I am trying to wrap up a few fixed income points-
If they give you the current term structure if the yield curve and then also tell you that future rates will indeed come true as this curve predicts (ie I know how to do those forward rate calculations), what is the expected return of say the 5 year bond over the next 6 months?
I am unsure if the answer would be the current 5 year bonds yield over 2, OR the cur 6-month yield?
TY in advance!
It’s the current 6-month spot rate.
I’ve been wondering the same exact thing! S2000magician could you explain the intuition behind this?
U do know, s2000 doesnt like the word intuition!
Hahaha my bad. Reasoning? Mechanics? I just can’t seem for the life of me understand why all the bonds earn the same holding period return over a given time period if spot rates migrate to their forward rates.
I guess it does make sense…if things move ‘as currently predicted now’, why would any strategy over any time period outperform any other? It wouldn’t, I guess.
I cannot explain intuition because intuition means arriving at the correct answer without knowing why.
I can, however, give you a simple example that will spur your understanding (which, frankly, is a lot better than intuition).
Suppose that the 1-year spot rate today, s1,0, is 1% and the 2-year spot rate today, s2,0, is 2%. (I’m using the “,0” in the subscript to emphasize that this is a spot rate when we’re at time t = 0. In a moment we’ll need the 1-year spot rate when we’re at time t = 1, and I don’t want y’all to confuse the two.) Then the 1-year forward rate starting one year from today, 1f1, is calculated as:
(1 + s1,0)(1 + 1f1) = (1 + s2,0)2
(1 + 1f1) = (1 + s2,0)2 / (1 + s1,0)
1f1 = (1 + s2,0)2 / (1 + s1,0) − 1 = 1.022 / 1.01 − 1 = 3.0099%
Let’s say that you have a 2-year, annual-pay, $1,000 par bond with a 3% coupon rate. The value today is:
V0 = $30 / (1 + s1,0) + $1,030 / (1 + s1,0)(1 + 1f1) = $30 / 1.01 + $1,030 / (1.01)(1.030099)
= [$30 + $1,030 / 1.030099] / 1.01
In one year, the value (including the coupon payment you just received) will be:
V1 = $30 + $1,030 / (1 + s1,1)
If interest rates evolve according to the forward curve, then s1,1 = 1f1 =3.0099%, so,
V1 = $30 + $1,030 / 1.03009
So,
V1 / V0 = [$30 + $1,030 / 1.03009] / {[$30 + $1,030 / 1.030099] / 1.01}
= {[$30 + $1,030 / 1.03009] / [$30 + $1,030 / 1.03009]} × 1.01
= 1.01
= 1 + s1,0
For bonds with more than 2 periods till maturity, the same sort of thing happens.
For this part here, should $1,030 / (1 + 1f1) be $1,030 / [(1 + 1f1)(1 + s1,0)]?
At any rate, a huge thank you s2000! This was extremely illustrative and helpful.
Good catch! I’ll fix it.
By the way, the actual bond prices are:
V0 = $1,019.71
V1 = $1,029.90
V1 / V0 = $1,029.90 / $1,019.71 = 1.009993