Consider the following bonds:

$(1)$ A bond with maturity in three years, which pays out a yearly $3\%$ coupon.

$(2)$ A bond with maturity in two years, which pays out a yearly $5\%$ coupon.

$(3)$ A zero coupon$-$bond with maturity in four years.

All bonds have a face$-$value equal to SEK $1000$ and a yield to maturity equal to $4\%$; the

first interest coupons are due in one year from today.

We want to hedge a $$default$$ portfolio comprised of a long position in five bond of

the type in $(1).$

$3\mathrm{.}1\mathrm{.}1.$ How can we use the bonds in $(2)$ and $(3)$ to hedge the default portfolio against

changes in the YTM$,$ such that the present values of the hedge and the default portfolio are identical in absolute terms, but have opposite signs?

$3\mathrm{.}1\mathrm{.}2.$ The requirements imposed in $3\mathrm{.}1\mathrm{.}1$ guarantee that if the YTM on all bonds

remains unchanged, then the present values of the default portfolio and the hedge will

be identical, in absolute terms, albeit they will generally be different from today$$s ones.

Verify that this would be the case by calculating the value of each portfolio in one

year, in the scenario of unchanged YTM; do remember to include the cash payments

due on the bonds.

$3\mathrm{.}1\mathrm{.}3.$ Consider again the scenario in one year from now, with unchanged YTMs$,$ in

$3\mathrm{.}1\mathrm{.}2.$ Would the slopes of the default portfolio and the hedge also be identical, up to

their signs? Dear Expert, I am preparing to write exams on this question, unfortunately, I don't understand anything concerning this question. I want to appreciate this question by getting to know the meaning and reasons behind the calculations. I wish you can add calculations to your explanation please.