$6)$ Questions: Answer the following steps: Work must be shown. Photocopies, Cutting and Pasting from outside sources, etc. will not count. Please show your formulas and assumptions. These steps will lead you to the valuation of the Credit Default Swap.a$)$ Step $1$: Calculate Marginal Default Probabilities.

Reference$-$Case Study: Steel, Swaps, and Storms $-$ The Titan Manufacturing HedgeIt is July $14,2024$ and the global economy teeters on the brink of recession. Trade wars have escalated, and a series of climate disasters have disrupted supply chains worldwide. In this volatile landscape, Titan Manufacturing, once a stalwart of the American steel industry, finds itself in troubled waters.Sarah Chen, the newly promoted Head of Risk at GlobeTrust Investments, faces her first major test. She has the reputation as a brilliant quant in risk management strategies. GlobeTrust holds $$10$ million in Titan's bonds, a position that seemed rock$-$solid just a year ago. Titan Manufacturing is led by CEO Jack Morrison, a third$-$generation steel man struggling to adapt to new realities. Now, as storm clouds gather, Sarah must navigate treacherous financial waters to protect her firm's investment.Titan's latest earnings call was a disaster. CEO Morrison fumbled questions about rising energy costs and plummeting demand from the automotive sector. Rumors of layoffs at Titan's flagship plant in Pittsburgh are rife, and the stock has plunged $15\%$ in a month.Sarah's team has been working around the clock, analyzing Titan's financials and market conditions. As she stares at the data visualization on her screen, a notification pops up from Eliza who wants an update by end of day.$$ Eliza Thornton, who happens to be Sarah's mentor and GlobeTrust's Chief Investment Officer who has seen it all expects Sarah to make the right call.With the weight of a $$10$ million investment on her shoulders, Sarah decides to explore hedging with a $5-$year credit default swap $($CDS$).$ She reaches out to Carlos Mendes at FinShield Bank, knowing his expertise in pricing complex derivatives.As a result of the meeting, Carlos and she agree on the following:$-$ Notional Amount: $$10$ million$-$ Tenor: $5$ years$-$ Risk$-$free rate: $2\%-$ Recovery Rate: $40\%$To refine their default probability estimates, Sarah consults Dr$.$ Kapoor, Chief Economist, MetalMinds Consulting, which is the leading voice in the metals and mining sector. His insights are quite sobering as he stated that: "Titan's not just fighting market forces, they're battling technological disruption and climate policy shifts. Morrison's old$-$school approach is like bringing a knife to a gunfight."Based on Dr$.$ Kapoor's analysis, they derive cumulative default probabilities:Year $1$: $1\mathrm{.}2\%,$ Year $2$: $2\mathrm{.}5\%,$ Year $3$: $3\mathrm{.}8\%,$ Year $4$: $5\mathrm{.}0\%,$ Year $5$: $6\mathrm{.}1\%$As Sarah and Carlos finalize their calculations, breaking news flashes across financial networks. Titan has just secured a major government contract to supply steel for a nationwide infrastructure overhaul, part of a new climate resilience initiative. The market reacts instantly. Titan's stock jumps $8\%$ in after$-$hours trading. Sarah's phone buzzes $-$ it's Eliza. "Are we exposed or protected?" the CIO's text reads.Sarah quickly reconvenes with Dr$.$ Kapoor and Carlos. The economist urges caution: "The contract's a lifeline, but Titan's fundamentals are still weak. Morrison needs to use this opportunity to modernize operations, or they'll be back in hot water within a year." Carlos agrees. "The market's overreacting. This news doesn't change Titan's short$-$term debt servicing capabilities."With minutes to spare before her deadline, Sarah makes the call. She decides to proceed with the CDS purchase. Carlos offers a spread of $80$ basis points $($NOTE: The Spread of $80$ basis points does not necessarily represent fair value which you need to determine$).$ Sarah sends a detailed recommendation to Eliza, her finger hovering over the 'Send' button as doubt creeps in$.$ With a deep breath, she hits send.