Valuing Government Bonds: Term Structure, Pricing, and Sensitivity, Lecture notes of Finance

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FM212 - Principles of Finance

FM212 - Principles of FinanceJames Clark

Asset Pricing

Key Topics

^ Term Structure of Interest Rates
^ Pricing Government Bonds
^ Duration and Modified Duration
^ Measuring the Sensitivity of Bond Prices to Interest Rates
^ Bond Prices and Interest Rate Sensitivity
^ Spot Rates and Forward Rates
^ Theories of the Term Structure of Interest Rates

James Clark^

FM212 - Principles of Finance

FM212 - Principles of Finance Term Structure of Interest Rates UK Government Term Structure – 31^ James Clark

st^ th^ May and 30June 2016

FM212 - Principles of Finance Pricing Government Bonds Bonds are priced using spot rates. For example if you have a bond with n years tomaturity, an annual coupon of C and a face value F, then its price is determinedfrom:

(^ )^

(^ )^ (^

) 2

1 1 2

.............^1

n^ 1 1

n n n

C^ C^

C^ C^ F

PV^ r^ r

+ − r r −

=^ +^

+^ +

+^ +^

+^ +

The bonds that we consider for pricing throughout the slides are governmentbonds from countries such as the UK, U.S. where we assume they are defaultfree. If you’re pricing a stream of cash flows, each cash flow has its own relevantinterest rate depending on when it arrives.The PV of the entire stream is the sum of each cash flow discounted at theappropriate interest rate.^ James Clark

FM212 - Principles of Finance Pricing Government Bonds Arbitrage 1. An investment strategy that has a positive cash flow today and zero cash 1. An investment strategy that has a positive cash flow today and zero cashflows in the future in all states of nature.flows in the future in all states of nature.2. An investment strategy that has zero cash flows today and strictly non negative cashflows in all states of nature in the future with at least one positive future cash flow in a2. An investment strategy that has zero cash flows today and strictly nonfuture state of nature.negative cash flows in all states of nature in the future with at least one positivefuture cash flow in a future state of nature.Note definition 1. can be viewed the same as a zero cash flow today and aguaranteed positive cash flow in the future.If this is the case we can borrow the PV of the future positive cash flow today torealise the arbitrage cash flow now. To illustrate an example follows.^ James Clark

FM212 - Principles of Finance Pricing Government Bonds

t = 1 t = 0

r = 10% Borrow^ (^

£110 £110 £100 PV = =^ ) 1.

Repayloan with-£110interest 0

Example:^ Original arbitrage had zero cash flows today and a guaranteed cashinflow of £110 at t=1. £100 We now have a an investment strategy that has a positive cash flow today andzero cash flows in the future in all states of nature – definition 1. of arbitrage.^ James Clark

FM212 - Principles of Finance Pricing Government Bonds Consider the following bond data where coupons are annual and par value is£100. What should be the price of Bond A?^ Spot rates are implied through the prices of zero coupon bonds.

Bond^ Coupon rate

Maturity^ Price A^ 10%^

2 years B^ Zero Coupon

1 years^ £ C^ Zero Coupon

2 years^ £80 £100 1

£90^ =^1 r +^ £100 £80^ =^21 r +^ (^ )^2

1 1 £90^ 0.9 d =^ =^ = 1 £100^^ r^ +^1 ( )^

2

1 £80^ 0.8 d =^ =^ =^2 £100 1^ r^ +^2

James Clark

FM212 - Principles of Finance Pricing Government Bonds The price of Bond A (the 10% coupon bond) must be^ We can synthetically replicate Bond A (the coupon bond) using Bonds B and C(the zero coupon bonds).^ What is the replicating portfolio?^ The replicating portfolio consists of being long (buying) 0.1 units of Bond B andlong (buying) 1.1 units of Bond C.

1 1 2 (^ )^1

PV^ r^

^  r

^  = +^ ^

^ ^ ^

+^ + ^ ^

 ( ) ( )

10 110 10 0.9^1

110 0.8^97

PV^ d^ d =^ +^

=^ +^

What ensures using the spot rates to determine the price of the coupon bond isactually the market price?^ James Clark

FM212 - Principles of Finance Pricing Government Bonds^ (^ )^ (^

)

0.1 £90^ 1.1 £

Since being long 0.1 units of Bond B and long 1.1 units of Bond C gives the exactsame cash flows as Bond A the price of bond A must be the same price as 0.1units of Bond B and 1.1 units of Bond C.^ Two assets i.e. Bond A and the replicating portfolio(0.1 of Bond B and 1.1 ofBond C), which have exactly the same cash flows must have the same price.The no arbitrage price of Bond A has to be £97.^ Could you make an arbitrage profit if Bond A was priced at £98 in the market?^ James Clark

FM212 - Principles of Finance Pricing Government Bonds How can you make arbitrage profits? At t = 0 go short (sell) Bond A and go long (buy) the replicating portfolio i.e.go long (buy) 0.1 units of Bond B and go long (buy) 1.1 units of Bond C, andmake at t = 0 an arbitrage profit of £1. How did I know what strategy to use to make arbitrage profits?^ Let’s consider future cash flows when we go long or short an asset today.If we go long (buy) Bond A the future cash flows are positive and if we go short(sell) the replicating portfolio the future cash flows are negative. Likewise if wego short (sell) Bond A the future cash flows are negative and if we go long (buy)the replicating portfolio the future cash flows are positive.^ James Clark

FM212 - Principles of Finance Lecture 4 – Valuing Government BondsWe know if we go long one of the assets and short the other the future cashflows will sum to zero.^ How do we decide which one to go long and which one to go short?^ Buying (going long) an asset today results in a negative cash flow today.Selling (going short) an asset today results in a positive cash flow today.^ We want a net positive cash flow today. Buy cheap and sell high.^ Buy replicating portfolio (cheap) and sell Bond A (expensive).The full arbitrage strategy is therefore sell 1 unit of Bond A, buy 0.1 units ofBond B and buy 1.1 units of Bond C.

Pricing Government Bonds James Clark

FM212 - Principles of Finance

17

t = 0^

t = 2 t = 1 Bond A(short)^ £^

-£^

t = 0^

t = 2 t = 1 ReplicatingPortfolio(long)^ -£^

£^

£^

Lecture 4 – Valuing Government Bonds Our investment strategy yields an arbitrage profit of £1. It is arbitrage as ourinvestment strategy yields a guaranteed positive cash flow today and zero cashflows in the future in all states of nature i.e. this is a money machine.In reality we would short much more than one unit of Bond A to maximise thearbitrage profits before price movements eliminate the arbitrage opportunity.

Pricing Government Bonds James Clark

FM212 - Principles of Finance Duration and Modified Duration^ t^1 (^ ) C   t   y +  w = (^) t   PV of Bond     This is the price of the bond. We will represent the price of the bond by P whereP is calculated from

n^ Ct P = ∑^ t^1 y +^1 t =^ (^ )

here represents the C t cash flow at time t.

Macaulay Duration - Annual Coupon Bonds^ James Clark

FM212 - Principles of Finance Lecture 4 – Valuing Government Bonds

Duration and Modified Duration Macaulay Duration - Annual Coupon Bonds

C  ^ t^ t   n 1^ y +( )  D t (^) = ( ) (^) mac ∑  P (^1) t =     The weight is therefore the PV of the cash flow as a percentage of the bond’s price.Macaulay duration is simply the value weighted average maturity of a bond’scash flows.

n^ ( ) = wtD ∑ tmac = t^1 wt James Clark