【正文】 B. f. Portfolio of forwards: A swap (semiannual interest exchanges) can be viewed as a sequence of forwards with maturities: t1,t2,…,tn with a mon forward price. Define Pt(?) as the timet value of zerocoupon bond maturing at time ? for $1 face value. For ti?tti+1, 1. At ti+1: kk*, evaluated at t, (kk*)Pt(ti+1) 2. At ti+2: (ti+1)Q, evaluated at t, PVt,t(i+2){(ti+1)Q}=[k^(ti+1,ti+2)Q]exp[r(t,ti+2)(ti+2t)], where R^(ti+1,ti+2) is the forward rate (semiannual pounding) at time t over [ti+1,ti+2]. Why? 3. Similarly for ti+3,ti+4,… 4. The total value of the swap at time t: (kk*)exp(r(t,ti+1)(ti+1t)+?nj=i+1[k^(tj,tj+1)Q]exp[r(t,tj+1)(tj+1t)] Example: Continue the previous example R^(3m,9m)=2{exp[*(**)/()]1}=% R^(9m,15m)=2{exp[*(**)/()]1}=% V=()*+(4**100)*+ (**100)*= ? Variation of interest rate swaps Index amortized swaps: the notional principal is reduced over the life of the swap. Constant yield swaps: both parts are floating. For example, one part may be linked to the yield on the 30year Tbond and the other may be linked on the 10year Tnote. Ratecapped swaps: floating rate is capped. Putable and Callable swaps: one or both counterparties have the right to cancel the swap at certain times without additional costs. Forward swaps: the swap rate is set but the swap does not mence until a later date. ? Currency swaps Two parties exchange periodic interest payments and principals in two currencies. Example: Both A and B need to borrow USD50 million (or DEM equivalent of 84 million based on ) for threeyear. The financing rates facing them are summarized as follows: It is paratively cheaper for A to use the DEM debt. For B, USD borrowing will be cheaper. Why? 1. If A desires the DEM debt and B prefers the USD debt, there is no need for them to engage in a swap. 2. If A desires the USD debt and B prefers the DEM debt, A should still borrow DEM and B borrow USD. They can enter a currency swap to better both parties. USD DEM A B % % % % a. Interest payment flows %USD Company ??? Company ??? | |??? %DEM A ??? B %USD %DEM b. Initial principal flow 84m DEM Company ??? Company ??? | | ??? 84DEM A ??? B 50m USD 50m USD b. Terminal principal flow 84m DEM Company ??? Company ??? | | ??? 84DEM A ??? B 50m USD 50m USD b. Company A: Borrows DEM debt and enters the above swap. c. Company B: Borrows USD debt and enters the above swap. d. The results: 1. Company A: Beginning: Exchange DEM84 million for USD50 million, a fair transaction at the current exchange rate (DEM168/USD1). Inbetween: On a semiannual basis, receives *% and pays USD25m*% due to the swap, and pays DEM42m*% due to its DEM debt. The payment is USD25m*%+DEM42m*%, paring to USD25m*%. End: Exchange USD50m for DEM84m, not a fair exchange at the prevailing exchange rate. 2. Company B: Beginning: Exchange USD50 million for DEM84 million, a fair transaction at the current exchange rate (DEM168/USD1). Inbetween: On a semiannual basis, receives USD25m*% and pays *% due to the swap, and pays USD25m*% due to its USD debt. The payment is DEM42m*%, which is less than DEM42m*%, End: Exchange DEM84m for USD50m, not a fair exchange at the prevailing exchange rate. Swap through an intermediary %$ %$ Company ??? Swap ?? Company ??? | |??? %DM A ??? Dealer ??? B 9%USD %DM %DM The results 1. Company A: Beginning: Exchange DEM84 million for USD50 million, a fair transaction at the current exchange rate (DEM168/USD1). Inbetween: On a semiannual basis, receives *% and pays USD25m*% due to the swap, and pays DEM42m*% due to its DEM debt. The payment is USD25m*%, which is less than USD25m*%. End: Exchange USD50m for DEM84m, not a fair exchange at the prevailing exchange rate. 2. Company B: Beginning: Exchange USD50 million for DEM84 million, a fair transaction at the current exchange rate (DEM168/USD1). Inbetween: On a semiannual basis, receives USD25m*% and pays *% due to the swap, and pays USD25m*% due to its USD debt. The payment is DEM42m*%, which is less than DEM42m*% End: Exchange DEM84m for USD50m, not a fair exchange at the prevailing exchange rate 3. Swap dealer: On a semiannual ba