# Questions about the forward curves

Hello, I have some questions about the forward curves and I really need help:

My first question is: The curriculum states a relationship “For an upward- sloping yield curve—r(T* + T) > r(T*)—the forward rate rises as T* increases”, with "Denote the forward rate of a loan initiated T years from today with tenor (further maturity) of T years by f(T*,T)" and r(t) is the spot rate for tenor t years*. It means if spot rate r(3) > r(2) > r(1) then f(2,1) > f(1,1) or so. To illustrate, they give example with r(1) = 9%, r(2) = 10%, r(3)=11%, then f(2,1)= 13.03% > f(1,1) = 11.01%.

However, if I replace r(1) with another value, says, 6%, then f(1,1)= 14.15% > f(2,1), so the relationship does not hold as they stated???

My second question is about the graph of forward curve. The curriculum provides an example of graph:

I cannot understand how they construct the graph above.

• With the spot curve, I do not really know the value on X-axis is month or year. If X-axis is year, then it does not really reflect the values provided in the table below the graph, for example: table tells that if maturity = 20 years then spot rate = 3.35% BUT on the graph, the Y-value for X-value = 20 is only around 2%.
• With the forward curves, for example the July 2017 curve (T*=4 years from spot curve 31 July 2013), if the X-value of the forward curve, says = 20 years, I do not know whether the value of the Y-value is f(4,20) or f(4,16).

Thank you!!!