5.4. If an investor borrows$100 at 10% for one year and then invests the money at 10.5% for two years, the result is a cashoutflow of 100e°! = $110.52 at the end of year | and an inflow of 100e"'°5*? = $123.37 at the endof year 2. Because 123.37 = 110.52e!', a return equal to the forward rate (11%) is earned on$110.52 during the second year. Suppose next that the investor borrows $100 for four years at 11%and invests it for three years at 10.8%. The result is a cash inflow of 1006" 1083 _ $138.26 at theend of the third year and a cash outflow of 100e™!!** = $155.27 at the end of the fourth year.Because 155.27 = 138.26e"!'°, money is being borrowed for the fourth year at the forward rateof 11.6%.
Equation (5.1) can be written
R
 
omwnNea
‘i
25 7)TY THT,
This shows that if the zero curve is upward sloping between 7, and 7), so that R; > R,, thenRr > R>. Similarly, if the zero curve is downward sloping with R, < Ry, then Rr < Ro.100 CHAPTER 5
 
Taking limits as T, approaches 7; in equation (5.2) and letting the common value of the two beT, we obiainaR
Rr =R+T
where R is the zero rate for a maturity of T. The value of R- obtained in this way is known as the
instantaneous forward rate for a maturity of T. This is the forward rate that is applicable to a veryshort future time period that begins at time T.
5.6 FORWARD RATE AGREEMENTS
 
 
A forward rate agreement (FRA) is an over-the-counter agreement that a certain interest rate willapply to a certain principal during a specified future period of time. In this section we examine howforward rate agreements can be valued in terms of forward rates.
Consider a forward rate agreement in which it is agreed that a financial institution will earn aninterest rate Rx for the period of time between 7, and T, on a principal of L. Define:
R;: The forward LIBOR interest rate for the period between times 7, and 7,R: The actual LIBOR interest rate observed at time 7, for a maturity T,
We will depart from our usual assumption of continuous compounding and assume that the ratesRx, Re, and R are all measured with a compounding frequency reflecting their maturity. Thismeans that if 7, — T; = 0.5, they are expressed with semiannual compounding; if T, — T, = 0.25,they are expressed with quarterly compounding; and so on. The forward rate agreement is anagreement to the following two cash flows:?
Time 7: -—LTime Ty: +L[1 + Rx(, -T))I
To value the FRA, we first note that it is always worth zero when Ry = Ret This is because, asnoted in the previous section, a large financial institution can at no cost lock in the forward rate fora future time period. For example, it can ensure that it earns the forward rate for the time periodbetween years 2 and 3 by borrowing a certain amount of money for two years and investing it forthree years. Similarly, it can ensure that it pays the forward rate for the time period between years 2and 3 by borrowing a certain amount of money for three years and investing it for two years.We can now use an argument analogous to that used in Section 3.8 to calculate the value of theFRA for values of Rx other than Ry. Compare two FRAs. The first promises that the forward rateR, will be earned on a principal of L between times T; and 7>; the second promises that Rx will beearned on the same principal between the same two dates. The two contracts are the same exceptfor the interest payments received at time 75. The excess of the value of the second contract over
> In practice, an FRA such as the one considered is usually settled in cash at time T,. The cash settlement is the
present value of the cash flows or14+ Re(h - Th)
[+RR—-T)
* it is usually the case that Rx is set equal to Rr when the FRA is first initiated.Interest Rate Markets 101
the first is therefore the present value of the difference between these interest payments, orL(Rg — Rp (Tz — Tie?
where R> is the continously compounded zero rate for a maturity 7>.> Because the value of theFRA promising Rr is zero, the value of the FRA promising Rx is
V = L(Ry — Re NT> — Te 2” (5.3)
When an FRA specifies that the interest rate Rx will be paid rather than received, its value issimilarly given by
V = L(Rp— ReV(T, — Te? (5.4)
Example 5.1. Suppose that the three-month LIBOR rate is 5% and the six-month LIBOR rate is5.5% with continuous compounding. Consider an FRA where we will receive a rate of 7%, measuredwith quarterly compounding, on a principal of $1 million between the end of month 3 and the end ofmonth 6. In this case, the forward rate is 6% with continuous compounding or 6.0452% withquarterly compounding. From equation (5.3) it follows that the value of the FRA is
1,000,000 x (0.07 — 0.060452) x 0.25 x e70055*05 _ $9 399
Alternative Characterization of FRAs
Consider again an FRA that guarantees that a rate Rx will be earned between times 7, and 7). Theprincipal L can be borrowed at rate R at time 7, and repaid al time 7). When this transaction iscombined with the FRA, we see that the FRA is equivalent to the following cash flows:
Time 7,: 0Time 7T,: L{l + R(T — T))]Time 7,: —L[l + R(T, ~ T))]
 
L(Rx — RYT -T))
at time 7). The FRA is equivalent to an agreement where at time 7> interest at the predeterminedrate, Rx, is received and interest at the market rate, R, is paid. Comparing this result with equation(5.3), we see that the FRA can be valued by assuming that R = R,- and discounting the resultantcash flows at the risk-free rate. We have therefore shown:
1. An FRA is equivalent to an agreement where interest at a predetermined rate, Rx, isexchanged for interest at the market rate, R.
2. An FRA can be valued by assuming that forward interest rate is certain to be realized.These two results will be useful when we consider interest rate swaps in the next chapter.
5 Note that Rx, R, and R- are expressed with a compounding frequency corresponding to 7, — 1), whereas R, isexpressed with continuous compounding.102 CHAPTER 5
5.7 THEORIES OF THE TERM STRUCTURE
 
 
It is natural to ask what determines the shape of the zero curve. Why is it sometimes downwardsloping, sometimes upward sloping, and sometimes partly upward sloping and partly downwardsloping? A number of different theories have been proposed. The simplest is expectations theory,which conjectures that long-term interest rates should reflect expected future short-term interestrates. More precisely, it argues that a forward interest rate corresponding to a certain future periodis equal to the expected future zero interest rate for that period. Another idea, segmentation theory,conjectures that there need be no relationship between short-, medium-, and long-term interestrates. Under the theory, a major investor such as a large pension fund invests in bonds of a certainmaturity and does not readily switch from one maturity to another. The short-term interest rate isdetermined by supply and demand in the short-term bond market; the medium-term interest rate isdetermined by supply and demand in the medium-term bond market; and so on.
The theory that is in some ways most appealing is liquidity preference theory, which argues thatforward rates should always be higher than expected future zero rates. The basic assumptionunderlying the theory is that investors prefer to preserve their liquidity and invest funds for shortperiods of time. Borrowers, on the other hand, usually prefer to borrow at fixed rates for longperiods of time. If the interest raies offered by banks and other financial intermediaries corres-ponded to expectations theory, long-term interest rates would equal the average of expected futureshort-term interest rates. In the absence of any incentive to do otherwise, investors would tend todeposit their funds for short time periods, and borrowers would tend to choose to borrow for longtime periods. Financial intermediaries would then find themselves financing substantial amounts oflong-term fixed-rate loans with short-term deposits. Excessive interest rate risk would result. Inpractice, in order to match depositors with borrowers and avoid interest rate risk, financialintermediaries raise long-term interest rates relative to expected future short-term interest rates.This strategy reduces the demand for long-term fixed-rate borrowing and encourages investors todeposit their funds for long terms.
Liquidity preference theory leads to a situation in which forward rates are greater than expectedfuture zero rates. It is also consistent with the empirical result that yield curves tend to be upwardsloping more often than they are downward sloping.
5.8 DAY COUNT CONVENTIONS
 
 
We now examine the day count conventions that are used when interest rates are quoted. This is aquite separate issue from the compounding frequency used to measure interest rates, which wasdiscussed in Section 3.3. The day count defines the way in which interest accrues over time.Generally, we know the interest earned over some reference period (e.g., the time between couponpayments), and we are interested in calculating the interest earned over some other period.
The day count convention is usually expressed as X/Y¥. When we are calculating the interestearned between two dates, X defines the way in which the number of days between the two dates iscalculated, and Y defines the way in which the total number of days in the reference period ismeasured. The interest earned between the two dates is
Number of days between dates
—  —  * Interest earned in eriNumber of days in reference period nierest © reference periodInterest Rate Markets 103
Three day count conventions that are commonly used in the United States are:
1, Actual/actual (in period)2. 30/3603. Actual/360
Actual/actual (in period) is used for U.S. Treasury bonds, 30/360 is used for U.S. corporate andmunicipal bonds, and actual/360 is used for U.S. Treasury bills and other money marketinstruments.
The use of actual/actual (in period) for Treasury bonds indicates that the interest earned betweentwo dates is based on the ratio of the actual days elapsed to the actual number of days in the periodbetween coupon payments. Suppose that the bond principal is $100, coupon payment dates areMarch | and September 1, and the coupon rate is 8%. We wish to calculate the interest earnedbetween March 1 and July 3. The reference period is from March | to September 1. There are184 (actual) days in this period, and interest of $4 is earned during the period. There are124 (actual) days between March 1 and July 3. The interest earned between March | and July 3is therefore
124
Te4 x 4 = 2.6957
The use of 30/360 for corporate and municipal bonds indicates that we assume 30 days per monthand 360 days per year when carrying out calculations. With 30/360, the total number of daysbetween March | and September | is 180. The total number of days between March 1 and July 3 is(4 x 30) + 2 = 122. In a corporate bond with the same terms as the Treasury bond just considered,the interest earned between March 1 and July 3 would therefore be
1242711180 *
The use of actual/360 for a money market instrument indicates that the reference period is 360 days.The interest earned during part of a year is calculated by dividing the actual number of elapsed daysby 360 and multiplying by the rate. The interest earned in 90 days is therefore exactly one-fourth ofthe quoted rate. Note that the interest earned in a whole year of 365 days is 365/360 times thequoted rate.
Fa nVed VUELTA
 
 
The price quoted for an interest-bearing instrument is often not the same as the cash price youwould pay if you purchased it. We illustrate this by considering the way in which prices are quotedfor Treasury bonds and Treasury bills in the United States.
Bonds
Treasury bond prices in the United States are quoted in dollars and thirty-seconds of a dollar. Thequoted price is for a bond with a face value of $100. Thus, a quote of 90-05 indicates that thequoted price for a bond with a face value of $100,000 is $90,156.25.104 CHAPTER 5
The quoted price is not the same as the cash price that is paid by the purchaser. In general,Cash price = Quoted price + Accrued interest since last coupon date
To illustrate this formula, suppose that it is March 5, 2001, and the bond under consideration is an11% coupon bond maturing on July 10, 2009, with a quoted price of 95-16 or $95.50. Becausecoupons are paid semiannually on government bonds, the most recent coupon date is January 10,2001, and the next coupon date is July 10, 2001. The number of days between January 10, 2001,and March 5, 2001, is 54, whereas the number of days between January 10, 2001, and July 10,2001, is 181. On a bond with $100 face value, the coupon payment is $5.50 on January 10 andJuly 10. The accrued interest on March 5, 2001, is the share of the July 10 coupon accruing to thebondholder on March 5, 2001. Because actual/actual in period is used for Treasury bonds, this is54
The cash price per $100 face value for the July 10, 2001, bond is therefore$95.5 + $1.64 = $97.14Thus, the cash price of a $100,000 bond is $97,140.
Treasury Bills
As already mentioned, the actual/360 day count convention is used for Treasury bills in the UnitedStates. Price quotes are for a Treasury bill with a face value of $100. There is a difference betweenthe cash price and quoted price for a Treasury bill. If Y is the cash price of a Treasury bill that hasa face value of $100 and » days to maturity, the quoted price is
360(100 — ¥)Ar
This is referred to as the discount rate. It is the annualized dollar return provided by the Treasurybill expressed as a percentage of the face value. If for a 91-day Treasury bill the cash price, Y,were 98, then the quoted price would be (360/91) x (100 — 98) = 7.91.
The discount rate or quoted price is not the same as the rate of return earned on the Treasurybill. The latter is calculated as the dollar return divided by the cost. In the example just given, therate of return would be 2/98, or 2.04% per 91 days. This amounts to
2 3658 x 91 = 0.08186
or 8.186% per annum with compounding every 91 days.
5.10 TREASURY BOND FUTURES
 
 
Table 5.5 shows interest rate futures quotes as they appeared in the Wall Street Journal on March16, 2001. The most popular long-term interest rate futures contract is the Treasury bond futurescontract traded on the Chicago Board of Trade (CBOT). In this contract, any government bondInterest Rate Markets 105
 
Table5.5 Interest rate futures quotes from the Wall Street Journal on March 16, 2001.(Columns show month, open, high, low, settle, change, lifetime high, lifetime low, andopen interest, respectively.)
 
   
Jane 106-08 106-17 105-24 10504 + 3 1067 96-21 arsed |
  
Sept 105-15 106-02 106-12 105-22 + 2 106-04 9622 2618 | ire
Dee 9342 9845 93.95 9340 + 09 G6 -03 360INTEREST RATE ve CSS ait 2 tf co ieTrossury Bonds (CBT) $100,000: ots 32nd of 100% ont So: 2 un -2 MtMar 106-15 106-26 105-03 10814 + 3 106-30 o-08  avuar | Se > 2 ee We i+ |+ |+ |
seeees'BPRVRBRKSS:
 
    
   
Est vol 225(00; vol Wed 22 320 open at SSH, +61 Sent “ 2604Treasury Notes (a Ste 100; pts 32nds of 100% Dec ~ ~ BE a 2897Mar 06405 Tors 10618 06-395 +" 85 107-0! 9604” 52833 Est vol 792400: vol Wed 950,073: open int 4,336,283, +26,797,RT Oe EE IAS tAS IG By) SRRR) | Euro-Yen (CME}-Yen 100,000,000; nts of 100% eenEst vol 226,000; vol Wed 278,130; open int 553,652, +13,722. OPEN HIGH LOW SETTLE CHANGE AGH LOW10 tr Agene Notes (CB1?-$100,000; pts g2nde of 100% Mar 98.77 90.77 98.77 877 -— Of 9979 9607 30,170Hr 1OIS0 deve Oraw teeor Ta" Nez ahs | 10264 dune 9367 9988 9387 967 - Of 9989 9620 15401dune 101-17 102-02 101-14 101-27 + 115 02-105 96-22 46,569 87 9988 987 ga 99051208Fst vol 4000; vol Wed 900; open in $8,833, “+573. got Be Be ge ee - 1 Se ae ee5 Yr Trea Notes (CBT)-$£00,000; pts 32nds of 100% woe 9984 9085 9994 9985 - Gf 9985 9797 4Mar 105-05 05-155 105-01 05-115 + 80 05-155 100-11 36,713 dune 9893 9984 9983 9983 01 9985 9787 = t65dune 105-15 05-245 105-09 106-20 + 95 05-245 101-04 320523 8 O78 TT 878 Oi 9979 9784 «| (1432Est vot 127,000; vol Wed 117,378; open int 366,236, +5,584. ft Be Be Bd ee a8 772 Yr Treasury Notes (CBT)-$200,000; pts 32nds of 100% Est vol 4,308: vol Wed 7.187; open int 74,062, +1,499. .Mar 402-24 102-31 102-24 102-29 + ST 102-31 11-282 6,647 eer sy EED0,00D; pts of 10%fue 10226 03-002 (2.282 105-00 + 62 02-000 101-12 7ag42 | Short Starling (LIFFE) £600,000; pteEst vol 10,100, vo) Wied 6,628, open int 85,489, ~426, " wee ye ah ght 8 Be Se ae30 Day Federal Funds + (C8135 million; pts of 100% St 0495 S898 5493 O4G7 5 OG BhOd Soa) DOATe2Mar 94740 94.750 94790 94745 + .070' 95810 99.205 21090 | pe gag ceo gugo 9453 2 Og 231 88147fn Ob Sz Bi BM + Ree Be Oh | Mire He S16 he UE + OF mI TamB62 62 655 60 + 0B 9H? 9502 205 160 9483 9459 Sat ‘ 3Shall: 4 opm 872 Te fio Mp Sa ay MT Bowe Oe eeeMuni Bond Index (eat) $1,000; times Bond Buyer MBI dune 9857 9458 9455 9457 + 03 9458 9277 15276Mar 105415 106-20 105-14 105-17 vy 10527 9-03 G83 | Got gask OLBS Gas? GAM + Od BAER 9290 2't60dune 104-21 104-30 104-16 104-22 + 1 105-00 10-30 11,835 Dec 9450 9452 9449 S85} + 03 9452 9292 5297Est vot 850; vol Wed 1,203: open int 18,167, ~130. ” “OS 08 9443 93.01 5gInder Cose 1040, Vdd 50, We ee At gtaTreasury Bille (CME)-$1 mil.; pts of 100% Sept 445 + 03 9416 9395 Ott
 
De gas + 08 S90 8325 BBYee eH le CHANGE wei CHANGE hig | Est vol 154,788 vol wea 213:500 open int 852.507 +6981,
War ge ea ee Bet BAB rR Se | Long allt (UrFE) (Decimal)-£60,000; pts of 100%
ne BO we ot 38 ; Mar” 11660 11681 11658 1689 + al 185.74 11210 6.212
Eat val 6, ol Wad 780, ope int 180, 60. dune 11769 116.17 14717 11810 + At 19817 11490 57299
Paar ee gs Oran Pte oy 200% Est vol 26.7 vol Wed 32,380 open int 625t om
on
oS.&oSe
        
 
          
     
   
 
 
   
Ay 9845 9519 95.19 9518 + Mar 5 ee OSfy $532 9540 5a) O83 + RS BF aune 9543 9554 9543 9553 + Ss + obchy 95.50 9559 9549 9558 + S2 + 0$561 + B75 + 04S67 +. SO + Oovol open int 60,543, 267. $70 + 04CME) $4, Millon; pte of 100% Gt + Od06 9510" 9505 9503 + 02 491 S46 + 0 089529 9524 9528 + 0 472 $544 + (04 9601 9405 37455SM 9539 G43 + 0 457 HG + 03 945 9390 21.445G55 543 9553 + 447 33 + 03 9594 9391 16.503$59 95.55 95.509 + 12 4.41 B19 + 03 95.21 9360 14.77064 9550 9562 + 10 4.38 19 + 03 9519 9383 3663YBa 5 M5 + 10 455 Si + 03 9513 9379 9Ha 25 537 + 9 48 9508 + 03 9507 9373 2162$5.17 9503 9514 + 08 485 595 + 02 496 9384 1,10619 9483 493 + «09 507 : S494 + 03 8495 98.07 «304OTs 3460 M7 + 0 529 : ed 545,643 open int 1,668,500 +37, 266.9465 9457 9465 + 06 5H 3 Month Euroswiss LIFE SFr 1,000,000; pts of 100%9459 SA4B 9456 + OB Sad ine "0 S00 Es $595 Oh OTA 88SAL 944d 9448 + 0 582 Sept 9201 9712 97.01 9709 + 08 $712 9559 33,795S437 25 84 + OB 5.66 Dec 96% 97.02 9694 S701 + 07 97.02 945 18.082BG 2 MH + OB 5 M02 9703 97.07 9700 9706 + 07 9707 %5.509430 9416 27 + OB 573 June 96.97 96.99 9697 9699 + 05 9699 9582 5254M24 42 + Be 5D Sept mem RO BRS HIT 1.889412 5400 98D + 0B 592 Oe O68 +05 681 9659 1.623S12 9401 9808 + 06 581 Est vol 30,483 vol Wed 31.982 open int 177,670 +8540,406 9397 9403 + 08 597 Canadian Bankers Acceptance (ME}.c$1,000,0009401 9391 9397 + 07 603 Mar 95.26 95.35 52 95 + 006 95.05 315 69.430S39 93Bt o85 + OF une 9585 + O10 9588 9307 73,2099390 9362 9388 + 06 Seah 5 + O12 78 9305 4508$387 9378 9383 + 05 Dec 48 S65E + 042 9559 9297 192619382 B79 + 06 9536 95.31 9698 + 006 9535 9328 140029373 69 + 05 dune 9521 9515 95.21 + 005 9521 9295 13095see . 9372 + 05 Sept 95.07 + 0.04 9605 93.11 4736Ber + 4 Dec 5493 + 004 S4gi 9305 a8883 + 0 N03 5483 + 004 9485 9395 278O54 + 4 dune 9476 + 004 438 9395 166935