Yield Curves Flashcards
Bond to yield conversion
p_nt = -n*y_nt
ZCB bond returns
r_n,t+1 = y_nt - (n-1)(y_n,t+1-y_nt)+(n-1)(y_n,t+1-y_n-1,t+1)
Three components of ZCB bond returns
yield spread at purchase, change in constant maturity yield, carry
Forward rate definition
Y_n+1,t^{n+1}/Y_nt^{n}
Forward rate log definition–what is it interpreted as
f_nt = y_nt + (n+1)(y_n+1,t-y_nt)
Rate of change of ZCB prices with maturity, holding t
Forward rate relationship with returns
r_n,t+1 = f_n-1,t+1 - n\Delta y_n,t+1
Unconditional mean of bond returns equals–in terms of forwards
Unconditional average forward rates
Equal average simple bond returns across maturities implies:
f_n-1,t+1 -1/2\sigma^2(r_n,t+1)=f_n’-1,t+1-1/2\sigma^2(r_n’,t+1)
Differ by the differences in half the variance of bond returns.
Expectations hypothesis
Returns of bonds differ only in maturity (not time)
Implication of EH (two period and multiperiod)
1) Change in yield * (n-1)= spread
2) yield equals constant + average yield until n-1
Macaulay duration Properties (2)
1) minus the elasticity of a coupon bond’s price with respect to its gross yield;
2) minus the elasticity of prices with respect to yields
Returns as a function of Macaulay duration
r_c,n,t+1 \approx D_cny_cnt - (D_cn-1)y_c,n-1,t+1
duration adjusted change in yields
What does bond convexity mean?
Change in bond price is convex with respect to change in yield.
What causes bond convexity?
Modified duration declines with yield for ZCB (level of interest rates). Standard duration is constant, so linear.
