2Outline Measuring interest rates Zero rates Bond pricing Types of ratesMeasuring interest ratesZero ratesBond pricingDetermining Treasury zero ratesForward ratesForward rate agreementsDurationTheories of the term structure of interest ratesSummary2
3Types of Rates(1/3) Treasury Rates(公債利率) A government to borrow in its own currency.An investor earns on Treasury bills and Treasury bonds.It is usually assumed “risk-free rates”.
5Types of Rates(2/3) (2) LIBOR(倫敦銀行間放款利率) The London Interbank Offered Rate is a daily reference rate based on the interest rates at which banks borrow funds from other banks in the London wholesale money market .LIBOR is short term .Derivatives traders regard LIBOR rates as a better indication of the “true” risk-free rate than Treasury rates .Large banks also quote LIBID rates ( London Interbank Bid Rate ) and is the rate at which they will accept deposits from other banks.
7Types of Rates(3/3) (3) Repo Rates(附買回利率) A Repurchase agreement (also known as a repo or Sale and Repurchase Agreement) allows a borrower to use a financial security as collateral for a cash loan at a fixed rate of interest.The difference between the price is the interest it earns. We called the interest rate is repo rate.The most common type of repo is an overnight repo.
8Measuring interest rates(1/4) Suppose that an amount A is invested for n years at an interest rate of R per annum.If the rate is compounded m times per annum, the terminal value of the investment isA (1+ R/m)mn(1)Compound timesFuture value (R=10%)Every year (m=1)110.00Half year (m=2)110.25Season (m=4)110.38Monthly (m=12)110.47Weekly (m=51)110.518
9Example1. A pension fund manager invests $10 million in a debt obligation that promises to pay 7.3% per year for four years. What is the future value of the $10 million?To determine the future value of any sum of money invested today, we can use the future value equation, which is:
10Measuring interest rates(2/4) With continuous compounding, it can be shown that an amount A invested for n years at rate R grows toAeRn(2)Compounding a sum of money at a continuously compounded rate R for n years involves multiplying it by eRn. Discounting it at a continuously compounded rate R for n years involves multiplying e-Rn.Which of the following amounts is closest to the end value of invest-ing $3000 for ¾ years at s continuously compounded rate of 12%?
11Measuring interest rates(3/4) Suppose that Rc is a rate of interest with continuous compounding and Rm is the equivalent rate with compounding m times per annum. From the results in equations (1) and (2), we haveeRc= ( 1+ Rm/m )mnRc = m ln(1+ Rm/m)(3)Rm=m (eRc/m - 1 )(4)These equations can be used to convert a rate with a compounding frequency of m times per annum to a continuously compounded rate and vice versa.
13Example(1) Consider an interest rate is quoted as 10% per annum with semiannual compounding . From equation (3) with m=2 and Rm=0.1 ,the equivalent rate with continuous compounding is2ln (1+ 0.1/2) = Or % per annum(2)Suppose that a lender quotes the interest rate on loan as 8% per annum with continuous compounding , and that interest is actually paid quarterly . From equation (1) with m=4 and Rc=0.08 , the equivalent rate with quarterly compounding is4(e0.08/4 -1)=0.0808Or 8.08 per annum. This means that on a $1,000 loan ,interest payments of $20.20 would be required each quarter
14Zero rates Definition: 1. The n-year zero-coupon interest rate is the rate of interest earned on an investment that starts today and lasts for n years. All the interest and principal is realized at the end of n years.2. Sometimes also referred to as the n-year spot rate.
15Bond Pricing(1/4)Suppose that a 2-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annum semiannually.
16Bond Pricing(2/4) Theoretical price of the bond (理論價格) Continuous 3e-0.05*0.5 ＋ 3e-0.058*1.0 ＋ 3e-0.064*1.5 ＋ 103e-0.068*2.0＝98.39ContinuousCompoundingAe-Rn
17Bond Pricing(3/4) (2) Bond Yield (債券報酬率) A bond’s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. Assumed: y is the yield on the bond.3e-y*0.5 ＋ 3e-y*1.0 ＋3e-y*1.5 ＋ 103e-y*2.0 ＝98.39y＝6.76% (trial and error )
18Bond Pricing(4/4) (3) Par Yield (面額報酬率) The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value. Suppose that the coupon on a 2-year bond in our example is c per annum.c/2e-0.05*0.5＋c/2e-0.058*1.0＋c/2e-0.064*1.5＋(100+c/2)e-0.068*2.0＝100c＝6.87The 2-year par yield is 6.87% per annum with semiannual compounding.
19Determining Treasury zero rates(1/5) The most popular approach is known as the bootstrap method(拔靴法or 導引法).BondTime toAnnualBond CashPrincipalMaturityCouponPrice(dollars)(years)1000.2597.50.5094.91.0090.01.50896.02.0012101.6＊Half the stated coupon is assumed to be paid every 6 months.
20Determining Treasury zero rates(2/5) The bootstrap methodThe 3-month bond provides a return of 2.5 in 3 months on an initial investment of 97.5.With quarterly compounding, the 3-month zero rate is (4×2.5)/97.5＝10.256％(per annum).The rate is expressed with continuous compounding, it becomes 4 ln （1＋ /4）＝Similarly the 6 month and 1 year rates are ％ and ％ with continuous compounding.Six months(2x5.1)/94.9=10.748%2 ln （1＋ /2）＝One yearln （1＋10/90）＝
21Determining Treasury zero rates(3/5) The bootstrap methodTo calculate the 1.5 year zero rate, the payments are as follows:6 months : ＄4 ; 1 year:＄4 ; 1.5 years :＄104Suppose the 1.5-year zero rate is denoted by R.R＝10.681％Similarly the two-year rate is ％4e *0.5 ＋ 4e *1.0 ＋ 104e-y*1.5 ＝96e-y*1.5 =R=-ln( )/1.5 =6e *0.5 ＋ 6e *1.0 ＋ 6e *1.5 ＋ 106e-R*2.0 ＝101.6e-R*2.0 =R=-ln( )/2.0 =
22Determining Treasury zero rates(4/5) Zero Curve Calculated from the Data(Figure 1)Zero Rate (%)Maturity (yrs)10.12710.46910.53610.68110.808
23Determining Treasury zero rates(5/5) A common assumption is that the zero curve is linear between the points determined using the bootstrap method.The zero curve is horizontal prior to the first point and horizontal beyond the last point.By using longer maturity bonds, the zero curve would be more accurately determined beyond 2 years.
24Forward rates(1/5)Definition: the rates of interest implied by current zero rates for periods of time in the future.
25Forward rates(2/5) Calculation of Forward Rates (Table 5) Zero Rate forForward Rateann-year Investmentforn th YearYear (n)(% per annum)13.024.05.034.65.8184.108.40.206
26Forward rates(3/5) Calculation of Forward Rates (Table 5) The 4％ per annum rate for 2 years mean that, in return for an investment of ＄100 today, the receive100e0.04*2＝＄Suppose that ＄100 is invested. A rate of 3％ for the first year and 5％ for the second year gives at the end of the second year.100e0.03*1e0.05*1＝＄108.33When interest rates are continuously compounded and in successive time periods are combined, the overall equivalent rate is simply the average rate during the whole period.
27Forward rates(4/5) (5) (6) Formula of Forward Rates The forward rate for year 3 is the rate of interest that is implied by a 4％ per annum 2-year zero rate and a 4.6％ per annum 3-year zero rate.If R1and R2 are the zero rates for maturities T1 and T2 ,respectively, and RF is the forward interest rate for the period of time between T1 and T2 ,then(5)(6)
28Example Find the forward rate 3f4 Rf= R4 + (R4-R3) x T3 / (T4-T3) = 6.2%(1+ 0f4 )4 = (1+ 0f3)3 x (1+ 3f4 )(1+5%)4 = (1+4.6%)3 x (1+ 3f4 )= ≒ 0.062
29Forward rate agreements(1/4) Definition: FRA is an over-the-counter agreement that a certain interest rate will apply to either borrowing or lending a certain principal during a specified future period of time.
30Forward rate agreements(2/4) Notation DefineRk: The rate of interest agreed to in the FRARF: The forward LIBOR interest rate for the period between times T1 and T2 calculated today.RM: The actual LIBOR interest rate observed in the market at times T1 for the period between times T1 and T2L: The principal underlying the contract.
31Forward rate agreements(3/4) Cash flow formulaIf X company could earn RM from the LIBOR loan ,The FRA means that it will earn RK. The extra interest rate (which may be negative) that is earns as a result of entering into the FRA is RK-RM. The interest rate set at time T1 and paid at time T2. The extra interest rate therefore leads to a cash flow to company at time T2 ofSimilarly there is a cash flow to the Y company at time T2 of
32Example at the 3.25-year point. This is equivalent to a cash flow of Suppose that a company enters into an FRA that specifies it will receive a fixed rate of 4% on a principal of $1 -million for a 3-month period starting in 3 years . If 3-month LIBOR proves to be 4.5% for the 3-month period the cash flow to the lender will beat the 3.25-year point. This is equivalent to a cash flow ofat the 3-year point . The cash flow to the party on the opposite side of the transaction will be +$1,250 at the 3.25-year point or +1, at the 3-year point (All the example interest rates in this example are expressed with quarterly compounding . )
33Forward rate agreements(4/4) Valuation FormulasValue of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 isValue of FRA where a fixed rate is paid isRF is the forward rate for the period and R2 is the zero rate for maturity T2
34ExampleSuppose the LIBOR zero and forward rates are as in Table Consider an FRA where we will receive a rate of 6% ,measured with annual compounding , on a principal of $1-millon between the end of year 1 and the end of year 2. In this case , the forward rate is 5% with continuous compounding or 5.127% with annual compounding . The value of the FRA isWe see that an FRA can be valued if we1.Caculate the payoff on the assumption that forward rates are realized ,that is on the assumption that RM=RF . (RM= The actual LIBOR interest rate RF =The forward LIBOR interest )2.Discount this payoff at the risk-free rate .
35Duration(1/3)Definition: A measure of how long on average the holder of the bond has to wait before receiving cash payment.A zero-coupon bond that lasts n years has a duration of n years.A coupon-bearing bond lasting n years has a duration of less than n years.
36Duration(2/3)Duration of a bond that provides cash flow c i at time t i iswhere B is its price and y is its yield (continuously compounded) This leads to(12)
37ExampleFor the bond in table 4.6 , the bond price B is and the duration D is ,so that equation giveWhen yield on the bond increase by 10 basis point (=0.1%) , it follows that.The duration relationship predicts that= ,so that the bond price goes down to = How accurate is this ? When the bond yield increase by 10 basis point to 12.1% , the bond isWhich is (to three decimal place) the same as that predicted by the duration relationship .
38Duration(3/3) Modified Duration The preceding analysis is based on the assumption that y is expressed with continuous compounding. If y is expressed with a compounding frequency of m times per year, thenA variable D*, defined by
39ExampleThe bond in table 4.6 has a price of and a duration of The yield , express with semiannual compounding is %. The modified duration , D* is given byWhen the yield (semiannually compound) on the bond increase by 10 basis point (=0.1%) , we have The duration relationship predicts thatWe expect to be x 0.001= , so that the bond price goes down to = How accurate is this ? When the bond yield increase by 10 basis point to % , an exact calculation similar to that previous example shows that the bond price becomes This shows that the modified duration calculation gives good accuracy .
40Theories of the term structure of interest rates Expectations Theory: forward rates equal expected future zero rates.Market Segmentation: short, medium and long rates determined independently of each other.Liquidity Preference Theory: forward rates higher than expected future zero rates.
41Summary(1/2)Treasury rates are the rates paid by a government on borrowings in its own currency.LIBOR rates are short-term lending rates offered by banks in the interbank market.The n-year zero or spot rate is the applicable to an investment lasting for n years when all of the return is realized at the end.Forward rates are the rates applicable to future periods of time implied by today’s zero rates.
42Summary(2/2)FRA is an over-the-counter agreement that a certain interest rate will apply to either borrowing or lending a certain principal at LIBOR during a specified future period of time.Duration measures the sensitivity of the value of a bond portfolio to a small parallel shift in the zero- coupon yield curve.Liquidity preference theory can be used to explain the interest rate term structures that are observed in practice.