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The bonds which Greece has paid had been valued by market as junk once, just before there payment. Given that the observed market value is the net present value of the instrument, why were they so low?

Because the value to be received has been discounted by the market via a huge credit spread related to the issuer=Greece?

Or because they have been reflecting the optionality of Greece between paying this debt and defaulting for everything(=cost of default)? When you consider the probability (of survival) to be/get the amount K_up, and the probability of default to be/get K_down, it is valuable as an option (with its associated NPV/price). The cash flows to come to be considered as a barrier basket option (with multiple cash settlements).

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ETFs Playing Bigger Role in Junk-Bond Market: Funds are poised to overtake credit derivatives as method of speculating on high-yield debt. Bloomberg, September 17 treasuryandrisk.com/2012/09/17/… –  user7056 Sep 19 '12 at 9:49
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Bonds are traditionally valued using the discount curve, the credit spread to determine the probability of default and usually you still have to adjust the price using a spread because bonds can become illiquid.

If you ignore the last spread the price is sum of discounted cashflows except that at every point of time you need to compute the implied probability of default. If the bond defaults, the market assumes that you receive a recovery. By computing the value of the coupon leg conditionally on survival and the default leg for the default case, you should obtain a NPV which is much closer to the traded price of this security.

Usually you still need to adjust this price by a spread as the market can trade the security at a different price than your theoretical price and this is especially true for illiquid bonds. You could generalise this by saying that the market tries to anticipate what is the distribution of the expected cashflows and asks for a premium for uncertainty.

Assuming a flat rate, the basic NPV of the cashflows is: $$ B = P_T \frac{1}{(1 + r)^T} + \sum C_t \frac{1}{(1 + r)^t} $$

Some would try to assume that you need to add a credit spread over the discounting rate: $$ B = P_T \frac{1}{(1 + r + c)^T} + \sum C_t \frac{1}{(1 + r +c)^t} $$

This is still incorrect as the implied default probability from the CDS spread. You would want to compute: $$ B = P_T \frac{1}{(1 + r)^T)}surv(T) + \sum C_t \frac{1}{(1 + r +c)^t}surv(t) $$

where $surv(t)$ is a function the cumulative probability of survival at time t.

But you are still incorrect, the bonds actually pay you a recovery in the form of cash or new bonds in the case of default. Assuming that this recovery is represented by an amount $R$ and the instantaneous default probability at time t is represented by a function $q(t)$: $$ B = P_T \frac{1}{(1 + r)^T)}surv(T) + \sum C_t \frac{1}{(1 + r)^t}surv(t) + R \int_{t=0}^T \frac{1}{(1 + r)^t} \cdot q(t) $$

Even with all these adjustments, you will still not be matching the market price of a security, you can consider the difference is likely to be due to some liquidity issues or error in your estimation of your recovery or other parameters.

There are various basis methods which will yield different results, the best is to find which one the people who trade this market seems to be using and try to match the market in addition to calculate a fair theoretical value.

HTH

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Thank you, @BlueTrin. NPV is supposed to be the difference between an investment's market value and its cost. The market value has been very low. There are 3 options here: –  user7056 Sep 7 '12 at 12:09
    
Thank you, @BlueTrin. NPV is supposed to be the difference between an investment's market value and its cost. The market value has been very low. There are 3 options: 1. the cost of the investment is negative, 2. the NPV did not match the market value (it should be even lower than the market value), 3. this definition of NPV is no longer exact. What you are talking about is the expected (credit) gain, which favours option 3? –  user7056 Sep 7 '12 at 12:15
    
I will detail my answer during the week-end as I am at work, but ... when you do NPV, you implicitely assume that there is no default as you multiply the cashflows by the discount factors. This does not work in a risky world, as the discount factor needs to be multiplied itself by the survival probability. –  BlueTrin Sep 7 '12 at 12:17
    
by the defintion of the NPV, one is supposed to have a model which has the NPV of the bond equal to the market value. The way to do the calibration is via the (credit) spread, which is supposed to express in a continuous way the rating state/probability of default. But (from the accounting point of view) one does not discount the incoming cash flows based on the credit state of the borrower, because it has already entered in the interest rate that you are charging for giving the principal away :( –  user7056 Sep 7 '12 at 15:29
    
@user7056: what you just posted tells you that you should add a spread to the discounting rate. i.e the discount factor becomes 1/(1+r+c) instead of 1/(1+r) for each year. This is a very simplistic assumption because: the CDS does not translate in default probabilities exactly like you described and because in case of default on a bond, you usually do not receive 0, you usually get a recovery amount. –  BlueTrin Sep 7 '12 at 16:32
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