I conclude ‘the yield curve does not change’ and ‘the forward curve does not change’ have different meanings and I am lost again:-)
Who said ‘the forward curve does not change’?
And don’t afraid or worry if you don’t understand this problem. This problem is really difficult. But it’s not difficult because of its mathematical aspect itself (the problem is really easy). It is difficult because CFA Curriculum doesn’t have any clear definition/notation, so we don’t know what they mean (I was surprise that some writers of CFA Curriculum are PhD… and I suppose that they are PhD in Literature or History).
And “the more you are good in mathematics, the more hardly you learn technical sessions”. You might put yourself in a “high school background in mathematics, literature student” 's position in order to understand their question and to not have much “metaphysical questions”. Your CFA studying life might be easier.
Just a joke, maybe 
The spot curve is (uniquely) derived from the par curve, and the forward curve is (uniquely) derived from the spot curve.
If the yield curve (i.e., the par curve) doesn’t change, then the spot curve doesn’t change. If the spot curve doesn’t change, then the forward curve doesn’t change. Hence, if the yield curve doesn’t change, the forward curve doesn’t change.
There is a unique link between “spot curve”, “par curve” and “forward curve”,I agree.
But when they (CFA) mention the “yield curve”, they can mean directly “spot curve” (aka “spot yield curve”), for example https://en.wikipedia.org/wiki/Yield_curve#Construction_of_the_full_yield_curve_from_market_data
Maybe this is an example, when CFA talk about something, we don’t really know exactly what they mean , and we must guest. You guested “the forward yield curve”, I guested “the spot yield curve”. And just by change, " the forward curve doesn’t change" or " the spot curve doesn’t change" give the same results.
Fair enough: “the” yield curve is the spot curve.
But it really doesn’t matter. What matters is that if any one of the three – par curve, spot curve, forward curve – is unchanged, the other two are also unchanged.
I agree that I approach this topic rather mathematically, but the fixed income is quite precise subject and I don’t see any other way to mastering it than clear understanding of concepts. And I am clearly missing on something here. The examples kindly provided by the members of the forum are not helpful, and not because I resist validity of the curriculum, but because I don’t understand what it says. Thank you.
Ok, here’s the reason why I keep asking questions on this topic: If forward curve is unchanged, future spot rates have to evolve as prescribed by the forward curve. But then the future spot yield curve is not what it was in previous periods (unless all yields are the same, flat curve): for example, last year’s forward yield f(1,1) becomes this year’s spot yield r(1). Hence spot curve changes. Reverse is true as well: unchanged spot curve implies changed forward curve.
Another angle (from curriculum reading 42, example on spot rates implied by forward curve): “if spot rates evolve as implied by the current forward curve, the return of a bond over a one-period is always the one-year rate (the risk free rate over the one period”. For any maturity bond. Then why would riding the yield curve strategy be better than buy and hold? Both strategies are supposed to earn the risk free rate.
With all due respect, that’s exactly wrong.
If the forward curve is unchanged, then the spot curve is unchanged, so future spot rates in fact _ haven’t evolved _ as prescribed by the forward curve.
Suppose that today’s 1-year spot rate is 1% and today’s 2-year spot rate is 2% (both EARs, not BEYs, to keep things simpler). Today the 1-year forward rate starting 1 year from now is 3.01%. Suppose that one year from today the spot curve is _ unchanged _: then the 1-year spot rate is 1%, not 3.01%.
I think we define a change in a yield curve differently.
We read from level 2015 Curriculum Volume 5 Reading 42 Section 2.3 Yield Curve movement and the Forward Rate, p.232:
“This section establishes several important results concerning forward prices and the spot yield curve in anticipation of discussing the relevance of the forward curve to active bond investors. The first observation is that the forward contract price remains unchanged as long as future spot rates evolve as predicted by today’s forward curve. Therefore, a change in the forward price reflects a deviation in the spot curve from that predicted by today’s forward curve.”
This is followed by the mathematical proof (formulas (7-10)). In particular, it says “Now suppose that after time t, the new discount function is the same as the forward discount function implied by today’s discount function (formula (8))”. And finally, “Equation (10) shows that the forward contract price remains inchanged as long as future spot rates are equal to what is predicted by today’s forward curve”.
The logic of proof shows that the curriculum calls the forward price of k-year bond delivered at time t and priced at time 0 F(0,t,k) unchanged , if after passing of time T, F(0,t,k) = F(T,t-T,k). What you call an unchanged curve, is different. In fact, you state that the forward price is unchanged if F(0,t,k) = F(T,t,k) after passing of time T.
Just skimming your post to go directly to formulas, I found that to write F(T,t-T,k) is not correct. Maybe you means F(0,t,k) = F(T,t + T,k)
and we state that the forward price is unchanged if F(0,t,k) = F(T,t +T ,k) after passing of time T.
Just recall your definition: F(x,t,k) - forward price of k-year bond delivered at time t and priced at time x (t>x)
And In my opinion, you should write definitions of “yield curve”, “spot curve”, “forward curve”, “forward”, “expected price”, “future price”, “forward price”,…
After that, you can resolve two problems you posted: the first one is to establish relations of the yield curves (spot forward par .), the second is to study the behaviors of yield curves (spot, par, forward) when the sharp of one of curve is not constant(eg: upward slopping,…). The answer of the second problem will help you understand the riding the yield curve strategy.
The main difficulty is not to resolve the two problems, the main difficulty is in writing the definition, in transforming layman’s term in mathematical objects.
See, my, or rather, the curriculum’s definition of unchanged forward price implies that a forward of 10-year bond delivered in 3 years in 2 years becomes a forward of 10-year bond delivered in 1 year. If these two forwards are equal, the price is unchanged.
As for definitions of yield curve, spot curve, forward curve they are of course what you have described which coincides with the curriculum. They are uniquely defined by each other and I do not see any ambiguity in their definitions.
The first one: F(2,3,10), the second one F(0,1,10). Hence, F(0,t,k) = F(T,t + T,k) with t= 1, T = 2, k =10
And if you have definitions, the answers are straightforward (you can read the post N°2 and N°8 for answer of the riding yield curve strategy).
I think there should be t = 3, T = 2, k = 10. t is time to delivery.
Ok, with your help I think I am getting there I totally agree on connection between the curves and their definitions. The bootstrapping procedure is quite clear as well. The gray area starts with passing of time.
The right question to ask would be: what happens to a forward that delivers 10-year bond in 3 years after 2 years (hint: it becomes a forward that delivers 10-year bond in 1 year)? Can we compare this forward and that forward and talk about the change of value? If these two forwards are indeed different objects then the meaning of a change is not clear at all.
If the forward _ curve _ doesn’t change, the forward price will change.
It’s not that we define a change in the yield curve differently; it’s that you’re confusing a change in the forward curve with a change in the forward price.
OK, I think I finally got it: Let’s say we have currently an upward sloping yield curve. The forward curve is above it. Four possibilities from here:
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the yield curve in 3 years is the same as today (unchanged). Then bond prices in 3 years are expected to be higher than what current forward curve suggests. Therefore 7-year bond will be priced higher than what the forward curve suggests. At the same time, buying 3 year bond today will provide the return exactly what the forward curve suggests. Conclusion: Buy a 10 year bond and sell in 3 years as 7 year bond. Return will be better (if you believe in the scenario)
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the yield curve follows the forward curve for 3 years (goes up). All bonds will have the same annual return each year. No difference between buy-and-hold and riding the curve.
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the yield curve goes down (let’s say just a shift). Even better than the case 1. Ride the curve.
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the yield curve in 3 years rises above the forward curve (a shift). Buying a 3 year bond and holding it to maturity is better than riding the curve.
Whew! It took me long!