Spot vs Forward: Average Speed vs Current Speed

The spot rate \(r(T)\) is the average of forward rates \(f(\tau)\) over \([0,T]\)

\(r(T)\;=\;\dfrac{1}{T}\displaystyle\int_0^T f(\tau)\,d\tau\)
Think of \(f(\tau)\) as the speedometer reading and \(r(T)\) as the trip's average speed. The forward rises above the spot whenever rates are accelerating; the area under \(f\) up to \(T\) equals \(T\!\cdot\!r(T)\).
2.00%
5.00%
+1.50%
3.0y
5.0y
Selected T
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Forward f(T) "current speed"
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Spot r(T) "average"
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Area = T·r(T)
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f(T) − r(T)
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