## LIBOR vs OIS forward rates

Some time ago, a single yield curve was used both for discounting and for projecting of cash flows when pricing fixed income securities. “Projecting” means application of martingale property of LIBOR forward rates by using current snapshot of LIBOR forward curve as if it will stay constant in future. The discount coefficients where calculated from forward LIBOR curve.
The following relation was often used to calculate the value of a floating leg of an interest rate swap:
$PV_{float}(t) = B(t,T_i) - B(t,T_n)$ (floating cash flows occur at $T_i, T_{i+1}, ..., T_n$)
The hedging argument was as follows:

• Long 1 ZCB maturing at $T_i$ and short 1 bond maturing at $T_n$
• At time $T_i$ invest \$1 received into LIBOR($T_i$)
• At time $T_{i+1}$ receive \$1 + year_fraction*LIBOR($T_i$) and invest 1\$ into LIBOR($T_i$)
• Keep doing …
• At time $T_n$ receive \$1 + year_fraction*LIBOR(\$latexT_{n-1}\$) and repay shorted bond with \$1.
• What you are left with is your floating leg.
There is a problem with this argument. LIBOR is only a published benchmark rate. It is not always true that that agents can easily borrow or invest at LIBOR; so it may not accurately measure the actual time value of money. OIS (overnight index swap) rate better measures time value of money for short time horizons. OIS curve is used for discounting while forward rate curve is used for “projecting”.

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