Comparing Rates Of Different Maturities

[JMars7424]

The Murkiness Of Comparing Rates Of Different Maturities​

​

Consider 2 zero-coupon bonds. One that matures in 11 months and one that matures in 12 months. They both mature to $100.

Scenario A: The 11-month bond is trading for $92 and the 12-month bond is trading for $90.

What are the annualized yields of these bonds if we assume continuous compounding?1

Computing the 12-month yield

r = ln($100/$90) = 10.54%

Computing the 11-month yield

r = ln($100/$92) * 12/11 = 9.10%

This is an ascending yield curve. You are compensated with a higher interest rate for tying up your money for a longer period of time.

But it is very steep.

 

[gsarm1987]

@JMars7424 looks correct. annualized, continuously compounded yields are calculated correctly. About steepness, we take the spread between longer and shorter term. Correct

 

[David Harper CFA FRM]

Agree with @gsarm1987 and this ("You are compensated with a higher interest rate for tying up your money for a longer period of time") is interesting to me because I just wrote a fresh PQ set yesterday so term structures are top of mind, see https://forum.bionicturtle.com/threads/p1-t3-22-31-term-structure-theories.24340/

So the implied one-month forward rate, F(11/12, 12/12) = (10.54% * 12/12 - 9.10% * 11/12) / (12/12 - 11/12) = 26.37%. Steep indeed!

Realistically, we can assume it's true that "you are compensated with a higher interest rate for tying up your money for a longer period of time." Under liquidity preference theory, the 26.37% includes compensation for the (slightly!) longer maturity. But, merely as a theoretical matter, pure expectations would say that forward is an unbiased predictor of the (expected) future spot rate. It's the difference between: F(11/12, 12/12) = E[S(11/12, 12)] under expectations versus F(11/12, 12/12) > E[S(11/12, 12)]. Put another way: an upward-sloping term structure justifies liquidity preference, but it does not necessarily imply liquidity preference.

In this way, there is a truth to the "murkiness" mentioned in the title. On one level, the calculations are not murky: especially continuous compounding is super elegant how it exhibits time-additivity (!). Okay, but once we perform the calculations, there is a "murky" aspect to the interpretation, by which I mean that, given this data, we cannot parse exactly the component(s) of the term premium. FWIW.

P.S. Probably the "murky" refers to annualizing challenges because ... a few thoughts:

• If we refer to holding periods (ie, not annualized)
  • The 11 month bond has a cc HPR of 8.34%, and
  • The implied one-month forward is 26.37% / 12 = 2.20%; such that (due to elegant time additivity of CC )
  • the 12-month return = 8.34% + 2.20% = 10.54%; ie., same as one year bond. This is the great thing about cc.
• But in general we still want to express rates in per annum (aka, annualized terms) and use the x/12 multiplier to scale in the exponent. To compare the rates, we should use the annualized numbers

Here's a puzzle for you:
If we invest the 92 for 11 months and roll-over the $100 for the final month, we end up with 100*exp(6.37%*1/12) = $102.22; but the other way, we end up with $100.00. Is the forward rate wrong?

Another question (and maybe a matter of my opinion): given these two bonds, is there a convenient way to express the single, simple truth of the implied term structure?