Valuation is a process of finding the fair value of a security at a particular moment in time in a specified economic environment. Good valuation depends on the ability to reverse-engineer its cash flow structure (cash flow uncertainties, optionality, and path dependency) as a function of time-based systematic risk factors. According to the option pricing theory, the fair price of a security is defined as a mathematical expectation of all conceivable discounted future payoffs.
Because valuation models use information from the current economic environment (yield curves, foreign exchange rates, observed market prices, implied volatilities, and various credit spreads) as input. Therefore, prices of fixed income securities would change due to the inputs fluctuate. In this case, the job of risk management is to assess the risk associated with the market moves. Here, the knowledge of probability associated with potential losses is combined with the estimates of their magnitude to arrive at a "worst-case loss." In contrast to valuation that attempts to model the bulk of the probabilistic distribution of future returns, risk management is concerned with potential losses which found at the left tail of the future distribution of random returns.
Traders in the market use Taylor series expansions to study the price sensitivity of fixed income securities. The first-order approximation of the price/yield function is known as delta, or duration (as elasticity measure). The second-order approximation is called delta-gamma or duration-convexity.
Modified duration - a measure of price sensitivity of a fixed income security to changes in its yield-to-maturity. It is linked to the first-order term of Taylor series expansion. Modified duration is defined as the negative of the percentage change in price, given 100 basis point change in yield:
Modified Duration = - (1/P) (dP/dy)
Modified Convexity = 1/P (d
2P/dy
2,) - estimates the degree of nonlinearity of the price/yield curve.
Macaulay Duration = (1+y) x Modified Duration - measure price volatility, i.e. larger the Macaulay duration, the more volatile the price of the bond. This simply implies higher risks for securities whose cash flows are concentrated in the future.
For n-year bond that pays fixed annual cash flows CF
1, ..., CF
n and its yield-of-maturity,
P = SUM ( CF
t / [1+y]
t) t=1..n
Modified Duration = 1/ (P x (1+y)) x SUM ( (t x CF
t) / [1+y]
t) t=1..n
Approximation of percentage change in price for a given change in yield is given by:
delta P / P ~= -Modifed Duration x delta y + Modified Convexity / 2 x (delta y)
2.
Extension of modified duration methodology uses annualized spot rate r
t at time t rather than yield-to-maturity. Even so, the fair value may still be different from that on the market because of the numerous assumptions built into the generation of static cash flows. In order to reconcile the values, practitioners introduced the concept of zero volatility spread (ZVO), defined as an additional constant element of discounting that forces the fair value to equal the market price:
P = SUM ( CF
t / [1+r
t+ZVO]
t) t=1..n
Liquidity risk is important among the most potent sources of market risk. However, due to the over-the-counter nature of the majority of fixed income markets, machanisms for recording completed transactions do not exist. Thus, traders use subjective judgement to set the bid/ask spreads for each asset class.
