Derivation of yield curve construction formula with tenor basis spread

If LIBOR is truly risk-free, the swap of 3-month LIBOR and 6-month LIBOR should be tradable with zero spread. However, LIBOR takes into account the credit risk of banks, and tenor swaps are exactly the transactions that swap the risk of 3-month LIBOR and 6-month LIBOR. After the financial crisis, the risk of bankruptcy, that is, the difference in tenor, was conscious, and when swap 3-month LIBOR and 6-month LIBOR, the 3-month LIBOR side began to trade with a non-negligible spread.

After all, after supplementing the discount factor and tenor spread in 0.25 year increments, the future 3-month LIBOR can be calculated as follows (Nakahara, 2013).

\begin{eqnarray} L_{forward}^{3MWorld}(t,t_{i},t_{i+0.25})=\frac{1-DF(t,t_{i+0.25})-TS^{3M-6M}(t,t_{i+0.25})\times\sum_{j=1}^{4i+1}{DF(t,t_{\frac{j}{4}})\times\frac{1}{4}}}{DF(t,t_{i+0.25})\times\frac{1}{4}}+\frac{-\sum_{j=1}^{4i}{L_{forward}^{3MWorld}(t,t_{\frac{j-1}{4}},t_{\frac{j}{4}})\times DF(t,t_{\frac{j}{4}})\times\frac{1}{4}}}{DF(t,t_{i+0.25})\times\frac{1}{4}}． \end{eqnarray}

However,

  • $L(t,T)$: Observable LIBOR at time point $t$ maturity $T$
  • $L_{forward}^{3MWorld}(t,t_{0},t_{0.25})=L(t,t_{0.25})$
  • $L_{forward}^{3MWorld}(t,t_{i},t_{i+0.25})$: A 3-month LIBOR with a reset date of $t_{i}$ and a maturity of $t_{i+0.25}$ as seen from the point in time $t$
  • $TS^{3M-6M}(t,T)$: Tenor spread when swapping 6-month LIBOR for 3-month LIBOR at time $t$ and maturity $T$

In this article, we try to derive the equation (1). At this time, the evaluation formula of the JPY tenor swap is expressed as follows (Takahashi, 2010)．

\begin{eqnarray} \sum_{n=1}^{N}\delta_{n}(E_{t}[L(T_{n-1},T_{n}+\tau_{N}]P_{t,T_{n}})=\sum_{m=1}^{M}\delta_{m}E_{t}[L(T_{m-1},T_{m}]P_{t,T_{m}}． \end{eqnarray}

However,

  • $N=2M$
  • $L(T_{n-1},T_{n})$: LIBOR with a reset date of $T_{n-1}$ and a maturity of $T_{n}$
  • $P_{t,T_{n}}$: Zero coupon bond price with maturity of $t$ at time $ T_{n}$
  • $\delta_{n}$: Interest calculation period
  • $\tau_{N}$: Tenor spread when swapping 6-month LIBOR for 3-month LIBOR at time $t$ and maturity of $T (= T_{N} = T_{M})$

Now, we want to derive 3 months LIBOR, that is, $E_{t}[L(T_{N-1}, T_{N}]$, therefore expand the left side of the expression (2) as follows.

\begin{eqnarray} \sum_{n=1}^{N}\delta_{n}(E_{t}[L(T_{n-1},T_{n}+\tau_{N}]P_{t,T_{n}})=\sum_{n=1}^{N-1}\delta_{n}(E_{t}[L(T_{n-1},T_{n})]P_{t,T_{n}}+\delta_{N}E_{t}[L(T_{N-1},T_{N})]P_{t,T_{N}}+\sum_{n=1}^{N}\delta_{n}\tau_{N}P_{t,T_{n}}． \end{eqnarray}

On the other hand, the relationship between Libor and the discount rate is

\begin{eqnarray} E_{t}[L(T_{m-1},T_{m})]=\frac{1}{\delta_{n}}\left(\frac{P_{t,T_{m-1}}}{P_{t,T_{n}}}-1\right)． \end{eqnarray}

Since it is represented by, the left side of the expression \label{tenor} can be expanded as follows.

\begin{eqnarray} \sum_{m=1}^{M}\delta_{m}E_{t}[L(T_{m-1},T_{m}]P_{t,T_{m}}=P_{t,T_{0}}-P_{t,T_{M}}=1-P_{t,T_{M}}． \end{eqnarray}

After all, JPY 3 months LIBOR, ie $E_{t} [L(T_{N-1},T_{N}]$, is given below.

\begin{eqnarray} E_{t}[L(T_{N-1},T_{N}]=\frac{1-P_{t,T_{M}}-\sum_{n=1}^{N}\delta_{n}\tau_{N}P_{t,T_{n}}}{\delta_{N}P_{t,T_{N}}}+\frac{-\sum_{n=1}^{N-1}\delta_{n}(E_{t}[L(T_{n-1},T_{n})]P_{t,T_{n}}}{\delta_{N}P_{t,T_{N}}}． \end{eqnarray}

Reference

  • Nakahara, G.，2013. LIBOR and OIS discount that can be seen in EXCEL (In Japanese)
  • Fujii, M.，Shimada, Y. and Takahashi, A.，2010. “A Note on Construction of Multiple Swap Curves with and without Collateral”， FSA Research Review Vol.6(March 2010)．