Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues

1

Prior to the yield curve: spot and forward rates

1.1 INTEREST RATES, PRESENT AND FUTURE VALUES, INTEREST COMPOUNDING

Consider a period of time, from t0 to t, in Figure 1.1.

Figure 1.1 Interest on a period of time, from t0 to t

$1 invested (or borrowed) @ i from t0 up to t gives $A. t is the maturity or tenor of the operation. $1 is called the present value (PV), and $A the corresponding future value (FV). i represents the interest rate or yield.

In this basic operation, no interest payment is made between t0 and t: in such a case, i is called a “0-coupon rate” or “zero” in short. Zeroes are also called “spot rates” as they refer to currently prevailing rates (at t0). Let us denote zt the current zero for a maturity t.

In the financial markets, the unit period of time is the year, and the interest rates, or yields, are expressed in percent per annum (% p.a.), that is, per year. In the US market, interest rates may also be expressed on a semi-annual basis (s.a.) with respect to the market of US bonds paying semi-annual coupons. Database providers, such as Bloomberg or Reuters, do well in always specifying whether the rates they mention are expressed on an annual or a semi-annual basis.

If the maturity t = 1 year, and z1 the corresponding zero rate expressed in % p.a., the relationship between PV and FV is

(1.1)

meaning that the future value FV is the sum of the present value ...