Infinite-Maturity Zero-Coupon Bond Yields and Discount-Price Limits
Summary
The document considers the limiting yield and price of a zero-coupon bond as maturity tends to infinity. For a flat, constant rate, the discount price tends to zero when the rate is positive, making the limiting yield itself less important to the price limit. With a maturity-dependent curve, the limiting behavior instead depends on how the yield varies with maturity.
It points to long-term curve construction in insurance, including the Ultimate Forward Rate and Smith–Wilson approach, and discusses Nelson–Siegel-style parameterization. In that model, the long-maturity yield approaches its level parameter when the decay parameter is positive; the bond price then tends to zero if that limiting yield is positive. The discussion is conceptual rather than a full derivation, and the zero-price conclusion depends on the stated rate conditions. The original question and answers do not establish a universal limiting yield for every curve specification.
Key ideas
- A constant positive yield makes a zero-coupon bond's price tend to zero as maturity grows.
- For a maturity-dependent curve, the asymptotic yield depends on the curve's functional form.
- The Nelson–Siegel specification approaches its level parameter at long maturities under the stated parameter condition.
- An Ultimate Forward Rate and Smith–Wilson fitting are cited as tools for long-term curve construction.
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Full text
# What is the yield on an infinitely lived ZCB?
# What is the yield on an infinitely lived ZCB?
I guess the price of a Zero-Coupon Bond with infinite maturity should go to zero, what about its yield? I am asking this because I was dealing with the yield curve and its asymptotic properties when $t\to\infty$
## Answer by crunch (score 2)
https://quant.stackexchange.com/a/16765
This is something that banks don't do very well (in my opinion), but we can look to the insurance industry for help.
- Insurance liabilities often span decades, and the regulation has come up with something called the Ultimate Forward Rate (or UFR). It's currently a hotly debated topic with the advent of Solvency II (insurance regulation) coming into effect on 01/01/2016. This is because the UFR is not always set with an eye to long term interest rates, but more by looking at an appropriate liability discount rate.
- The insurance industry preferred curve fitting approach is the Smith-Wilson model, which has the UFR as an input.
Hopefully this is a useful starting point for your research.
In the end, the actual value of the yield of an infinitely lived bond is irrelavant. As long as your infinite-year forward rate is reasonable (i.e. not $ \infty $), then $\lim_{t \rightarrow \infty} e^{-rt} = 0$ anyway.
## Answer by user25064 (score 2)
https://quant.stackexchange.com/a/16766
while it is true that $$\lim_{T\to\infty} Z(t, T) = \lim_{T\to\infty} e^{-r(T-t)} = 0$$ this is when $r$ is independent of time to maturity, a flat and constant yield curve. In practice, we use yield curves which vary depending on what day they are estimated and what maturity the ZCB is. If in fact $r(t, T)$ depends on today and the maturity then the properties of that function are going to determine what the limit is. Of course, any model that allows for a non-zero price for an infinite maturity ZCB is admitting arbitrage.
Commonly, the Nelson Siegel and Nelson Siegel Svensson (original paper) models are used, in that case $$ r(t, T) = \beta_0 + \beta_1{1-\exp(-(T-t)/\tau)\over(T-t)/\tau} + \beta_2\left({1-\exp(-(T-t)/\tau)\over(T-t)/\tau}-\exp(-(T-t)/\tau)\right)$$
In the case of this model $\lim_{T\to\infty}r(t, T) = \beta_0$ whenever $\tau > 0$ and so $$\lim_{T\to\infty}Z(t, T) = \lim_{T\to\infty} e^{-r(t, T)(T-t)}=0$$ whenever $\beta_0 > 0$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.