How Yield and Cash-Flow Timing Determine Bond Price and Duration

# How required yield affects price of the bond and how the durations changes


# How required yield affects price of the bond and how the durations changes












can somebody answer, those two theoretical questions?

- How does the bond price depend on the desired yield (market interest rates)?

- How the duration changes if we have a shorter / longer maturity and if we have a bond with a coupon / without a coupon

My answer for the second question would be, if we have longer maturity and the bond with coupon then the duration will be smaller that the duration for the shorter maturity fir the bond with coupon. When we have the bond without coupon, the duration will be larger for the longer maturity.

Is that correct? But I do not know answer for the first one.

Can anyone please help me?

## Answer by Kevin (score 2, accepted)

https://quant.stackexchange.com/a/50519

I start with some general information and answer your questions below. I'll assume continuous compounding.

The bond price is given by \begin{align*} P_B=\sum_{i=1}^n c_{t_i} \cdot e^{-y\cdot t_i}, \end{align*} where $c_{t_i}$ denote the $n$ coupon payments occurring at time points $t_i$ and $y$ the yield-to-maturity. Note that the last payment $c_{t_n}$ included both, the final coupon and the bond's face value.

The (Macaulay) duration is given by \begin{align*} D &= -\frac{1}{P_B} \frac{\partial P_B}{\partial y} \\ &= \frac{1}{P_B} \sum_{i=1}^n t_i\cdot c_{t_i}e^{-y\cdot t_i}. \end{align*} The duration may be interpreted as weighted average of the time points $t_i$ when the coupon payments occur with unit ''year'' and ''weights'' $\frac{c_{t_i} e^{-y\cdot t_i}}{P_B}$. Note that these weights add up to one. Alternatively, the duration approximates the change in the bond price given a change in the interest rate, i.e. $\Delta P_B\approx -D P_B \Delta y$. Using discrete compounding, you'll need the modified duration here. For larger changes in the interest rate, you may want to include the bond's convexity.

As a special case, consider a zero-coupon bond with $c_{t_i}=0$ for $i<n$ and $c_{t_n}=1$. Thus, $P_B=e^{-y t_n}$ and $D=t_n$. As there are no payments, the weighted averages of the coupon payment dates is simply the bond's maturity: that is how long you have to wait until you receive cash flows. Note that the bond price can be solved for the bond's yield explicitely, i.e. $y=-\frac{1}{t_n}\ln(P_B)$. This is in general not possible for coupon bonds whose yield-to-maturity is typically found numerically.

Now, let's talk about your questions...

- The bond price $P_B$ is monotonically decreasing in the yield $y$. The higher $y$, the lower the present value of the individual coupon payments, $c_{t_i}e^{-y t_i}$. Hence, we observe a negative/inverse relationship between bond price and bond yield.

- The duration of a zero-coupon bond is simply the bond's maturity. Hence, the shorter the time to maturity, the lower the bond's duration. This makes sense. A zero-coupon bond which matures quite soonish is hardly sensitive towards interest rate changes. A coupon-bearing bond has an even lower duration than a zero-coupon bond since you already receive some cash flows (coupon payments) prior to the bond's maturity.