Term structure of Interest Rates

Question 1

Calculate ZCB prices from annually compounded spot rates

Answer 1

Pt=(1+st)^-t

Question 2

Calculate forward rates from ZCB prices

Answer 2

(Pt-1/Pt)-1

Question 3

Calculate par rates from ZCB prices

Answer 3

(1-Pt)/(P1+P2+…+Pt)

Question 4

at where t=0

Answer 4

0

Question 5

Pt where t=0

Answer 5

1

Question 6

How to convert par yields into forward rates/ spot rates

Answer 6

To convert a par yield curve into spot or forward curves, start by
inductive computation of annuities, then difference these to get zero
coupon bond prices. To convert zero coupon bond prices into yields,
calculate internal rates of return (spot yields) or ratios of consecutive
bonds (forward yields).

Question 7

What is a check to ensure youre calculations are correct

Answer 7

f1=g1=s1

Question 8

Why might Changes in spot rates be smaller than forward rates jumps?

Answer 8

ZCB are obtained from spot rates raised to power of the term so for longer terms a small change on spot rates is magnified in the ZCB prices and hence in forward rates.

Question 9

Define the I spread

Answer 9

YTM on that asset minus the weighted average of government bond spot yields using the promised cash flows of the risky bond as weights

Hence: Weighted average of spot curve +I spread = YTM

Question 10

Define the Z spread

Answer 10

Term independent addition tot he government bond spot curve such that the market price of the risky bond is its market price.

Hence PV at( spot curve +Z spread) = PV at YTM

Question 11

What are the promise cash flows of a 5 year bond with coupons 8%

Answer 11

0.08 each year and extra 1 at final year so:
0.08,0.08,0.08,0.08,1.08

Question 12

What is the market price of a risky bond

Answer 12

PV of cash flows discounted at YTM

Question 13

If the spot curve is flat how do I spread and Z spread relate

Answer 13

Equal

Question 14

Does non decreasing spot curve imply a non decreasing forward curve?

Answer 14

Spot rate plus one is the geometric mean of the forward rates plus one
So no non decreasing spot curve does not imply a non decreasing forward curve
Spot curve is a smoothed version of the forward curve
If the f’s jump up and stay at a new level it will take the s’s a while to catch up as they are an overage. So if f increase and overshoots before returning down a little the spot rates can smooth this out

Question 15

How to convert forward curve PVs to ZCB

Answer 15

P0=1
P1=P0/(1+f1)
P2=P1/(1+f2)
etc

Question 16

How to calculate Pv of cash flows using ZCB prices

Answer 16

Multiple ZCB by cash flows and sum up

Question 17

What are an advantage and disadvantage of Smith Wilson method over nelson siegel curve?

Answer 17

Smith wilson is guaranteed to have a unique solution as solving linear equations .
Smith wilson approach does require two additiona parameter inputs the ultimate forward rate and alpha which have to be specified - disadvanatge or a way of limiting behaviour

Question 18

How to interpolate par cruve

Answer 18

• assume linear between points

Question 19

Compare Smith wilson fit with fit obtained by interpolating and bootstrapping

Answer 19

Interpolation and bootstrap: easier to execute but may produce jumps
SW produces a smooth continuous forward curve and is in principle straightforward enough with some details that need specifying.
SW needs additional parameters - either giving extra flexibility or are arbitrary