Part $3$ Butterfly Strategy

A common strategy in fixed income is the butterfly strategy; it profits from a parallel shift in the yield curve. In this problem, you will construct a portfolio to execute the butterfly strategy by buying$/$selling bonds in the correct proportions to profit from a parallel shift in the yield curve. In addition, you will be asked to compute the profit$/$loss on the strategy for a shift in the curve.

Consider the following zero$-$coupon curve with annually compounded yields.

Maturity Yield

$2-$years $5\%$

$5-$years $5\%$

$10-$years $5\%$

The strategy calls for the investor to buy long$-$ and short$-$dated bonds. In this case, the investor would buy the $2-$year and $10-$year bonds while selling short the $5-$year bond.

Strategy requirements $($constraints$)$: The strategy must satisfy two requirements to be a butterfly strategy:

$($ i $)$ It should be $($modified$)$ duration neutral. Meaning that the change in the value of the long and short position implied by MoD must offset. Specifically, the product of MoD and total bond value for the long position must equal the product of MoD and total bond value for the short position. This constraint means that the portfolio is hedged against small levels changes in the yield curve, which are not part of a parallel shift in the curve.

$($ ii $)$ It must be a zero$-$cost portfolio. That is$,$ the proceeds from selling short the mid$-$term bond must fully fund the purchase of long and short$-$dated bonds.

$1.$ Question:

Approximate the modified duration for the $2-$year bond using a change in yield $(\backslash$Delta y$)$ of $0\mathrm{.}0001$

$2.$ Approximate the modified duration for the $5-$year bond using a change in yield $(\backslash$Delta y$)$ of $0\mathrm{.}0001$

$3.$ Approximate the modified duration for the $10-$year bond using a change in yield $(\backslash$Delta y$)$ of $0\mathrm{.}0001$

$4.$ Find the market value of your position in the $5-$year bond if you shorted $$1,000$ of face value in this bond to implement the Butterfly Strategy.

$5.$ Use the following notation to write down the $2$ mathematical expressions that describe the two constraints which define the butterfly trading strategy.

X $=$ the market value of the $2-$year bond to buy for the strategy

Z $=$ the market value of the $10-$year bond to buy for the strategy

Include any explanation you think may be relevant