Week 1
Basics
2.1, 2.2, 2.4, 2.9, 5.1
2.7 ex 2, 4, 5, 7
5.7 ex 1, 3
Price of zero-coupon bond with principal value of 100: Pz(t,T) = 100 x Z(t,T)
Z(t,T) = 100/payoff
Interest rate depends on compounding frequency:
- Given r, higher compounding frequency -> higher payoff
- Given payoff, higher compounding frequency -> lower r
rn(t,T): annual compounding rate
Continuous compounding: n very large
Term structure of interest rates/spot curve/yield curve, at a certain time t defines the relation
between the level of interest rates and their time to maturity T-t.
Short maturity T-bills have no coupons.
Coupon bond at time t, coupon rate c, semi-annual coupon payment and payment dates T1, T2, ..., Tn = T. discount
factors Z(t, Ti) for each date Ti. Value of coupon bond:
Bootstrap methodology: with sufficient data we can obtain the discount factors for every maturity
Given date t, n coupon bonds coupon ci, maturities Ti, 6-month intervals, i = 1, 2, ..., n.
Or, interpolate the discount factors: Splines- or Nelson-Siegel method to fir the term structure.
,
,Semi-annual floating rate bond, spread s, coupon payments c(Ti) at T1 = 0.5, T2 = 1, T3 = 1.5, ..., Tn = T
- c(Ti) = 100 x (r2(Ti – 0.5) + s) / 2
- r2(Ti-1, Ti): 6-month treasury rate at time Ti-1, with T0 = 0
- Each coupon date: reset date
- Coupon at t = 0.5 depends on today’s interest rate r2(0, 0.5) (and c(1) depends on interest rate at t = 0.5:
r2(0.5, 1, etc.)
- Spread is a fixed payment on the bond, so
- The (ex-coupon) of a floating rate bond, with s = 0 and maturity T, at any reset date, is equal to its face value:
PFR(Ti+1) = 100
Price of a (semi-annual) floating rate bond outside of reset dates:
- Ti < t < Ti+1
- Pay off 100+c(Ti+1) at next reset date Ti+1
- c(Ti+1) = 100 x r2(Ti) / 2 is known at t > Ti
- Z(t, Ti+1): discount factor from t to Ti+1
-
- At reset dates: Z(Ti, Ti+1) = 1 / (1 + r2(Ti) / 2), so ex-coupon price of a floating rate bond (with s = 0) on any
reset date is its face value PFR(Ti-1) = 100
Fix interest rate on a $100 loan, at t, from T1 to T2, choose forward rate fc(t, T1, T2), based on compounding
frequency n.
- Cashflow at T1 = 100
- Cashflow at T2 =
- No cashflow at time t, so 0 value at time t:
F(t, T1, T2): forward discount factor
Forward curve: plot of f(0, T, T+Δ) against T, for fixed Δ
The relation between the spot- and forward curve can be used to get the spot rate curve from forward rates:
, Spot rates r(0,T) and forward rates f(0, T, T+Δ) both express the same term structure and are both based on the
same discount factors Z(0, T)
Basics
2.1, 2.2, 2.4, 2.9, 5.1
2.7 ex 2, 4, 5, 7
5.7 ex 1, 3
Price of zero-coupon bond with principal value of 100: Pz(t,T) = 100 x Z(t,T)
Z(t,T) = 100/payoff
Interest rate depends on compounding frequency:
- Given r, higher compounding frequency -> higher payoff
- Given payoff, higher compounding frequency -> lower r
rn(t,T): annual compounding rate
Continuous compounding: n very large
Term structure of interest rates/spot curve/yield curve, at a certain time t defines the relation
between the level of interest rates and their time to maturity T-t.
Short maturity T-bills have no coupons.
Coupon bond at time t, coupon rate c, semi-annual coupon payment and payment dates T1, T2, ..., Tn = T. discount
factors Z(t, Ti) for each date Ti. Value of coupon bond:
Bootstrap methodology: with sufficient data we can obtain the discount factors for every maturity
Given date t, n coupon bonds coupon ci, maturities Ti, 6-month intervals, i = 1, 2, ..., n.
Or, interpolate the discount factors: Splines- or Nelson-Siegel method to fir the term structure.
,
,Semi-annual floating rate bond, spread s, coupon payments c(Ti) at T1 = 0.5, T2 = 1, T3 = 1.5, ..., Tn = T
- c(Ti) = 100 x (r2(Ti – 0.5) + s) / 2
- r2(Ti-1, Ti): 6-month treasury rate at time Ti-1, with T0 = 0
- Each coupon date: reset date
- Coupon at t = 0.5 depends on today’s interest rate r2(0, 0.5) (and c(1) depends on interest rate at t = 0.5:
r2(0.5, 1, etc.)
- Spread is a fixed payment on the bond, so
- The (ex-coupon) of a floating rate bond, with s = 0 and maturity T, at any reset date, is equal to its face value:
PFR(Ti+1) = 100
Price of a (semi-annual) floating rate bond outside of reset dates:
- Ti < t < Ti+1
- Pay off 100+c(Ti+1) at next reset date Ti+1
- c(Ti+1) = 100 x r2(Ti) / 2 is known at t > Ti
- Z(t, Ti+1): discount factor from t to Ti+1
-
- At reset dates: Z(Ti, Ti+1) = 1 / (1 + r2(Ti) / 2), so ex-coupon price of a floating rate bond (with s = 0) on any
reset date is its face value PFR(Ti-1) = 100
Fix interest rate on a $100 loan, at t, from T1 to T2, choose forward rate fc(t, T1, T2), based on compounding
frequency n.
- Cashflow at T1 = 100
- Cashflow at T2 =
- No cashflow at time t, so 0 value at time t:
F(t, T1, T2): forward discount factor
Forward curve: plot of f(0, T, T+Δ) against T, for fixed Δ
The relation between the spot- and forward curve can be used to get the spot rate curve from forward rates:
, Spot rates r(0,T) and forward rates f(0, T, T+Δ) both express the same term structure and are both based on the
same discount factors Z(0, T)
