## Types of yield curve provided

### Nominal zero coupon yields (spot interest rates)

For the data presented on the Bank’s website, the nominal government spot interest rate for n years refers to the interest rate applicable today (‘spot’) on an n year risk-free nominal loan. It is the rate at which an individual nominal cash flow on some future date is discounted to determine its present value. By definition it would be the yield to maturity of a nominal zero coupon bond^{2} and can be considered as an average of single period rates to that maturity.^{3} Conventional dated stocks with a significant amount in issue and having more than three months to maturity, and GC 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 from the Bank’s nominal yield curve. Spot interest rates from the commercial bank liability curves are equivalent rates implicit in the yields on the LIBOR-related instruments used in the curves’ construction. LIBOR rates are for uncollateralised lending within the interbank market. They are not risk free and contain a credit premium to reflect that. SONIA rates should be subject to limited credit risk as the contracts settle overnight. OIS contracts are also structured so that they involve minimal counterparty risk, such that OIS interest rates should contain very little compensation for credit risk.

### Nominal forward rates

Forward rates are the interest rates for future periods that are implicitly incorporated within today’s spot interest rates for loans of different maturities. For example, suppose that the interest rate today for borrowing and lending money for six months is 6% per annum and that the rate for borrowing and lending for 12 months is 7%. Taken together, these two interest rates contain an implicit forward rate for borrowing for a six-month period starting in six months’ time. To see this, consider a borrower who wants to lock in today’s rate for borrowing £100 for that period. He can do so by borrowing £97.09^{4} for a year at 7% and investing it at the (annualised) six-month rate of 6%. In six months’ time he receives back this sum plus six months’ interest at 6% (£2.91) which gives him the £100 of funds in six months’ time that he wanted. After a year he has to pay back £97.09 plus a year of interest at 7% (£103.88). In other words, the borrower ensures that his interest cost for the £100 of funds he wants to borrow in six months’ time is £3.88. He manages to lock in an annualised interest rate (the forward rate^{5}) of 7.77% now for borrowing in the future.

In this example, we considered six-month forward rates. We can consider forward rates that rule for different periods, for example 1-year, or 3-month or two-week forward rates. In the limit, as the period of the loan considered tends to zero, we arrive at the instantaneous forward rate. Instantaneous forward rates are a stylised concept that corresponds to the notion of continuous compounding, and are commonly used measures in financial markets. Instantaneous forward rates are the building block of our estimated yield curves, from which other representations can be uniquely derived.^{6}

### Real spot and forward rates

The return on a nominal bond can be decomposed into two components: a real rate of return and a compensation for the erosion of purchasing power arising from inflation. For conventional government nominal zero coupon bonds, such as those in the example above, the nominal return is certain (provided it is held to maturity) but the real return is not (because inflation is uncertain). An index-linked zero coupon bond would have its value linked to movements in a suitable price index to prevent inflation eroding its purchasing power (so its ‘real value’ is protected). For such a zero coupon bond the real return would be certain if the bond were held to maturity. A real debt market provides information on the ex ante real interest rates faced by borrowers and lenders who want to avoid the effects of inflation. In practice, there are factors that mean index-linked gilts do not offer complete inflation protection, and the UK index-linked gilt market is not as liquid as that for conventional UK gilts. Nevertheless, this market allows us to calculate real spot and forward rates analogous to the nominal spot and forward rates described above.

### Implied inflation rates

We have seen that the index-linked gilt market allows us to obtain real interest rates and the conventional gilt market allows us to obtain nominal interest rates. These nominal rates embody the real interest rate plus a compensation for the erosion of the purchasing power of this investment by inflation. The Bank uses this decomposition (commonly known as the Fisher relationship) and the real and nominal yield curves to calculate the implied inflation rate factored in to nominal interest rates. This is often interpreted as a measure of inflation expectations, although some care is required in doing so.^{7} As with nominal and real interest rates, we can think of ‘spot’ implied inflation rates (subject to the caveats in footnote 8) as the average rate of inflation expected to rule over a given period. Similarly forward implied inflation rates can be interpreted as the rate of inflation expected to rule over a given period which begins at some future date. In the limit, we can calculate instantaneous forward implied inflation rates just as with real and nominal rates.