EXAM Prep Flashcards
preparing for the final exam
Spline vs Yield Curve (NS) methods
ie discount function fitting vs spot rate fitting
- Both methods are static curve fitting procedures, nothing to say about how the term structure (yield curve) will change over time.
- Fitting the yield curve directly results in model coefficients that have a direct economic interpretation, it is more parsimonious than spline methods and can result in effective hedging schemes especially at the short end of the yield curve.
- Yield curve methods exhibit systematic concavity at the long end due to lack of flexibility and hence they price long term bonds with less accuracy.
- Spline methods do not suffer from this lack of flexibility.
- Subject to continuity constraints, individual segments can move almost independently of one another.
- Spline methods should be used for relative value analysis.
Drawbacks of bootstrapping
Disadvantage of bootstrapping combined with interpolation is that it often leads non smooth yield curves with irrelgular kinks
NS parameter interpretation
lim T to infinity, r(0,T)=theta zero
lim T to zero , r(0,T)=theta zero + theta one
- theta zero can be interpreted as the long run interest rate and theta one can be interpreted as the negatie spread
- The factor multiplied by theta two becomes large for intermediate values of T and falls to zero as T approaches zero or infinity, therefore it represents the curvature factor.
- lambda is the scale parameter
Properties of Duration
Durations can be shown wieghted average time, but it doesn’t always apply, for instance zeros have longer duratin than its maturity, and Duration of a floating rate bond is T(i+1) - t, where t is time now and it is in between reset dates.
Duration depends on coupon rate, when coupon rate increases duration falls, this can be thought in 2 ways:
1) lower average time of cash flow payments: the higher the coupon the larger are the intermediate coupons relative to last one, Thus the average time gets closer to today.
2) Lower sensitivity to interest rates: the higher the coupon rates the larger are the cash flows in the near future, and the cash flow sooner are less sensitive to changes in interest rates.
What can you tell about about Barbell Bullet strategy? (butterfly spread)
It seems in the long term the strategy seems to yield positive convexity value, but actually it bleeds value as time passes, such that it offsets the positive convexity
Explain how the Campbell-Shiller and Fama-Bliss regressions are used as tests of the expectations hypothesis. Discuss the results of these regressions when applied to US yield curve data.
r(t,T) =
=[r(t, t + 1) + Et(r(t + 1, T) × (τ – 1)] / τ
{expected future yield}
+ λ / τ {risk premium}
– (τ – 1)^2Vt(r(t + 1, T)) / 2τ
{convexity}
Given that risk premium for holding 1 year horizon is equal to the convexity term, we derive
Et[r(t+1,t+τ) – r(t,t+τ)] = [r(t,t+τ) – r(t,t+1)] / (τ-1)
- This means that the expected change in yield depends on the slope of the term structure, this is called the expectation hypothesis (EH).
- Yet this does not hold in empirical testing, in fact the exact opposite relation seems to hold
Campbell and Shiller 1991 tested it by runing the following regression.
[r(t+1,t+τ) – r(t,t+τ)] = α + β [r(t,t+τ) – r(t,t+1)] / (τ-1) + ɛ(t+1)
US zero coupon yield curve data used 1964-2006, and tested α=0, β=1
the result indicated significant negative β, which means positive yield curve predict decrease in future yield and vice versa.
This violation of the EH implies that the log-risk
premium (LRP) must depend on the slope of the term structure, where LRP is:
LRPt(τ) = λ – (τ – 1)2 × Vt(r(t + 1, T)) / 2
• So we should have that:
Et[r(t+1,t+τ) – r(t,t+τ)] = [r(t,t+τ) – r(t,t+1)] / (τ-1) – LRPt(τ)
Changes in long term term yields are inversely related to the slope of the term structure, then LRPt(τ) must be positively related to the slope.
The implication is if term structure is strongly sloped up, doesnt necessarily means market is expecting higher future rates, rather quite opposite, investors require higher risk premium to hold longer term bonds. In turn it implies on average we expect a high capital gain in long term zero coupon bond in next year. Capital gain in zero occur due to yield decrease.
So they found empirically strongly sloped term structure predicts lower future yield on average.
Fama Bliss used
E[ log(Pz(t+1,T) / Pz(t,T)) - log( 100/ Pz(t,t+1))]=LRPt(τ)
EH implies LRPt(τ)=0, however Fama Bliss showed this is not 0.
they run following regression:
LERt(τ) = α + β[f (t, t + τ – 1, t + τ) – r(t, t + 1)] + ε(t)
Note that LRPt(τ) = Et[LERt(τ)]
• Expectation hypothesis would say that α = β = 0, which is not the case β is significantly different from zero and positive
• Excess log return is in fact predictable: when the forward spread is strongly positive, that is, the term structure is positively sloped, on average investments in long-term bonds generate a higher return compared to short term bonds
Explain how Treasury Inflation Protected Securities (TIPS) protect investors against inflation risk. Derive a pricing formula for a standard coupon bearing TIPS.
wip
Expectation hypothesis
The expectation hypothesis implied, the assumption that positively slopped term structure of interest rate
