Deterministic Rates and the Link Between Spot, Forward, and Bond Yields
Summary
The document explains how zero-coupon bond prices connect to instantaneous forward rates, short rates, and the bond’s yield. A bond price can be written as the exponential of the negative integral of forward rates over the bond’s term. Under risk-neutral pricing, it can also be written using the accumulated short rate. When the short rate is deterministic, the expectation in that pricing expression reduces to a deterministic discount factor.
Differentiating the bond price with respect to maturity shows that the forward rate observed at time t equals the future short rate at that date, given the deterministic-rate assumption. Thus the yield to maturity, expressed as an average over the term, is also the average of those short rates. This equivalence depends on rates being deterministic; the explanation does not establish the same relationship for stochastic rates or address details such as compounding conventions beyond the continuous form used.
Key ideas
- A zero-coupon bond price is determined by integrating instantaneous forward rates across its term.
- With deterministic short rates, risk-neutral discounting reduces to the exponential of the accumulated short rate.
- The forward rate observed at time t equals the short rate at the corresponding future date under the stated assumptions.
- The continuously compounded yield is the term average of both the forward rates and deterministic future short rates.
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# Bond Prices in terms of short and forward rates
# Bond Prices in terms of short and forward rates
Of course, a pure discount bond price $P(t,T)$ may be stated in terms of its yield $R(t,T)$ as $$ P(t,T) = e^{-R(t,T)(T-t)}. $$ Let's assume both the (instantaneous) short rate $r(t)$ and (instantaneous) forward rate $f(t,T)$ are deterministic functions. The relationships to discount bond prices are \begin{align} r(t) & = -\frac{\partial}{\partial T} \log P(t,t), \\ f(t,T) & = -\frac{\partial}{\partial T} \log P(t,T). \end{align} From this, it is clear that $$ P(t,T) = \exp\left(-\int_t^T f(t,u) \, du\right) \qquad (1). $$
On the other hand, the bond price is often stated in terms of risk-neutral expectations using the short rate, such as $$ P(t,T) = E_Q\left(\exp\left(-\int_t^T r(u) \, du\right) \mid \mathcal{F}_t\right), $$ and since I am assuming $r(t)$ is deterministic, it should be that $$ P(t,T) = \exp\left(-\int_t^T r(u) \, du\right) \qquad (2). $$ Comparing Eqns (1) and (2), it seems like $$ \int_t^T r(u) \, du = \int_t^T f(t,u) \, du. $$
Does this even make sense? Furthermore, since $P(t,T) = e^{-R(t,T)(T,t)}$, we would get $$ R(t,T) = \frac{1}{T-t}\int_t^T f(t,u) \, du = \frac{1}{T-t}\int_t^T r(u) \, du. $$ That is, the yield is both the average of the instantaneous forward rate (this is true), and the average of the instantaneous spot rate. Is this latter statement true?
## Answer by Gordon (score 1)
https://quant.stackexchange.com/a/21802
Since the interest rate is deterministic, for $t< u \le T$, \begin{align*} f(t, u) &= -\frac{\partial}{\partial u} \ln P(t, u)\\ &=-\frac{\partial}{\partial u} \ln \left(E\left(\exp\left(-\int_t^u r(s)\, ds \right) \mid \mathcal{F}_t\right) \right)\\ &=-\frac{\partial}{\partial u} \ln \left(\exp\left(-\int_t^u r(s)\, ds\right) \right)\\ &=\frac{\partial}{\partial u}\int_t^u r(s)\, ds \\ &= r(u). \end{align*} Consequently, \begin{align*} \frac{1}{T-t}\int_t^T f(t, u)\, du = \frac{1}{T-t}\int_t^T r(u)\, du. \end{align*}
## Answer by dm63 (score 0)
https://quant.stackexchange.com/a/21896
This says that if rates are deterministic, the spot rate follows the forward rates that are initially observed. Makes sense.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.