How does one calculate carry-roll-down theoretically assuming expectations of short-term rates are realised

Question

I am not asking for an explanation that is hugely quantitative, but rather one that is more intuitive.

I am aware that there are different assumptions that one could take when it comes to carry-roll-down, such as forward rates being realised and yields being unchanged.

But I am also reading that a better assumption would be if expectations of short-term rates are realised. How could one theoretically do this? I am aware that expectations of short-term rates are linked to forward rates, but not sure how one would set this out. I am reading that it is hard to implement because they would have to specify expectations of rates in the future and then describe how forward rates are formed relative to the expectations.

This is the quote from Tuckman when he is comparing with the other assumptions that I am trying to understand:

"A more conceptually appealing scenario for computing carry-roll-down is that expectations of short-term rates are realized. This is much more difficult to implement than the other scenarios presented in this section because an investor has to specify expectations of rates in the future and then describe how forward rates are formed relative to these expectations"

The bit that confuses me the most is the last sentence - what does he mean when he says "and then describes how forward rates are formed relative to these expectations?"

Answer 1

Carry and roll-down are intuitively relatively simple concepts. Imagine you trade a 10y Swap, where you pay fixed and receive 6m floating rates. Imagine that:

• Your fixed rate is: $r_{10}$
• Your first 6m floating coupon is $c_0$ (which gets fixed at inception of the Swap trade, because the floating rates are fixed "in advance" and paid six months in arrears)
• The swap curve is upward-sloping: meaning that: $r_1<r_2<r_3<...<r_{10}$ (where $r_1$ is the fixed rate of a 1y swap, $r_2$ is the fixed rate of a two year swap, etc).

Carry in the world of Swap-trading refers to how much it costs you to hold your position after you've entered into it: when you trade the swap and the first floating coupon had been fixed, holding the position for the first six months (all else staying constant) will cost you simply the present value of: "$c_0 - r_{10}$".

(in our example, the carry will be negative, because the curve is upward sloping, meaning that $c_0$ is smaller than $r_{10}$, and you pay the fixed (so if you hold the swap until the first cash-flow materializes, you are guaranteed to have a negative carry of "$c_0 - r_{10}$", if you assume that the yield-curve stays exactly the same as when you had entered the trade). Exercise: convince yourself that forward-starting swaps have zero carry (why? Because no floating cash-flows are fixed at inception)

The roll-down is simply how all your future cash-flows revalue as the swap maturity shortens (i.e. as you "roll down the curve"). Think of it like this: in 6 months from trade inception, your 10y swap will become 9.5 year swap. At trade inception, you committed to paying a fixed rate equal to $r_{10}$, and because the curve is "upward sloping", the fixed rate on a 9.5 year swap, i.e. $r_{9.5}$ (at trade inception) is lower than $r_{10}$. If you freeze this yield curve at trade inception and assume it'll look exactly the same in 6 months, you will be sitting on a 9.5 year swap, but still paying fixed rate $r_{10}$ that is higher than $r_{9.5}$: so your roll-down will also be negative.

In conclusion:

• whether carry and roll-down are negative or positive depends on the shape of the swap curve.
• Carry is just the difference between your fixed rate and the first floating coupon (annualized, often expressed in bps per day or bps per month).
• Roll-down is the difference between your fixed rate and the next (liquid) fixed-rate point on the swap curve (shorter in maturity)

Carry on Bonds:

• You buy the bond at inception, the money you use to buy the bond needs to be funded at some funding rate $r_{funding}$ (assume you roll-over the funding bi-weekly via your treasury)
• The bond accrues interest (assume at the rate of yield, i.e. $y$)
• If there is a liquid repo market for the bond, you can lend the bond out and earn extra bps on the repo (assume $r_{repo}$ as what you make (assume bi-weekly roll-over))

Assuming you don't reinvest any proceeds, your total bond carry $C$ per month will be (bond notional = $N$) (all rates annualized):

$$C=N\left(-2(r_{funding})+y+2r_{repo}\right)\frac{1}{12}$$

Answer 2

Not exactly answering your question, let us walk through a simplified example.

Suppose you have an interest rate swap. Suppose you use the same interest rate curve build from 3mo and 6mo LIBORs, and 1Y, 1Y6M, 2Y, 3Y... swap rates to project floating leg's coupons and to discount future cash flows. (Using ED futures to build the curve leads to boring technical issues.)

You calculate the fair price of the swap on day T-1. That includes the accrued on fixed and floating legs.

Not exactly what you asked, instead of assuming that 1 day forwards are realized, let us build an interest rate cure on T+0 using the same LIBORs and swap rate quotes from T-1. I.e. assume that the markets have not moved at all: whatever the 5Y swap rate was on T-1, it is the same on T+0 (with 1 day later maturity). You calculate the fair price of the swap on day T+0 using this rolled curve. The change in the accrued on fixed and foating legs is the carry. The rest of the change in fair value due to the passage of time is the rolldown.

If you're trying to build a P&L Explain, where you attribute the change in fair value from T-1 to T+0 to the change in the observable market rates, then calculating the rolldown like this will leave less unexplained P&L than if you reprice on T+0 just re-using the curve from T-1 (view-less shift, double-counting the P&L contribution of realized forwards in the contribution of IR change). I.e. discount factors something like $D(T+0,t)=D(T-1,t)/D(T-1,T+0)$?

Another way around this double-counting might be combine viewless shift (the forwards are realized) with some IR-time cross-gamma to cancel out the double-counting.