interest rates finance (derivative securities) 312 tuesday, 8 august 2006 readings: chapter 4
TRANSCRIPT
Interest RatesInterest RatesFinance (Derivative Securities) 312
Tuesday, 8 August 2006
Readings: Chapter 4
Types of RatesTypes of Rates
Treasury Rates• Short-term government securities
LIBOR• London Interbank Offer Rate• Rate applicable to wholesale deposits
between banksRepo Rates
• Repurchase agreements
Measuring RatesMeasuring Rates
Compounding frequency is unit of measurement
Increased frequency leads to continuous compounding• $100 grows to $100eRT when invested at a
continuously compounded rate R for time T• $100 received at time T discounts to $100e–RT
at time zero when continuously compounded discount rate is R
Conversion FormulaConversion Formula
Rc : continuously compounded rate
Rm: same rate with compounding m times per year
R m
R
m
R m e
cm
mR mc
ln
/
1
1
Bond PricingBond Pricing
Relies on interest rates on zero-coupon bonds (zero rates)• Interest is realised
only at maturity date
Maturity(years)
Zero Rate(% cont comp)
0.5 5.0
1.0 5.8
1.5 6.4
2.0 6.8
3 3 3
103 98 39
0 05 0 5 0 058 1 0 0 064 1 5
0 068 2 0
e e e
e
. . . . . .
. . .
To calculate the price of a two year coupon bond paying 6% semi-annually:
Bond YieldBond Yield
Single interest rate that discounts remaining CFs to equal the price
Using the previous example, solve the following equation for y:
y = 0.0676 or 6.76%
3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .
Par YieldPar Yield
Coupon rate that equates a bond’s price to its face value
Using previous example:
g)compoundin s.a. (with get to 876
1002
100
222
0.2068.0
5.1064.00.1058.05.005.0
.c=
ec
ec
ec
ec
Par YieldPar Yield
If: m = no. of coupon payments per year
P = present value of $1 received at maturity
A = present value of an annuity of $1 on each coupon date
then:
cP m
A
( )100 100
Calculating Zero RatesCalculating Zero Rates
Bond Time to Annual Bond
Principal Maturity Coupon Price
(dollars) (years) (dollars) (dollars)
100 0.25 0 97.5
100 0.50 0 94.9
100 1.00 0 90.0
100 1.50 8 96.0
100 2.00 12 101.6
Bootstrap MethodBootstrap Method
2.5 can be earned on 97.5 after three months
3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding, and 10.127% with continuous compounding
Similarly the 6-month and 1-year rates are 10.469% and 10.536% with continuous compounding
Bootstrap MethodBootstrap Method
To calculate 1.5-year rate, solve:
to get R = 0.10681 or 10.681%
Similarly the two-year rate is 10.808%
9610444 5.10.110536.05.010469.0 Reee
Zero CurveZero Curve
Zero Rate (%)
Maturity (yrs)
10.127
10.469 10.536
10.681
10.808
9
10
11
12
0 0.5 1 1.5 2 2.5
Forward RatesForward Rates
Future zero rates implied by the current term structure
Zero Rate for Forward Rate
an n -year Investment for n th Year
Year (n ) (% per annum) (% per annum)
1 10.0
2 10.5 11.0
3 10.8 11.4
4 11.0 11.6
5 11.1 11.5
Calculating Forward Calculating Forward RatesRates
Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded
The forward rate for the period between times T1 and T2 is:
R T R T
T T2 2 1 1
2 1
Slope of Yield CurveSlope of Yield Curve
For an upward sloping yield curve:
Fwd Rate > Zero Rate > Par Yield
For a downward sloping yield curve:
Par Yield > Zero Rate > Fwd Rate
Forward Rate Forward Rate AgreementsAgreements
Agreement that a fixed rate will apply to a certain principal during a specified future time period
Equivalent to agreement where interest at a predetermined rate, RK , is exchanged for interest at the market rate
Can be valued by assuming that the forward interest rate will be realised
Forward Rate Forward Rate AgreementsAgreements
Let:• RK = interest rate agreed to in FRA
• RF = forward LIBOR rate for period T1 to T2 calculated today
• RM = actual LIBOR rate for period T1 to T2 observed at T1
• L = principal underlying the contract
Forward Rate Forward Rate AgreementsAgreements
If X lends to Y under the FRA, then:• Cashflow to X at T2 = L(RK – RM)(T2 – T1)• Cashflow to Y at T2 = L(RM – RK)(T2 – T1)
Since FRAs are settled at T1, payoffs must be discounted at [1 + RM (T2 – T1)]
Value of FRA is the payoff, based on forward rates, discounted at R2T2
• ValueX = L(RK – RF)(T2 – T1)e–R2T2
• ValueY = L(RF – RK)(T2 – T1)e–R2T2
Theories of Term Theories of Term StructureStructure
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