The Nobel Memorial Prize for Robert F. Engle

Constructing “Raw” Yields

Let us first fix ideas and establish notation by introducing three key theoretical constructs and the
relationships among them: the discount curve, the forward curve, and the yield curve. Let P_t (τ) denote
the price of a τ-period discount bond, i.e., the present value at time t of $1 receivable τ periods ahead, and
let y_t (τ) denote its continuously-compounded zero-coupon nominal yield to maturity. From the yield
curve we obtain the discount curve,

PtW = e"^,'^f",

and from the discount curve we obtain the instantaneous (nominal) forward rate curve,
ft(τ) = PP ((τ)/ Pt (τ).

The relationship between the yield to maturity and the forward rate is therefore

y( (^τ) ^{= 1}
τ
0

which implies that the zero-coupon yield is an equally-weighed average of forward rates. Given the yield
curve or forward curve, we can price any coupon bond as the sum of the present values of future coupon
and principal payments.

In practice, yield curves, discount curves and forward curves are not observed. Instead, they
must be estimated from observed bond prices. Two popular approaches to constructing yields proceed by
estimating a smooth discount curve and then converting to yields at the relevant maturities via the above
formulae. The first discount-curve approach to yield construction is due to McCulloch (1975) and
McCulloch and Kwon (1993), who model the discount curve with a cubic spline. The fitted discount
curve, however, diverges at long maturities instead of converging to zero. Hence such curves provide a
poor fit to yield curves that are flat or have a flat long end, which requires an exponentially decreasing
discount function.

A second discount-curve approach to yield construction is due to Vasicek and Fong (1982), who
fit exponential splines to the discount curve, using a negative transformation of maturity instead of
maturity itself, which ensures that the forward rates and zero-coupon yields converge to a fixed limit as
maturity increases. Hence the Vasicek-Fong model is more successful at fitting yield curves with flat

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