Treasury bills and Gilts
CORPORATE FINANCE - 1
UNIVERSITY OF EXETER
MARK FREEMAN, 2004
Week 2
1
Risk-free, tax free, projects
In today’s class, we shall concentrate on the valuation of riskfree 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
2
US fixed income securities Q2 2004
Treasuries
Mortgage-backed
Fed Agencies
Municipal
Corporate
Money market
Asset backed
Total value = USD 22,787,000,000,000 or USD 22.8tn.
Source: The bond market association
Week 2
3
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
Lowest Accepted Yield
Average Yield
Highest Accepted Yield
Avg Rate of Discount (%)
Average price per £100 nominal (£)
Tail (In Yield Terms)
Amount Tendered For (£)
Amount On Offer (£)
Cover
Amount Allocated (£)
Week 2
ISIN Code: GB00B024NB10
4.720000
4.734680
4.740000 (94.80% allotted)
4.678843
98.820675
0.005320
6,336,000,000.00
1,000,000,000.00
6.34
1,000,000,000.00
4
US Treasury bills
Details of US T-bill auctions can be found
http://www.publicdebt.treas.gov/of/ofaicqry.htm
Security
Term
Auction
Date
Issue
Date
Maturity
Date
13-WEEK
13-WEEK
13-WEEK
13-WEEK
13-WEEK
13-WEEK
10-04-2004
09-27-2004
09-20-2004
09-13-2004
09-07-2004
08-30-2004
10-07-2004
09-30-2004
09-23-2004
09-16-2004
09-09-2004
09-02-2004
01-06-2005
12-30-2004
12-23-2004
12-16-2004
12-09-2004
12-02-2004
Week 2
Discount Investment
Rate %
Yield %
0.000
1.710
1.685
1.640
1.635
1.580
0.000
1.741
1.716
1.671
1.663
1.607
Price
Per
$100
0.000000
99.567750
99.574069
99.585
99.587
99.601
at
CUSIP
912795RU7
912795RT0
912795RS2
912795RR4
912795RQ6
912795RP8
5
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
Investment Yield
Average Rate of Discount
Average Yield
Bank Discount 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
6
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 27-092004.
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
$100
360
Days to maturity
$0.43225 x
$100
360
91
=
1.710%
In the UK, the convention is to use the actual number of days
in the year, rather than 360.
UK Discount rate=
4.678844%
£100 - £98.820675 x
£100
Week 2
365 =
92
7
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
=
UK bill =
Week 2
$0.43225 =
$99.567750
0.4341%
£1.179325 =
£98.820675
1.1934%
8
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 + r )t −1
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
9
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
10
Bank Discount Yield vs. Money Market
Yields
Now,
Bank Discount Yield
=
100 – P x
100
360
t
Money Market Yield
=
100 – P x
P
360
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 =
= 1.717%
360 x 1.710%
360 – (91 x 1.710%)
Which agrees with the calculations above.
Week 2
11
Gilts
Information on the UK Gilts market is also given on the
www.dmo.gov.uk website.
For a description of the Gilts in
http://www.dmo.gov.uk/gilts/f1gilts.htm
Conventional
Gilts
ISIN Codes
Redemption Dividend
Date
Dates
Total
amount
in Issue
£mn
nominal
issue
go
to
Amount Central Govt
held in
Holdings
stripped
(DMO &
form
CRND) at 31
(£mn) at August 2004
3 Sept
2004
Shorts: (maturity
up to 7 years)
6¾% Treasury 2004 GB0008889619
9½% Conversion GB0008987777
2005
26-Nov-04
26
May/Nov
6,597
18-Apr-05 18 Apr/Oct
4,469
-
473
-
104
157
312
8½% Treasury 2005 GB0008880808
07-Dec-05
7 Jun/Dec
10,486
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
12
Interpreting the data
Consider the 7¾% Treasury 2006 that matures on 08-Sep2006
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
13
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
14
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+ r )t −1 formula that is more economically correct. So, the
first job is to adjust the redemption yield to get a true sixmonthly 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
3.875
3.875
3.875
103.875
+
+
+
= 105.66
1.02376 1.023762 1.023763 1.02376 4
This agrees to the quoted price to within a rounding error.
Week 2
15
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
Capital Gain or Loss
Coupon Income
Reinvestment Income
Total Return
105.66
100 – 105.66
4 x 3.875
3.875 x [(1.02376)3-1]
3.875 x [(1.02376)2-1]
3.875 x [(1.02376)1-1]
105.66
-5.66
15.5
0.283
0.186
0.092
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
16
Are Gilts risk-free?
It is worth distinguishing between two types of risk
1)
2)
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!).
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
17
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,
£1
+
Ar % =
N
(1 + r )
£1
(1 + r ) 2
+L+
£1
(1 + r ) N
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:

1 
1
%
r

A
= 1 −

N
r  (1 + r ) N 

Week 2

18
Annuity values
So, for example, if N = 3 and r = 5%, then the annuity
value, A5% = 2.723.
3
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).
(3.875* A2.376% ) +
4
Week 2
100
1.023764
= (3.875*3.773) + 91.03 = 105.66
19
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 semiannual redemption yield of y. The price of the bond is given
by:
Or
P
P/100
= C * A(N, y%) + 100(1+y)-N
=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
20
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")
2½% Treasury GB0009031096
Undated
Central Govt
Total
Holdings (DMO
amount in & CRND) at 31
Issue £mn
August 2004
nominal
(£mn)
22
1 Apr/Oct
493
3½% War GB0009386284
Undated
1 Jun/Dec
ISIN
Codes
Redemption Dividend
Date
Dates
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
21
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:
P r% =
1
r
Pricing undated bonds immediately after a coupon payment
is now straightforward. Convert the interest yield into a sixmonthly 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
22
Questions – Week 2
CORPORATE FINANCE - 1
UNIVERSITY OF EXETER
MARK FREEMAN, 2004
Week 2
23
Question set 2
1)
2)
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.
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)
4)
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.
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?
Week 2
24