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# YouTubeT4-31: Fixed income: Carry roll down

#### Nicole Seaman

Staff member
Subscriber
Financial Risk Manager (FRM, Topic 4: Valuation and Risk Models, Fixed Income, Bruce Tuckman Chapter 3, Returns, Spreads and Yields). The Carry-Roll-Down is the price change in the bond due exclusively to the passage of time. It is only one component of a bond's total profit and loss (P&L). The bond's total P&L equals Price Appreciation plus Cash Carry (i.e., coupon). Price Appreciation equals Carry-Roll-Down plus Price Change due to Shift in Rates (market risk) plus Price Change due to spread narrowing/widening (credit risk). Last edited:

#### Esteban5185

##### New Member
Does an unfunded derivative such as an Interest Rate Future have carry?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
H @Esteban5185 interest question. For me, it depends what is meant exactly by "carry" as I perceive it has multiple definitions (or rather, it depends on the context). The above video is illustrated carry-roll-down: price change in the bond due solely to the passage of time, without respect to (because they are captured as other components) interest rate shifts (aka, market risk change) or credit spread widening/narrowing (ie, credit risk change).

As an interest rate future is a futures contract on an commodity, I would define "carry" as cost of carry (COC) and therefore would consider any backwardation situation as a positive carry. In backwardation (aka, inverted futures curve), F(t++) < F(t+) < S(0) such that, if the forward curve does not shift, the roll return is positive: as the contract matures, the spot price is increasing. Theoretically, COC says F(t) = S(0)*exp[(r+u-q - y)*t] so that we'd expect backwardation when (income + convenience) > (financing + storage cost), or put simply, the benefits of ownership exceed the costs. I hope that's interesting.

#### Branislav

##### Member
Dear @David Harper CFA FRM , if you can explained more intuitively this part:
"If the term structure slopes upward on average and yet remains unchanged on
average, it must be that the upward-sloping shape is completely explained by investors’
requiring a risk premium that increases with term..".
In details, do not understand part ..slopes upward on avg and yet remains stable on avg...?
Thanks a lot in advance as usual

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Branislav Did i say that, I must have said that somewhere? Well, our baseline but unrealistic theory of the term structure is pure expectations. Pure expectations says that forward rates predict future spot rates without any other factors. So if the current, say, six month spot (interest) rate is 2.0% and the predicted future spot rate is also 2.0%, then the forward rate must be 2.0% because it represents the "pure expectation" of the future spot rate. In this case, the forward rate curve must be flat, and so too therefore, the spot rate curve. So imagine we are in this world, where today the spot rate is 2.0% and next month, quarter and year, the spot rate will also be 2.0%. In a world where the future spot rate is constant and where pure expectations applies, the term structure must be flat (cannot be upward sloping).

Now introduce some upward slope to the term structure: upward slope in the spot rates implies even greater upward slope in the forward rates. Now our forward rates are predicting future spot rates greater than 2.0%. But if the term structure is static, then our short term six-month spot rate remains at 2.0%; e.g., if we go forward in time one year, the six-month spot rate will still be 2.0% yet the term structure is upward sloping (i.e., this is the assumption of unchanged term structure )! What happened? the greater-than-2.0% forward failed to exactly predict the realized 2.0% short rate. Why did it over-estimate? Because investors must have imposed a risk premium. Put another way, If the term structure is upward sloping starting at 2.0%, then the forward rates must be higher than 2.0%, and if pure expectations applies, then the future spot rate must go higher than 2.0%. If they do not, the pure expectations failed to account for everything.

So I am not sure exactly how/where I voiced it, but what I think i meant is: under an assumption of unchanged term structure, and where the term structure happens to upward-sloping, the upward slope must be explained by a risk premium demanded by investors.