Term structure of interest rates

In summary, the term structure of interest rates is upward sloping. The term structure of interest rates corresponds to the forward rates that are available for a given period of time. The curve can be expressed as a product of forward rates, so for each 3 month interval, i = 1 to 20, the 5 year interest rate = ∏ (1+ri). The longer duration of the zero rate makes it have a higher yield than a coupon bearing bond.
  • #1
the4thamigo_uk
47
0
I came across this question in chapter 4 of Hull 'Options Futures and other Derivatives'. I have the answer but I am not sure what the explanation is. Could anyone help?

The term structure of interest rates is upward sloping. Put the following in order of magnitude :

a) the 5 year zero rate
b) the yield on a 5 year coupon bearing bond
c) The forward rate corresponding to the period between 4.75 and 5 years in the future

The answer is c > a > b, but why?
 
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  • #2
C = the curve can be expressed as a product of forward rates, so for each 3 month interval, i = 1 to 20, the 5 year interest rate = ∏ (1+ri). If the curve is upward sloping, i20 > i1

A = the zero has a longer duration than a coupon bearing bond so will have a higher yield than a similar maturity coupon bond
 
  • #3
Thanks BWV.

The explanation for C makes a lot of sense now. I am not so clear about the explanation for A though. I can intuitively see that if I lock money away for a longer period I should expect a greater return, but how does the duration formula show this?

For the two bonds I have something like the following :

P_z = F/(1+R)^n
P_c = C(1/(1+r) + 1/(1+r)^2 + ... + 1/(1+r)^n) + F/(1+r)^n

Now P_z and P_c are not expected to be equal, and I can choose C to be anything I like, so I have complete flexibility to change P_c and C to give me an r > R or r < R.
 
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  • #4
forget about the duration formula for a second (although the duration of a zero is equal to its maturity while a coupon bond is always less)

so if you think about your formula for the coupon bond and the interest rate each coupon payment would command in the market if you sold it - i.e. if the bond pays semiannually the first coupon payment would be discounted at the 6 month interest rate, the second at the one year rate etc. the present value of all these payments makes up, along with the discounted value of the principal repayment the total of the bond value. The interest rate on the zero would be equal to the discount rate of the principal repayment at maturity which would be higher than the rate applied to any of the semiannual payments
 
  • #5
Ok I got it now... thanks a lot for your help
 

1. What is the term structure of interest rates?

The term structure of interest rates refers to the relationship between the maturity or length of time of a debt instrument and the corresponding interest rate. It shows how interest rates vary for different maturities, typically ranging from short-term (less than one year) to long-term (more than ten years).

2. How does the term structure of interest rates affect the economy?

The term structure of interest rates is a crucial indicator of the overall health of the economy. It reflects the market's expectations for future interest rates and can impact borrowing costs, investment decisions, and consumer spending. A steep or upward sloping yield curve usually indicates a healthy economy, while an inverted or downward sloping yield curve can signal an impending recession.

3. What factors influence the term structure of interest rates?

Several factors can influence the term structure of interest rates, including inflation expectations, economic growth, central bank policies, and supply and demand for different maturities of debt securities. Market forces also play a significant role in determining the shape of the yield curve, as investors continuously reassess and adjust their expectations for future interest rates.

4. How can investors use the term structure of interest rates?

Investors can use the term structure of interest rates to make informed decisions about their portfolio allocations. For example, if they believe that interest rates will rise in the future, they may want to invest in shorter-term bonds to avoid locking in lower interest rates for an extended period. Conversely, if they expect interest rates to fall, they may opt for longer-term bonds to lock in higher rates.

5. What is the difference between a flat, upward sloping, and inverted yield curve?

A flat yield curve occurs when there is a minimal difference between short-term and long-term interest rates. An upward sloping yield curve, also known as a normal yield curve, is when long-term interest rates are higher than short-term rates. An inverted yield curve, also known as a negative yield curve, is when short-term rates are higher than long-term rates. These different shapes of the yield curve reflect market expectations and can provide insights into the future direction of interest rates and the economy.

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