# 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.

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.