What is the difference between pull to par and roll down in both mathematics and conceptual?

Question

I don't really understand the difference. Shouldn't roll down and pull to par be the same technically? If a bond is trading as a discount it "increases" in value because everyday gets closer to par, and it rolls into another issue which has gotten closer? I feel that pull to par is incorporated within roll down because of this.

Answer 1

Pull-to-par just says that a bond's (clean) price will converge towards its face value as the bonds approaches maturity. There is nothing really interesting about pull-to-par - a bond's (clean) price has to converge to its face value, because a bond with just a few days to maturity is essentially a short-term cash deposit.

Look at it this way - the price of an $n$-year zero coupon bond is

$$ p_n = 100/(1+y)^n\approx 100 \times (1 - ny) $$

The approximation is okay for small $n$, i.e. when we are close to maturity. This clearly converges to 100 as $n$ approaches zero, no matter what the yield is. This is pull-to-par.

Roll-down is a statement about the capital appreciation or depreciation on a bond, assuming that the shape of the yield curve doesn't change. For example, say that you have the yield curve below (for simplicity, say it is the yield curve for zero coupon bonds).

The yield of the five-year bond is 4.08%, and so its price is $100 / (1.048)^5 = \$79.10$. In a years time, assuming that the yield curve is unchanged, it will be a four-year bond, with a yield of 3.7%, so its price will be $100 / (1.037)^4 = \$86.47$.

Therefore the return from holding the bond, assuming that its yield doesn't change, is

$$ R = \frac{86.47}{79.10} - 1 = 9.32\% $$

Note that this is much higher than the bond's yield, which is $4.08\%$! The difference between the bond's yield, and the expected return assuming no change in the yield curve, is the roll-down. In this case the roll-down is

$$ 9.32\% - 4.08\% = 5.25\% $$

so the roll-down can be a very significant contributor to the return on a bond, especially in environments with steep yield curves and low yield volatility.

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One way to understand carry, yield and roll-down is to look at the return on zero-coupon bonds. If the yield curve for a bond maturiting in $n$ years, at time $t$ is $y_{n,t}$ then the prices of zero coupon bonds are

$$ p_{n,t} = \frac{1}{(1 + y_{n,t})^n} $$

One year later, at $t+1$, that bond is priced using the yield $y_{n-1,t+1}$ (because after one year has passed, the bond has a tenor one year shorter) so its price is

$$ p_{n-1,t+1} = \frac{1}{(1 + y_{n-1,t+1})^{n-1}} $$

The return from holding the bond over the year is

$$ \begin{align} R & = \frac{\frac{1}{(1 + y_{n-1,t+1})^{n-1}}}{\frac{1}{(1 + y_{n,t})^n}} - 1 \\ & = \frac{(1 + y_{n,t})^n}{(1 + y_{n-1,t+1})^{n-1}} - 1 \\ & \approx ny_{n,t} - (n-1)y_{n-1,t+1} \\ & = y_{n,t} + (n-1)(y_{n,t} - y_{n-1,t}) - (n-1)(y_{n-1,t+1}-y_{n-1,t}) \\ & = y_{n,t} + (n-1)(y_{n,t} - y_{n-1,t}) - (n-1)\Delta y_{n-1,t} \end{align} $$

The first term is the yield return, the second is the roll-down return and the final term is the duration return, i.e. the return due to changes in yield between $t$ and $t+1$.

Looking at the roll-down return, you can see that roll-down is larger for bonds that have higher duration (i.e. larger $n$) and larger for bonds that are on a steep part of the yield curve (i.e. $y_{n,t} - y_{n-1,t}$ is large).

Answer 2

Pull-to-par says that the bond's price will gradually converge toward par (100% of face value) when yield is unchanged. This process is also known as accretion for a bond trading at a discount (since its price gradually goes higher toward par) and amortization for a bond trading at a premium (since its price gradually declines toward par). Pull-to-par says nothing about the shape of the yield curve.

Rolldown is all about the shape of the yield curve. If the yield curve is upward sloping, you "roll down" the yield curve (i.e., yield goes down) as time passes, resulting in capital gains. If yield curve is downward sloping, you "roll up" the yield curve with the passage of time (i.e., yield goes up).

Consider a 10-year zero coupon bond trading at a yield of 10%. Its initial price is $100 / (1 + 10\%)^{10} = 38.55$ (assuming annual compounding). After a year, its price, assuming the same yield, becomes $100 / (1 + 10\%)^9 = 42.41$. This increase is price is pull-to-par at work.

Now assume that the yield curve is upward sloping, such that the 9-year yield is 9%. Further, let's assume that the yield curve did not change over the year. Even though the yield curve is unchanged, because our original 10-year zero coupon bond has rolled down to the 9-year point, its yield is now 9% instead of 10%. The price is therefore $100 / (1 + 9\%)^9 = 46.04$. This is rolldown at work.