Microsoft has issued a zero-coupon bond. The zero-coupon bond has a price in the market today of

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Question:

Microsoft has issued a zero-coupon bond. The zero-coupon bond has a price in the market today of 97.5 dollars. The zero-coupon bond pays 100 dollars in exactly one year from now. To be precise, Microsoft promises to pay 100 dollars in one year's time. Of course, there is the possibility that Microsoft may default on its promised payment. In the event of default, liquidators would be appointed to sell off Microsoft's assets and then distribute the proceeds to bond-holders so they get some of their money back. Analysts estimate that Microsoft's assets will generate a payment of $60, per every $100 promised, in one year's time in the event of Microsoft defaulting.

Microsoft's bonds trade freely in the market and you can sell them short if you have to (without any extra costs). You can also borrow or lend risk-free at 2% per annum (continuously compounded). There are no transactions costs.

You are working as a trader at Goldman Sachs. You get a phone call from a customer who has significant exposure to the risk of Microsoft defaulting and she is really worried about this. She needs a hedge. She asks if, today, you will sell her a security to help her hedge her default risk. She wants to buy a security (let us call it a DIGITAL DEFAULT contract) from you today which pays nothing at all if Microsoft does NOT default but will pay her 100 dollars exactly one year from now if Microsoft does default (in essence, she wants to buy a kind of insurance contract) in the intervening time period.

The aim of this question is to establish at what price should you sell this DIGITAL DEFAULT contract to your customer. You assume the absence of arbitrage.

a) Set up a portfolio consisting of a short position in the DIGITAL DEFAULT contract and a position in Microsoft's bond which is risk-free to Goldman Sachs.

b) What is the value of this portfolio in dollars today? in one year's time?

c) In the absence of arbitrage, and using your answer to part (b), at what price would you sell the DIGITAL DEFAULT contract to your customer?

d) By using the risk-neutral valuation principle, check your calculation in part (c). What is the riskneutral probability Microsoft defaulting?