Why Term Bonds Carry Interest-Rate Risk Premiums
Summary
The document explains why a fixed-maturity zero-coupon bond is not locally risk-free even when its promised cash flow is known. In a one-factor short-rate model, the money account earns the prevailing short rate through continuous rollover, while a term bond’s market price changes as expectations about future short rates shift. A decline in rates can raise a bond’s price because the discounted value of its fixed future payment rises; rate changes can also move the price in the other direction.
This unpredictable mark-to-market movement exposes a bondholder to interest-rate risk, providing intuition for why its expected return need not equal the instantaneous short rate. The question also states that a standard model implies a common local risk premium per unit of interest-rate exposure across maturities. The answer gives intuition rather than a derivation, and it does not specify the size or sign of the premium in a particular model or market.
Key ideas
- A term bond’s promised payment can be fixed while its current market price remains uncertain.
- Changes in expected future short rates alter the discounting of a bond’s cash flow and therefore its price.
- The money account rolls over at the current short rate, unlike a term bond whose value responds to rate movements.
- The model described associates the same local risk premium per unit of rate exposure with bonds of different maturities.
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Full text
# Short-rate models: Risk-premium of $T$-bonds
# Short-rate models: Risk-premium of $T$-bonds
Following "Arbitrage Theory in Continuous Time" by Thomas Bjork, a standard one-factor short-rate model is of the form \begin{align*} dr_t = \mu(t,r_t)dt + \sigma(t,r_t)dW_t. \end{align*} The only exogeneously given asset is then the locally risk-free money account with dynamics \begin{align*} dB_t = r_tB_tdt, \quad\text{ or }\quad B_t = e^{\int_0^tr_sds}, \end{align*} which can be shown to be equivalent to investing in a self-financing rolling over trading strategy, which at each time $t$ consists entirely of bonds with maturity at $t+dt$.
Now, from the $r_t$ dynamics one can derive the dynamics for the price of a $T$-bond, $p(t,T)$: \begin{align*} \frac{dp(t,T)}{p(t,T)} = \alpha_T(t)tdt + \sigma_T(t)dW_t, \end{align*} and the punchline is that for any maturities $S>t$ and $T>t$: \begin{align*} \frac{\alpha_T(t)-r_t}{\sigma_T(t)}=\frac{\alpha_S(t)-r_t}{\sigma_S(t)}, \end{align*} which means that the local risk-premium for exposing yourself to interest-rate risk is the same whether you invest in a bond with maturity $S$ or maturity $T$.
I understand the mathematics, but I am looking for an intuitive answer to the following question:
Why is there a risk-premium for investing in $T$-bonds?
Why don't we have $\alpha_T(t)=r_t$? Intuitively I feel like the $T$-bonds are just as locally risk-free as the money account, which, as mentioned above, is just like investing continuously in bonds that are about to mature.
## Answer by spaceisdarkgreen (score 2, accepted)
https://quant.stackexchange.com/a/35830
Imagine you hold a zero coupon bond with a certain maturity $T$ and the short rate follows a process like you specified.
You might know deterministically what the cash bond pays this period, but you don't know how the interest rate itself is going to change. If the interest rate goes down, then the expectation of future rates goes down and the expected amount of interest you'd receive rolling the cash account to time $T$ goes down, so the term discount factor goes down and the term bond price goes up.
Thus your bond gains (or loses) value in an unpredictable way this period, depending on what interest rates do. So there is local risk in owning a bond.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.