expensive computational resources are needed in your framework.
But
Price is better economic metric, it represents an explicit amount of cash payable for an instrument. YTM is a transformation (and not a great one) so determining the expected yield is probably not a good metric. I would perform the analysis to derive an expected price. From the expected price you can imply the ytm of the expected price. In my opinion,
$$ \text{ytm of expected price} = y(p_1P_1 + p_2P_2) $$
is better than the,
$$ \text{expected ytm} = p_1y_1(P_1) + p_2y_2(P_2) $$
And less resources are needed for this in your framework.
This is quite easy to visualise given the convexity of the ytm curve relative to price. In the scenario below 3.95 is the ytm of expected price, whereas 4.0 is the expected ytm: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.