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A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:. DURATION IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME PERIODS, t, WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE.

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A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

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  1. A PURE DISCOUNT BOND DOES NOT PAY CUPONS UNTIL ITS MATURITY; C = 0:

  2. DURATION IS THE WIEGHTED AVERAGE OF COUPON PAYMENTS’ TIME PERIODS, t, WEIGHTED BY THE PROPORTION THAT THE DISCOUNTED CASH FLOW, PAID AT EACH PERIOD, IS OF THE CURRENT BOND PRICE.

  3. DURATION INTERPRETED AS A MEASURE OF THE BOND PRICE SENSITIVITY

  4. The negative sign merely indicates that D changes in opposite direction to the change in the yield, r. Next we present a closed form formula to calculate duration of a bond:

  5. Coupon Rate

  6. DURATION OF A BOND PORTFOLIO V = The total bond portfolio value Pi = The value of the i-th bond Ni = The number of bonds of the i-th bond in the portfolio Vi = Pi Ni = The total value of the i-th bond in the portfolio V = ΣPiNi The total portfolio value. We now prove that: DP = ΣwiDi .

  7. is the weighted average of the durations of the bonds in the portfolio. The weights are the proportions the bond value is of the entire portfolio value.

  8. Example: Consider a portfolio of two T-bonds: D= (.1673)(10.4673) +(.8327)(12.4674) D = 12.1392

  9. IMMUNIZING BANK PORTFOLIO OF ASSETS AND LIABILITIES TIME 0ASSETSLIABIABILITIES $100,000,000 $100,000,000 (LOANS) (DEPOSITS) D = 5 D = 1 r = 10% r = 10% TIME 1 r => 12% BUT IF DA = DL THEY REACT TO RATES CHANGES IN EQUAL AMOUNTS. THE BANK PORTFOLIO IS IMMUNIZED , i.e., IT’S VALUE WILL NOT CHANGE FOR A “small” INTEREST RATE CHANGE, IF THE PORTFOLIO’S DURATION IS ZERO or: DP = DA - DL = 0.

  10. A 5-YEAR PLANNING PERIOD CASE OF IMMUNIZATION IN THE CASH MARKET BONDCFVMrDP A $100 $1,000 5 yrs 10% 4.17 $1,000 B $100 $1,000 10 yrs10% 6.76 $1,000 4.17WA + 6.76WB = 5 WA + WB = 1 WA = .677953668. WB = .322046332. VP = $200M implies: Hold $135,590,733.6 in bond A, And $64,409,266.4 in bond B. Next, assume that r rose to 12%. The portfolio in which bonds A and B are held in equal proportions will change to:

  11. [1 - 4.17 (.02/1.1)] 100M = $92,418,181.2 [1 - 6.76 (.02/1.1)] 100M = $87,709,090.91 TOTAL = $180,127,272.7 INVEST THIS AMOUNT FOR 5 YEARS AT 12% y CONTINUOUSLY COMPOUNDED YIELDS: $328,213,290. ANNUAL RETURN OF: The weighted average portfolio changes to: AFTER 5 YEARS AT 12%: $331,267,162. ANNUAL RETURN OF:

  12. Before turning to the futures markets we elaborate on the common practice of: Repurchase Agreements An integral part of trading T-bills and T-bill futures is the market for repurchase agreements, which are used in much of the arbitrage trading in T-bills. In a repurchase agreement -- also called an RP or repo -- one party sells a security (in this case, T-bills) to another party at one price and commits to repurchase the security at another price at a future date. The buyer of the T-bills in a repo is said to enter into a reverse repurchase agreement., or reverse repo. The buyer’s transactions are just the opposite of the seller’s. The figure below demonstrates the transactions in a repo.

  13. Transactions in a Repurchase Agreement Date 0 - Open the Repo: T- Bill Party A Party B PO Date t - Close the Repo T-Bill Party A Party B P1= P0(1+r0,t ) Example: T-bill FV = $1M. P0 = $980,000. The repo rate = 6%. The repo time: t = 4 days. P1= P0 [(repo rate)(n/360) + 1] = 980,000[(.06)(4/360) + 1] = 980,653.33

  14. A repurchase agreement effectively allows the seller to borrow from the buyer using the security as collateral. The seller receives funds today that must be paid back in the future and relinquishes the security for the duration of the agreement. The interest on the borrowing is the difference between the initial sale price and the subsequent price for repurchasing the security. The borrowing rate in a repurchase agreement is called the repo rate. The buyer of a reverse repurchase agreement receives a lending rate called the reverse repo rate. The repo market is a competitive dealer market with quotations available for both borrowing and lending. As with all borrowing and lending rates, there is a spread between repo and reverse repo rates.

  15. The amount one can borrow with a repo is less than the market value of the security by a margin called a haircut. The size of the haircut depends on the maturity and liquidity of the security. For repos on T-bills, the haircut is very small, often only one-eighth of a point. It can be as high as 5% for repurchase agreements on longer-term securities such as Treasury bonds and other government agency issues.Most repos are held only overnight, so those who wish to borrow for longer periods must roll their positions over every day. However, there are some longer-term repurchase agreements, called term repos, that come in standardized maturities of one, two, and three weeks and one, two, three, and six months.Some other customized agreements also are traded.

  16. INTEREST RATE FUTURES The three most traded interest rate futures are: TREASURY BILLS (CME) $1mil; pts. Of 100% EURODOLLARS (CME) $1mil; pts. Of 100% TREASURY BONDS (CBT) $100,000; pts. 32nds of 100%

  17. CONTRACT SPECIFICATIONS FOR: 90-DAY T-BILL; 3-Month EURODOLLAR FUTURES SPECIFICATIONS 13-WEEK 3-Month EURODOLLAR US T-BILL TIME DEPOSIT SIZE $1,000,000 $1,000,000 CONTRACT GRADE new or dated T-bills CASH SETTLEMENT with 13 weeks to maturity YIELDS DISCOUNT ADD-ON HOURS(Chicago time) 7:20 AM-2:00PM 7:20 AM - 2:00PM DELIVERY MONTHS MAR-JUN-SEP-DEC MAR-JUN-SEP-DEC TICKER SYMBOL TB EB MIN. FLUCTUATION .01(1 basis pt) .01(1 basis pt) IN PRICE ($25/pt) ($25/pt) LAST TRADING DAY The day before the 2nd London business day first delivery day before 3rd Wednesday DELIVERY DATE 1st day of spot month Last day of trading on which 13-week T-bill is issued and a 1-year T-bill has 13 weeks to maturity

  18. Transactions in a Cash-and-Carry Arbitrage. PO (MONEY) Repo Market Arbitrageur T-Bill Short Position F0,t PO (MONEY) T-Bill T-Bill Dealer Futures Market Date 0 Transactions in a Cash-and-Carry Arbitrage. P0(1+r0t) Repo Market Arbitrageur T-Bill Receive F 0,t Deliver T-Bill F 0,t > P0(1+r0,t) Futures Market Date t

  19. Transactions in a Reverse Cash-and-Carry Arbitrage. PO (MONEY) Repo Market Arbitrageur T-Bill Long Position F0,t T-Bill P0 T-Bill Dealer Futures Market Date 0 Transactions in a Reverse Cash-and-Carry Arbitrage P0(1+r0,t) Repo Market Arbitrageur T-Bill Take Delivery T-Bill Pay F 0,t F 0,t < P0(1+r0,t) Futures Market Date t

  20. Transactions in a Cash-and-Carry Arbitrage PO =$954,330.56 Repo Market Arbitrageur 182-day T-Bill Short Position FOt = $976,011.75 182-day T-Bill P0= $954,330.56 T-Bill Dealer Futures Market Date 0 Transactions in a Cash-and-Carry Arbitrage P0(1+r0,t) = $975,561.14 Repo Market Arbitrageur 91 day T-Bill Deliver 91-day T-Bill F 0,t = $976,011.75 Profit = $450,61 Futures Market Date t

  21. Transactions in a Reverse Cash-and-Carry Arbitrage PO = $954,330.56 Repo Market Arbitrageur 182-day T-Bill Long Position FO,t = $973,809.04 182-day T-Bill PO = $954,330.56 T-Bill Dealer Futures Market Date 0 Transactions in a Reverse Cash-and-Carry Arbitrage P1 = $975,561.13 Repo Market Arbitrageur 91 day T-Bill Take Delivery 91-day T-Bill F 0,t = $973,809.04 PROFIT = $1,752.09 Futures Market Date t

  22. LET THE YIELD ON THE SHORT-TERM BILL BE 8%: THEORETICAL RATE = 11.74% IS LESS THAN 12.50% = ACTUAL MARKET RATE REVERSE CASH-AND-CARRY

  23. To minimize dV/dr with respect to N, set: Next, we use the following substitutions for

  24. Recall that:

  25. Normally, the ratios of the yields sensitivities to the interest rate, r, are assumed to be zero. Thus: This is the price sensitivity hedge ratio.

  26. EURODOLLAR FUTURES These are futures on the interest earned on Eurodollar three-month time deposits. The rate used is LIBOR - London Inter-Bank Offer Rate. These time deposits are non transferable, thus, there is no delivery! Instead, the contracts are CASH SETTLED.

  27. Arbitrage with Eurodollar Futures

  28. Quasi Arbitrage or, how to borrow capital using Eurodollar Futures.

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