october 19, 2010math 2510: fin. math. 2 1 agenda recap: yield to maturity (or: to redemption). ct1,...

October 19, 2010 MATH 2510: Fin. Math. 21

Agenda

Recap: Yield to maturity (or: to redemption). CT1, Unit 13, Sec. 4.2.

Par yield. CT1 Unit 13, Sec. 4.3. And the solution of Q9 from the April 2009 CT1-exam.

Messing w/ youd head: Something that isn’t the yield curve.

Estimating the yield curve by bootstrapping: Hull Chapter 4, Sec. 5


More practical complications in yield curve estimation.

CT1 Unit 13, Sec. 5: Risk or sensitivity measures; duration.


Yield to Maturity (or: to Redemption)

Consider a bond with cash-flows ct at times t = 1, 2, …, T, and price P. Its yield to maturity, i, is the solution to the equation:

This is a non-linear equation; must be solved numerically.

.)1(1

T

tt

t

i

cP


Par Yield

The par yield , ycn, is the bullet bond coupon rate that makes a n-term bullet bond trade at par, i.e. have a price of £1 per £1 notional. Or in symbols:

With a non-flat yield curve, par yield and yield to maturity is not the same. (Artimetic vs. geometric effect.)

.1

)()(11

1

n

j j

nn

n

j jnnP

PycPycP


April 2009 CT1-Exam Q9

[different slides, old hand-out]


Teaser: A Graph I Got From Bloomberg

Mid-Oct. 2010 yields to maturity of UK government bonds (y-axis) for different maturities (x-axis).


Questions

(0. How do you like the scaling?)

1. Why is that not the (zero coupon spot) yield curve?

2. What do we do about that? How do we estimate the yield curve?


Estimating the Yield Curve

Consider the small principal/notional-100 bullet bond market from Hulls’s Table 4.3:

Time to maturity Coupon Price

0.25 0% 97.5

0.50 0% 94.9

1.00 0% 90.0

1.50 8% 96.0

2.00 12% 101.6


Let’s assume that coupons are paid semi-anually. work with continuously compounded zero

coupon (spot) rates

We can now determine – ”estimate” - the yield curve by working from short to lond maturities. This is called bootstrapping.


The Bootstrap Method

The 3 month rate solves the equation

This means 10.127% with continuous compounding.

Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding.

5.97100 25.025.0 Re


To calculate the 1.5 year rate we solve

to get R1.5 = 0.10681 or 10.681%. Notice how the previously calculated rates are used.

Similarly the 2 year rate is 10.808%

9610444 5.10.110536.05.010469.0 5.1 Reee


The Resulting Yield Curve

Determining Zero Coupon Rates: Hull's Bootstrapping Example in Chapter 4, Sec. 5

0.1

0.101

0.102

0.103

0.104

0.105

0.106

0.107

0.108

0.109

0 0.5 1 1.5 2 2.5

Maturity

Inte

rest

rat

e

Zero coupon rate


Yield Curve Estimation in Practice

Yield curve estimation (also known as ”yield curve stripping”) is the back-bone/the ”meat and potatoes” of any bank’s fixed income department.

Complication: More cash-flow dates than bonds. Solution: Use some interpolation scheme. (Piecewise constant, linear, Nelson-Siegel.)


Complication: Which products to use? Particularly in focus ”post Credit Crunch”.

This will be the topic of Course Work #2.


Duration

Consider this situation: You … … are a pension fund. … have to pay out £30m in 20 years … have collected £12m in premiums and

invested the money in 5-year (zero coupon) bonds


Assume the interest rate is 5% per year.

Your assets (the zero coupon bonds) are worth £12m; you own 15.32m of them (15.32 * (1.05)-5=12)

Your liabilities (the future pension payments) have a present value of £30m* (1.05)-20=£11.31m

So all is well; you are nice and solvent. (You might report a solvency percentage of (12 – 11.31)/11.31 ~6%.)


Now the interest rate changes to 4%.

You assets are now worth £15.32m * (1.04)-5=£12.59m, whereas the present value of your liabilities is £30m* (1.04)-20=£13.69m. Thus: You are no longer solvent.

What just happened here? And how do we avoid such nasty surprices?


Calculate not only the present values of our positions, but also their sensitivity to interest rate changes.

This is what duration (CT1, Unit 13, Sec. 5) is about.


Set-up:

Cash-flows at tk

Yield curve flat at i (or continuously compounded/on force form: )

Present value of cash-flows:

ktC

kt

kk t iCA )1(


Effective duration

The effective duration (or: volatility, or: modified duration) is defined as

So: Duration is a sensitivity to shifts in the yield

A

iCtA

iAv

k

k

t

k tk

)1()1(1

:


Macauley Duration

The Macauley duration (or: discounted mean term ) is defined by

So duration can also be intepreted as a weighted average of payment dates.

(Also: Sensitivity to shifts in the force of interest.)

viA

iCt k

k

t

k tk )1()1(

:

october 19, 2010math 2510: fin. math. 2 1 agenda recap: yield to maturity (or: to redemption). ct1,...

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