Help with a proof of the yield curve.

In summary: If it does, then the yield curve defined by r(t) = 1/t integral r(s) ds from 0 to t is a nondecreasing function.In summary, the yield curve defined by r(t) = 1/t integral r(s) ds from 0 to t is a nondecreasing function if P(αt) ≥ (P(t))^α for all 0<=α<=1 and P(t) = exp{-integral r(s) ds from 0 to t}, where r(s) is the spot rate function. This means that the inequality must hold for all values of t, not just for t > 0.
  • #1
Kuma
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Homework Statement



I need to show that the yield curve defined by

r(t) = 1/t integral r(s) ds from 0 to t is a nondecreasing function iff:

P(αt) ≥ (P(t))^α, for all 0<=α<=1 , t>= 0

and P(t) is defined as:

P(t) = exp{-integral r(s) ds from 0 to t}

and r(s) is the spot rate function. So basically the yield curve is the average of all the spot rates until time t.

Homework Equations





The Attempt at a Solution



So the definition of a nondecreasing function is that if f(b) > f(a) for all b>a.
Now
P(t) = exp{-t r(t)}
so P(αt) = exp{-αt r(αt)}

Now I have to show that P(αt) ≥ (P(t))^α results in r(t) being nondecreasing.
Doing some simplification I came up with the following inequality

∫r(s) ds from 0 to αt ≥ α∫r(s) ds from 0 to t
= r(αt) ≥ r(t)

so that will be nondecreasing as long as that is satisfied for all αt > t

But that means that α >= 1 which is not the case?
 
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  • #2




Your analysis is correct, but you have overlooked an important detail. The inequality P(αt) ≥ (P(t))^α must hold for all values of α from 0 to 1, not just for αt > t. This means that your inequality must hold for all values of t, not just for t > 0. Therefore, your conclusion that α ≥ 1 is incorrect. Instead, you should consider the case where t > αt and see if the inequality still holds.
 

1. What is a yield curve?

A yield curve is a graphical representation of the relationship between the interest rates and the time to maturity for a set of bonds. It shows the yield or expected return on a bond over its entire lifespan.

2. Why is the yield curve important?

The yield curve is an important indicator of the state of the economy and the market's expectations for future interest rates. It can also provide insights into inflation expectations and investors' risk appetite.

3. How is the yield curve calculated?

The yield curve is calculated by plotting the yields of bonds with different maturities on a graph. Typically, the yields of 3-month, 2-year, 5-year, 10-year, and 30-year Treasury bonds are used to create the curve.

4. What does a steep or flat yield curve indicate?

A steep yield curve, where long-term bond yields are significantly higher than short-term bond yields, typically indicates a strong economy with expectations of future inflation. A flat yield curve, where short-term bond yields are similar to long-term bond yields, may indicate a weakening economy and expectations of lower interest rates in the future.

5. What are the limitations of the yield curve?

While the yield curve is a useful tool for understanding the state of the economy, it is not a perfect indicator and can have limitations. For example, it may not accurately predict future interest rates or recessions, and it may be influenced by external factors such as central bank policies and market sentiment.

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