Locking in a Forward rate
Given the price of zero-coupon bond maturing at \( T_{1} \) (i.e the price now of receiving a risk-free \( $1 \) at time \( T_{1} \)), denoted by \( B_{1}^{T} \), and the price of a zero coupon bond maturing at \( T_{2} \), denoted by \( B_{2}^{T} \),what is the implied annual interest rate spanning the period \( T_{1} \) to \( T_{2} \). Show that this forward interest rate can be locked in now at no cost by a combination of positions on the above two zero coupon bonds.
Assuming continuous compounding, a zero coupon bond with maturity at T years will have a present value of \( e^{-rT} \) where r is the annual interest rate for the given period.
Let \( r_{1} \& r_{2} \) be the spot rates for times \( T_{1} \& T_{2} \) respectively and let \( F_{1}^{2} \) be the forward rate from \( T_{1} \) to \( T_{2} \). We have
\( B_{1}^{T} = e^{-r_{1}T_{1}} \)
\( B_{2}^{T} = e^{-r_{2}T_{2}} \)
Let us look at the investing \( B_{2}^{T} \) over a period of time \( T_{2} \). We now consider making this investment in 2 different ways. We use the no-arbitrage principle to argue that the two investment methods should yield the same return.
The first strategy in the table below invests \( B_{2}^{T} \) in a zero coupon bond maturing at \( T_{2} \). When the bond matures we will receive \( 1 \) dollar.
The second strategy shown in the table invests \( B_{2}^{T} \) in a zero coupon bond maturing at \( T_{1} \) and rolls out the return on that bond for another period of \( T_{2} - T_{1} \) at the forward rate \( F_{1}^{2} \)
Time Bond 2 Bond 1 + Forward
inflow outflow inflow outflow
\( 0 \) \( B_{2}^{T} \) \( B_{2}^{T} \)
\( T_{1} \) \( B_{2}^{T}e^{r_{1}T_{1}} \) \( B_{2}^{T}e^{r_{1}T_{1}} \)
\( T_{2} \) 1 \( B_{2}^{T}e^{r_{1}T_{1}}e^{F_{1}^{2}(T_{2} - T_{1})} \)
We can argue that the two strategies should have the same outcome at the end or it will result in an arbitrage opportunity. We can use this equality to formulate the forward rate as below.
\( B_{2}^{T}e^{r_{1}T_{1}}e^{F_{1}^{2}(T_{2} - T_{1})} = 1 \implies e^{F_{1}^{2}(T_{2} - T_{1})} = e^{r_{2}T_{2} - r_{1}T_{1}} \implies F_{1}^{2} = \frac{r_{2}T_{2} - r_{2}T_{2}}{T_{2} - T_{2}} \)
Locking in a forward rate today means that we do not make any payment today and we enter into a contract to invest e^{-F_{1}^{2}} at time \( T_{1} \) and receive \( 1 \) at time \( T_{2} \).
Consider a portfolio of two zero coupon bonds \( Z_{1} \) expiring at time \( T_{1} \) and \( Z_{2} \) expiring at time \( T_{2} \). Purchase \( Z_{2} \) and finance it by selling \( Z_{1} \), which ensures we do not spend any money today. At time \( T_{1} \), \( Z_{1} \) matures and you will pay the principal for \( Z_{1} \) at maturity. At time \( T_{2} \), you receive \( 1 \). Look at the table below for the workings.
Time Purchase \( Z_{2} \) + Sell \( Z_{1} \)
inflow outflow Total Cash Flow
\( 0 \) \( B_{2}^{T} \) \( B_{2}^{T} \) = \( 0 \)
\( T_{1} \) \( 0 \) \( B_{2}^{T}e^{r_{1}T_{1}} \) = \( -e^{r_{1}T_{1} - r_{2}T_{2}} = -e^{-F_{1}^{2}(T_{2}-T_{1}} \)
\( T_{2} \) \( 1 \) \( 0 \) = \( 1 \)
As can be seen in the table above, the initial cash-flow is \( 0 \) and the cash flow at time \( T_{1} \) is \( e^{-F_{1}^{2}(T_{2} - T_{1})} \) and we have locked in the forward rate.
