Week 2 1

Treasury bills and Gilts

CORPORATE FINANCE - 1 UNIVERSITY OF EXETER MARK FREEMAN, 2004

Week 2 2

Risk-free, tax free, projects

In today’s class, we shall concentrate on the valuation of risk-free cashflows in the absence of taxation. The introduction of risk and tax will come later in the course. While this may seem over simplistic, it will enable us to

understand much of what occurs with government fixed income securities. So, by the end of today’s class, we should be in a position to read and interpret bond market data. Further, as most companies want to raise debt as well as equity, we need to be aware of the other types of debt finance against which our project is competing – including

government debt. Therefore understanding fixed income securities will both help us with the fundamentals of project evaluation and debt financing.

Week 2 3

US fixed income securities Q2 2004

Total value = USD 22,787,000,000,000 or USD 22.8tn.

Source: The bond market association

Treasuries

Mortgage-backed

Fed Agencies

Municipal

Corporate

Money market

Asset backed

Week 2 4

UK T-bill auctions

T-bills, remember, are short-term Government borrowing. For institutional details of trading in UK Gilts and money market instruments, there is an excellent governmental website (www.dmo.gov.uk). The money is raised through auction. These happen each Friday:

92 Day TBill due 04/01/2005 ISIN Code: GB00B024NB10

Lowest Accepted Yield 4.720000

Average Yield 4.734680

Highest Accepted Yield 4.740000 (94.80% allotted)

Avg Rate of Discount (%) 4.678843

Average price per £100 nominal (£) 98.820675

Tail (In Yield Terms) 0.005320

Amount Tendered For (£) 6,336,000,000.00

Amount On Offer (£) 1,000,000,000.00

Cover 6.34

Amount Allocated (£) 1,000,000,000.00

Week 2 5

US Treasury bills

Details of US T-bill auctions can be found at http://www.publicdebt.treas.gov/of/ofaicqry.htm

Security Term

Auction Date

Issue Date

Maturity Date

Discount Rate %

Investment Yield %

Price Per $100

CUSIP

13-WEEK 10-04-2004 10-07-2004 01-06-2005 0.000 0.000 0.000000 912795RU7 13-WEEK 09-27-2004 09-30-2004 12-30-2004 1.710 1.741 99.567750 912795RT0 13-WEEK 09-20-2004 09-23-2004 12-23-2004 1.685 1.716 99.574069 912795RS2 13-WEEK 09-13-2004 09-16-2004 12-16-2004 1.640 1.671 99.585 912795RR4 13-WEEK 09-07-2004 09-09-2004 12-09-2004 1.635 1.663 99.587 912795RQ6 13-WEEK 08-30-2004 09-02-2004 12-02-2004 1.580 1.607 99.601 912795RP8

Week 2 6

The four T-bill yields

There are four ways in which the rate of return on T-bills are quoted

US markets UK markets CFA / My terms Discount Rate Average Rate of Discount Bank Discount Yield

Investment Yield Average Yield Money Market Yield

CD Equivalent Yield

Holding Period Yield

Effective Annual Yield

“True” Yield to Maturity

You need to be able to distinguish between these four yields.

Week 2 7

Yields on T-bills - 1

Consider the US data given above. A key number is the

auction price of 99.567750 for the thirteen-week bill of 27-09-

2004.

What this means is that, for every $99.567750 you lend to the

government today, you will be paid $100 on Dec 30 2004.

That is, you will receive $0.43225 more than you invested

(the “discount”) per $100 of face value. The Bank Discount

Yield is then calculated as follows:

Discount rate = Discount x 360

$100 Days to maturity

= $0.43225 x 360 = 1.710% $100 91 In the UK, the convention is to use the actual number of days in the year, rather than 360. UK Discount rate= £100 - £98.820675 x 365 =

4.678844% £100 92

Week 2 8

Yields on T-bills - 2

Much more important is the “coupon equivalent”. First we

calculate the Holding Period Yield to the t-bill:

Holding Period Yield = Selling price – Buying price

Buying price

= Discount

Auction price

So, for the US and UK bonds

US bill = $0.43225 = 0.4341%

$99.567750

UK bill = £1.179325 = 1.1934% £98.820675

Week 2 9

Annual T-bill yields

There are two ways of annualizing T-bill yields. The first is

the “correct” way of doing this is to use the 1)1( −+ tr

formula, which gives the Effective Annual Yield).

US bill = (1.004341)365/91 – 1 = 1.7526%

UK bill = (1.011934)365/92 – 1 = 4.8192% However, the bond market convention is to use an annualised

rate calculated using the rt formula. This gives the Investment Yield or Average Yield. Coupon equivalent = Rate of return * (Days in Year/Days to maturity US bill = 0.4341% * (365/91) = 1.741%

UK bill = 1.1934% * (365/92) = 4.73468%

Week 2 10

Investment Yield vs. Money Market Yields

With Investment Yields, we annualize by taking

Days in Year / Days to Maturity

As with the Bank Discount Yield, the Money Market Yield

takes

360 / Days to Maturity

So, for the Money Market Yield:

US bill = 0.4341% * (360/91) = 1.717%

Week 2 11

Bank Discount Yield vs. Money Market

Yields

Now,

Bank Discount Yield = 100 – P x 360

100 t

Money Market Yield = 100 – P x 360

P t

where t is the number of days until maturity. By substituting

out for P it is easily proven that

Money Market Yield = 360 x Bank Discount Yield

360 – (t x Bank Discount Yield)

So that, in the above example for the US 91-day bill,

Money Market Yield = 360 x 1.710% = 1.717%

360 – (91 x 1.710%)

Which agrees with the calculations above.

Week 2 12

Gilts

Information on the UK Gilts market is also given on the www.dmo.gov.uk website. For a description of the Gilts in issue go to http://www.dmo.gov.uk/gilts/f1gilts.htm

Conventional

Gilts ISIN Codes Redemption

Date Dividend

Dates Total

amount

in Issue

£mn

nominal

Amount

held in

stripped

form

(£mn) at

3 Sept

2004

Central Govt

Holdings

(DMO &

CRND) at 31

August 2004

Shorts: (maturity

up to 7 years)

6¾% Treasury 2004 GB0008889619 26-Nov-04 26 May/Nov

6,597 - 473

9½% Conversion 2005

GB0008987777 18-Apr-05 18 Apr/Oct 4,469 - 104

8½% Treasury 2005 GB0008880808 07-Dec-05 7 Jun/Dec 10,486 157 312

7¾% Treasury 2006 GB0008916024 08-Sep-06 8 Mar/Sep 3,955 - 441

7½% Treasury 2006 GB0009998302 07-Dec-06 7 Jun/Dec 11,807 162 275

4½% Treasury 2007 GB0034040740 07-Mar-07 7 Mar/Sep 11,500 0 21

8½% Treasury 2007 GB0009126557 16-Jul-07 16 Jan/Jul 4,638 - 371

Week 2 13

Interpreting the data

Consider the 7¾% Treasury 2006 that matures on 08-Sep-2006 From last week we know that this bond pays coupons every six months of £7.75 / 2 = £3.875. These occur on 8th

March and 8th September each year. Now consider the price of this bond on the 9th September 2004 – immediately after a coupon payment. This information is given on the DMO website at http://www.dmo.gov.uk/gilts/f2gilts.htm Stock ID Stock Name ISIN Code Clean Price Dirty Price Yield 9HCV04 9 1/2 Conversion 2004 GB0002212982 100.570000 104.151967 4.711379

6TTY04 6 3/4 Treasury 2004 GB0008889619 100.470000 102.432636 4.424946

9HCV05 9 1/2 Conversion 2005 GB0008987777 102.870000 106.633661 4.610762

10HEX05 10 1/2 Exchequer 2005 GB0003270005 105.750000 105.464674 4.703747

8HTY05 8 1/2 Treasury 2005 GB0008880808 104.540000 106.746284 4.679800

7TTY06 7 3/4 Treasury 2006 GB0008916024 105.640000 105.682818 4.752351

Week 2 14

Pricing the Bond

Notice that on this day the “Clean” and “Dirty” prices are almost the same. We will examine the difference between these numbers in more detail next week. The Interest Yield or Current Yield is given by the annual

coupon rate divided by the current price: 7.75 /105.68 = 7.33%. This, though, ignores any capital change in price of the bond, which is why the DMO website does not bother to quote it. The Redemption Yield or Yield to Maturity is more interesting as it includes capital gains and losses

Week 2 15

Understanding the Yield to Maturity

As a matter of convention (this is also true for the US), the redemption yield has been calculated on a six-monthly basis and then annualised using the rt formula rather than the

1)1( −+ tr formula that is more economically correct. So, the

first job is to adjust the redemption yield to get a true six-monthly interest rate (with bonds it is more useful to deal

with six-month rates than annual rates as this is the time interval between coupon payments). The easiest way of doing this is just to divide by 2. So, in our example the six monthly rate is 4.752351% / 2 = 2.376%. There are four payments to come. We will receive £3.875 in six, twelve and eighteen months and $103.875 in two years.

Therefore the price of the bond should be

66.105402376.1

875.103302376.1

875.3202376.1

875.3

02376.1

875.3=+++

This agrees to the quoted price to within a rounding error.

Week 2 16

Breaking down the Yield to Maturity

Given that the Redemption Yield on the bond is 3.055% on a semi-annual basis, and given that we are investing 105.66 for four periods, this means that we need 105.66 * (1.02376)4 = 116.07 in four periods. Where does this come from?

We can break down the return into its component parts.

Capital Returned 105.66 105.66

Capital Gain or Loss 100 – 105.66 -5.66

Coupon Income 4 x 3.875 15.5

Reinvestment Income 3.875 x [(1.02376)3-1] 3.875 x [(1.02376)2-1] 3.875 x [(1.02376)1-1]

0.283 0.186 0.092

Total Return 116.06

Notice that the assumption here is that we re-invest the coupon income at the Yield-to-Maturity. This is not guaranteed as interest rates change over time. This is the main weakness of using Yield to Maturity.

Week 2 17

Are Gilts risk-free? It is worth distinguishing between two types of risk

1) Default risk. This is the risk that the government will refuse to pay either the coupons or the £100 on maturity of the bond. Gilts are virtually default risk free (governments print money!).

2) Inflation risk. This is the risk that the value of the coupons that we will receive is less, in purchasing power terms, than we had anticipated. Gilts are subject to

inflation risk.

Inflation risk is seen most clearly in the day-to-day fluctuations of bond prices. For this reason, Gilts cannot be considered to be risk-free assets, even though there is no default risk. The uncertainty

comes from not knowing the real rate of return that the bond will offer over its life because of inflation uncertainty.

Week 2 18

Annuities Rather than discounting each cashflow individually we can take a “short-cut” by using the annuity formula: this is

particularly useful for bonds that mature in many years time. Denote the interest rate per period (in this case, six months) to be r . If we receive £1 in payment at the end of each of the next N periods, then the present value of these cashflows is called the annuity value. That is,

Nrrr

rNA

)1(

1£

2)1(

1£

)1(

1£%

+++

++

+= L

Values for annuities for different N and r are provided at the back of nearly all standard finance textbooks. Further,

financial calculators have this as a function. The annuity formula is:

+−=

Nrr

rNA

)1(

111%

Week 2 19

Annuity values

So, for example, if N = 3 and r = 5%, then the annuity

value, .723.2%53

=A

Economically, what this is telling us is that, if the annual interest rate is 5%, then we should be indifferent between choosing between (a) £2.723 today or (b) £1 paid with

certainty in 12, 24 and 36 months time. Returning to the Tr 7 3/4pc ‘06 discussed above, we can consider this to be a four period annuity at 2.376% with payments of £3.875, plus a final payment of £100 in two years (which is four periods).

66.10503.91)773.3*875.3(402376.1

100)%376.2

4*875.3( =+=+A

Week 2 20

Gilts: premium or discount?

Consider a Gilt that has just paid a coupon. It has N coupons left to pay, a semi-annual coupon of C and a semi-

annual redemption yield of y. The price of the bond is given by:

P = C * A(N, y%) + 100(1+y)-N Or P/100 =c * A(N, y%) + (1+y)-N

Where A(N, y%) is an annuity factor and c = C/100 or the

semi-annual coupon rate as a proportion of the face value.

After a simple piece of algebraic manipulation it is easily

shown that this can be rewritten as:

P/100 = 1 + (c – y) * A(N, y%)

In other words, the bond will trade at the par value of £100 immediately after a coupon payment if the coupon rate is equal to the redemption yield. If the coupon rate is higher (lower) than the redemption yield then the bond will trade at a premium (discount).

Week 2 21

Undated bonds Some bonds never mature – that is, you will never receive the £100 at the end but they will go on paying coupons in

perpetuity. Undated Gilts (non

"rump")

ISIN

Codes

Redemption

Date

Dividend

Dates

Total

amount in

Issue £mn

nominal

Central Govt

Holdings (DMO

& CRND) at 31

August 2004

(£mn)

2½% Treasury GB0009031096 Undated 1 Apr/Oct 493 22

3½% War GB0009386284 Undated 1 Jun/Dec 1,939 30

Consider the price of the 2½ Treasury on 1st October 2004 2HTY 2 1/2 Treasury GB0009031096 52.520000 52.540604 4.760070

3HWR 3 1/2 War GB0009386284 74.840000 76.035355 4.676370

Week 2 22

Pricing undated bonds Since this bond never matures, there is no redemption yield. Therefore in this case (only!) it is the interest yield that is

relevant. The shortcut for pricing these bonds is to use the formula for perpetuities, which is a never-ending annuity. The perpetuity value is given by:

r

rP1% =

Pricing undated bonds immediately after a coupon payment is now straightforward. Convert the interest yield into a six-monthly rate by dividing it by 2. For the bond considered above this is 4.76% / 2 = 2.380%. Therefore the perpetuity value 1/r = 1/0.02380 = 42.02.

Each coupon payment is £1.25 per £100 so the total value of the bond is £1.25 * 42.02 = £52.52, which agrees, to within an approximation error, with the quoted clean and dirty prices.

Week 2 23

Questions – Week 2

CORPORATE FINANCE - 1 UNIVERSITY OF EXETER MARK FREEMAN, 2004

Week 2 24

Question set 2

1) Find the www.dmo.gov.uk report of the 91-day Treasury bill tender

(auction) of 2nd July 2004. From the given price, confirm the “Average

Yield” and “Average Rate of Discount”. What is the Effective Annual

Yield on this t-bill? Repeat with the 28-day Treasury bill tender of the

same date. 2) On 31st December 2004, if the interest rate for all maturities is 6%, how

much would you pay to receive £1 at the end of each calendar year up to

and including 31st December 2014. How much would you pay if the

payments came at the beginning of the year rather than the end of the year

(with final payment on 1st January 2014)? How would these answers change if the payments never ended?

For question 3 do not expect to get exact agreement, but your answer

should be very close (within 10p):

3) The Tr 8pc ’13 Gilt pays on coupons on the 27th March and 27th

September and matures on September 27th 2013. From the DMO

website, find the price and redemption yield on 27th September 2004.

Prove that these values are consistent. Break down the Yield to Maturity into its component parts.

4) The price of the 3½ War Undated Bond was £71.11 on 1/12/2003,

immediately after a coupon payment. What was the interest yield on this

gilt on this date?