Fixed Income

Downloads

Today’s work is in: matlab_lec09.m

Functions we need today: ycurve.m

Datasets we need today: data_bonds.m

Bonds

A bond is a security that pays a predefined amount at predefined future dates They care called fixed income securities because

the payments are (usually) fixed

Note that unlike stocks, bonds are (usually) free of cash flow risk Bonds still carry discount rate risk

Price

As with any security, the price of a bond is the discounted present value of all future cash flows

Suppose a bond pays $3 on Jan 1st 2009, $3 on Jan 1st 2010, and $103 on Jan 1st 2011

Suppose today is Jan 1st 2008 and you expect the discount rate to be 2% in 2008, 3% in 2009, and 4% in 2010

The bond’s price is

0652.10004.1*03.1*02.1

103

03.1*02.1

3

02.1

3

Yield

Note that in the above example the discount rate was not constant!

We can try to find a constant discount rate that would make the bond’s actual price equal to the discounted value of cash flows. This discount rate is called the Yield Note that the yield is not the “true” discount rate

For the above bond, the yield is

%98.20652.100)1(

103

)1(

3

1

322

yyyy

Institutional Details

In the U.S. most bonds have coupons which are paid every 6 months

Coupon payments are quoted as an annual percent of the principle ie if coupon is 3% bond pays $1.5 twice per year

At the time the bond matures, the owner receives the principal and one last coupon payment

A bond that matures on Jan 1st 2010 and has a 4% coupon rate has payments of: $102 on Jan 1st 2010, $2 on July 1st 2009, $2 on Jan 1st 2009 …

Institutional Details If I buy a bond with 2 months left until the coupon

payment, I will receive the full coupon payment I am entitled to all future payments

The price is the present value of all future payments, this is called the “dirty” price and is the true out of pocket cost to buyer This is the only number we should care about or know

Banks do not quote the dirty price but rather “clean” price ClnP = DrtP – AccrInt Accrued Interest = Coupon*(6-MonthUntilCoupon) Buyer pays DrtP=ClnP+AccrInt

Just after coupon is paid, AccrInt=0 so ClnP=DrtP To avoid unnecessary complications we will only look at

bonds close to after coupon paid so that ClnP≈DrtP

ConstructingDataset

This data set is being constructed on Oct 3, 2008 Data comes from Bloomberg screen on trading floor Could also get data from WSJ:

http://online.wsj.com/mdc/public/page/mdc_bonds.html?mod=mdc_topnav_2_3000

We will construct the vector maturity:>>maturity(1)=2009+4/12; We will construct the vector price:>>price(1)=100+33.5/36%prices quoted as fraction of 36%we are given bid and ask, take midpoint We will construct the vector coupon:>>coupon(1)=3+1/8;

Load Data

>>data_bonds;>>N=length(maturity); %number of bonds>>calendar=[2008+(9/12):(1/12):2030]';>>T=length(calendar);>>startdate=2008+(10/12)+((3/31)/12);

Construct CF

>>CF=zeros(T,N);%make matrix of cashflows>>for i=1:N;

[a Tmat]=min(abs(maturity(i)-calendar)); CF(Tmat,i)=100+coupon(i)*.5; for t=1:Tmat-1; if mod(t,6)==0 & calendar(Tmat-t)>startdate; CF(Tmat-t,i)=coupon(i)*.5; end; end;end;

Construct CF>> CF(1:20,1:6)

ans =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0101.5625 102.2500 1.6875 1.8125 2.0000 1.0625 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 101.6875 101.8125 2.0000 1.0625 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 102.0000 101.0625

Calculating Yield

>> for i=1:N;prerrbest=100000000;for j=1:1000;

y=0+.2*(j-1)/(1000-1); py=sum(CF(:,i)./((1+y).^(calendar-startdate))); prerr=abs(price(i)-py); if prerr<prerrbest; prerrbest=prerr; yieldbest(i)=y; end;

end;end;

>> plot(maturity,yieldbest,'.');>> xlabel('Maturity'); ylabel('Yield');

Calculating Yield

Note that yield increases with maturity, this is typical

However, this is not what is known as the yield curve (though it is related)

The Yield Curve

A zero-coupon bond is a bond that has no coupon payments and only pays the principal at maturity For such a bond yT=rT where r is the rate of return and T is

time until maturity Note that any other bond can be constructed with a portfolio

of “zeros” so if we know the prices (or yields) of “zeros” of every maturity we can price any other bond

The Yield curve is the yield on zero coupon bonds of every maturity Also called Term Structure of Interest Rates

Note that we have already broken up all of our bonds into “zeros” within the matrix CF

Now we just need a yield for each point in time which will make each bond price (almost) agree with the price of the cash flows

Constructingthe Yield Curve

We will not solve for a yield for every date but rather at a few different dates and then interpolate between them

%motnhs in which we will solve for yield>>yieldgriddate=[6 12 18 24 30 36 42 48 60 72 96 120 180 T]';%actual dates of those months>>yielddate=calendar(yieldgriddate);%initial guess (these are yields of bonds at various maturities

taken from printout)>>y0=[1.0123 1.0157 1.0150 1.0158 1.0175 1.0204 1.0228

1.0238 1.027 1.028 1.033 1.037 1.042 1.042]';

ycurve.m

function x=ycurve(yieldgrid,yielddate,calendar,startdate,CF,price,N,T);

%first interpolate to get yield for every date in dataset for t=1:T; [a b]=min(abs(yielddate-calendar(t))

+10000*(yielddate>=calendar(t))); if b==length(yielddate); b=b-1; end; mult=(calendar(t)-yielddate(b))/(yielddate(b+1)-yielddate(b)); yield(t,1)=yieldgrid(b)+mult*(yieldgrid(b+1)-yieldgrid(b));end;%next calculate the discounted present value of CF and compare to price for i=1:N; err(i)=price(i)-sum(CF(:,i)./(yield.^(calendar([1:T]')-startdate)));end; x=mean(abs(err));

Constructingthe Yield Curve

ycurve.m takes in any yield curve and return the pricing error from that curve, we want to make that error as small as possible

fminsearch.m can minimize errors, its inputs are a function, and a set of initial values; it will then search over values to find the minimum of the function

Sometimes you have to apply fminsearch.m more than once to get to “better” minimum

Constructingthe Yield Curve

>>y1=fminsearch(@ycurve,y0,[],yielddate,calendar,startdate,CF,price,N,T);>>y2=fminsearch(@ycurve,y1,[],yielddate,calendar,startdate,CF,price,N,T);>>y3=fminsearch(@ycurve,y2,[],yielddate,calendar,startdate,CF,price,N,T);%check minimum values:>> disp([ycurve(y0,yielddate,calendar,startdate,CF,price,N,T) ...

ycurve(y1,yielddate,calendar,startdate,CF,price,N,T) ... ycurve(y2,yielddate,calendar,startdate,CF,price,N,T) ... ycurve(y3,yielddate,calendar,startdate,CF,price,N,T)]);

3.1012 0.5914 0.5521 0.5518%plot the yield curve>>plot(yielddate,y3,'LineWidth',3); The yield curve is upward sloping, this is typical

This suggests expectations of higher interest rates in the future and/or more risk at longer maturities

Sometimes (rarely) yield curves are downward sloping This is called “inverted” and often precedes recessions Suggests expectations of lower interest rates and/or less risk at

longer maturities