Question: A contract pays
$$ P(T,T+\tau) - K$$
at $T$, where $K$ is fixed and $P(\cdot,S)$ is the price of a $S$-maturity zero-coupon bond (ZCB).
What is $K$ for which the contract's time $t$ price is null?
Answer:
Replication pricing:
At time $t$, we go long one $T+\tau$-maturity ZCB and short $ P(t,T)^{-1}P(t,T+\tau)$ $T$-maturity ZCB's.
Time $t$ cost of this position is $0$ as:
$$ (-1)\cdot P(t,T+\tau) + P(t,T)^{-1}P(t,T+\tau)\cdot P(t,T) = 0. $$
At time $T$, as the shorted bond matures, we have a flow of
$$ - P(t,T)^{-1}P(t,T+\tau). $$
But we are also expecting $1$ dollar flow at $T+\tau$, whose price at time $T$ is:
$$ P(T,T+\tau). $$
Hence, the $t$ price of payout (at time $T$)
$$ P(T,T+\tau) - P(t,T)^{-1}P(t,T+\tau) $$
is $0$. This is of course exactly our contract with
$$ K = P(t,T)^{-1}P(t,T+\tau). $$
Pricing under $T$-forward measure:
$$V_t = P(t,T)\mathbf{E}^{T}_t[P(T,T+\tau) - K]$$
Setting $V_t$ to $0$ implies:
$$K = \mathbf{E}^{T}_t[P(T,T+\tau)]$$
As $P(t,T+\tau)$ is a traded asset, under $T$-forward measure, process
$$ \left(P(t,T)^{-1} P(t,T+\tau)\right)_{t\geq 0}$$
is a martingale, which leads to:
$$\mathbf{E}^{T}_t[P(T,T)^{-1} P(T,T+\tau)] = P(t,T)^{-1} P(t,T+\tau).$$
Due to $P(T,T)=1$, we have:
$$K = \mathbf{E}^{T}_t[P(T,T+\tau)] = P(t,T)^{-1}P(t,T+\tau)$$
Pricing under money market account measure:
$$V_t = \beta_t\mathbf{E}_t[\beta_T^{-1} (P(T,T+\tau) - K)]$$
Setting $V_t$ to $0$ implies:
$$K = \mathbf{E}_t[\beta_T^{-1}]^{-1}\mathbf{E}_t[\beta_T^{-1} P(T,T+\tau)]$$
$$ = P(t,T)\mathbf{E}_t\left[\beta_T^{-1} \mathbf{E}_T[\beta_T \beta_{T+\tau}^{-1} ] \right] $$
$$ = P(t,T)^{-1}\mathbf{E}_t\left[ \mathbf{E}_T[ \beta_{T+\tau}^{-1} ] \right] $$
$$ = P(t,T)^{-1}\mathbf{E}_t\left[ \beta_{T+\tau}^{-1} \right] $$
$$ = P(t,T)^{-1}P(t,T+\tau), $$
using tower property of conditional expectations in the penultimate equality.
(Note: not necessarily a recent question, but expected to be asked - I flunked the replication pricing part that the interviewer was obviously enamored with; this is covered by both Brigo/Mercurio's book, in the context of FRA pricing, and by Andersen/Piterbarg's book, forward bond price.)