# yield curve question

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?

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