## Yields on government stocks

The flat yield on a security is the annual amount receivable in interest, expressed as a percentage of the clean price (i.e. the price excluding gross accrued interest). The gross redemption yield is the rate of interest which, if used to discount future dividends and the sum due at redemption, makes the present value equal to the market price of the stock. In calculating yields on stocks with possible alternative redemption dates, redemption is assumed at the latter date when the clean price is below face value and at the earlier date when the clean price is above face value.

The par yields in this table are derived from the Variable Roughness Penalty (VRP) model estimates of the term structure, described in an article in the November 1999 *Quarterly Bulletin* (page 384). This replaces earlier models: all data have been re-estimated. Conventional dated stocks with a significant amount in issue and having more than one year to maturity are used, together with General Collateral repo rates at the short end. Index-linked stocks, irredeemable stocks, double dated stocks, stocks with embedded options, variable and floating stocks are all excluded. Quoted values are averages of end-month data for 1970 to 1978 inclusive, and quarterly averages of all working days from 1979 onwards.

Yields can only be calculated for maturities where gilts exist. Hence for dates in the past where there was no bond longer than 20 years we do not quote a 20-year yield.

For War Loan, the data for 1970 to 1978 inclusive are the average of Wednesday observations; thereafter the figures are the averages of all working days.

The real gross redemption yields are calculated using the actual number of days until dividend and redemption payments are received. To calculate these yields for index-linked gilts of maturities greater than 8 months requires an assumption about the rate of inflation since the last published Retail Price Index (RPI). In line with market convention, a rate of 3% is assumed from April 1998 and a rate of 5% is used prior to that date. If the index-linked stock has less than or equal to eight months to maturity, the actual dividend and redemption proceeds are known. The yield then quoted is a nominal redemption yield.

Since 1994, the Bank has been estimating yield curves from the prices of both conventional and index-linked gilts. The yield on these instruments incorporates elements of real rates of return, inflation expectations and risk premia.

The model used to derive these estimates is a spline-based technique, which is designed for monetary policy analysis where smoothness of the curve is a key criterion. Six sets of estimates are generated: zero coupon real curves, zero coupon inflation curves, zero coupon nominal curves, implied forward real rates, implied forward inflation rates and implied forward nominal rates.

Yields can only be calculated for maturities where gilts exist. Hence for dates in the past where there was no bond longer than 20 years we do not quote a 20-year yield.

## Zero coupon nominal curves

The spot interest rate or zero coupon yield is the rate at which an individual cash flow on some future date is discounted to determine its present value. By definition it is the yield to maturity of a zero coupon bond and can be considered as an average of single period rates to that maturity. Conventional dated stocks with a significant amount in issue and having more than three months to maturity, plus General Collateral repo rates (at the short end) are used to estimate these yields; index-linked stocks, irredeemable stocks, double dated stocks, stocks with embedded options, variable and floating stocks are all excluded.

## Zero coupon real curves

Calculated from the prices of index-linked gilts, which were first issued following the 1981 budget, and comprised approximately 25% of the UK Government bond market at end-March 2001. Coupon payments and the redemption payment are revalued to reflect changes in the Retail Price Index, so preserving the real value of income and capital. Difficulties arise in calculating real zero coupon yields because:

- There are 11 index-linked stocks in issue (4 in 1982), which have a market value of £84 billion compared to a conventional market of £241 billion (at end-January 2003).
- There is an 8 month time lag in indexation, which means that the price of an index-linked gilt is a complicated function of both the nominal and real term structures.

## Zero coupon inflation curves

These are derived from real and nominal zero coupon curves using Fisher's identity (this equates the difference between the nominal and real yield curve at a particular maturity to a measure of inflation over the same period). They incorporate average expected inflation rates over specified periods of time, (for example, the average rate of market inflation expectations over a five-year period).

## Implied forward nominal and real rates

Implied forward rates are future one period interest rates that when compounded are consistent with the zero-coupon yield curve. They embody a forecast of the future short-term rate but also incorporate risk premia and other factors. Rates shown are instantaneous forward rates, i.e. the implied interest rates on future transactions with infinitesimal investment periods. In practice these can be identified with expected future overnight rates.

Implied forward nominal rates are calculated from the prices of conventional gilts, whereas implied forward real rates are calculated from index-linked gilts.

## Implied forward inflation rates

Real and nominal forward rate curves can be used to produce an implied forward inflation rate curve. This embodies the expected inflation rate at a specified point in the future, but also incorporates risk premia.

## Further references

More details on the Bank's current yield curve models are given in the Bank of England *Quarterly Bulletin* article: *'*New estimates of the UK term structure of real and nominal interest rates', Anderson & Sleath, November 1999. Further, more detailed descriptions and analysis were provided in a Bank of England Working Paper entitled 'New estimates of the UK real and nominal yield curves', John Sleath and Nicola Anderson, March 2001. These are available from Publications Group, Bank of England, Threadneedle Street, London, EC2R 8AH, to which enquires regarding these publications should be addressed. An overall view of yield curves, can be obtained by reference to 'Estimating and Interpreting the Yield Curve', written by Anderson, Breedon, Deacon, Derry and Murphy and published by Wiley Publishing. Index-linked bonds are discussed in detail in the book 'Inflation-Indexed Securities', written by Deacon and Derry and published by Prentice-Hall.

Data for a full set of curves fitted using the Bank approach are available via the Yield Curve page.

## Real gross redemption yields

### 4¼ % Euro Treasury Note 2001 / 4½ % Bank of England Euro Treasury Note 2004

Yield based on market observations at the close of business each day.

### 2½ % index-linked Treasury 2016

Yield based on the representative middle market prices at the close of official business.

### US 10 year Government bond

Yield based on recently issued, actively traded securities as defined in the Federal Reserve Bulletin.

### German 10 year Government bond

Yield is defined by the Deutsche Bundesbank.