Identification of Risk Factors

1. Identification of Risk Factors

2. Market Risk and Credit risk • Market risk is defined as the risk of fluctuations in portfolio values due to volatility in market price. • Absolute and Relative risk • Absolute risk is measured in dollars terms – used for bank trading portfolios. • Relative risk is measured relative to benchmark index – used for pension funds that are given the task of beating a benchmark group.

3. Source of Market Loss – Bond and Equity Bond A bond value is defined as the present value of the bond’s payments discounted by the yield to maturity. yield to maturityis the annual yield promised to investor who will hold the bond to maturity.

4. Bond Valuation Numerical Example A bond with $1,000 notional value, 8% coupon and 5 years to maturity is traded with 10% yield to maturity. Bond’s Payments

5. P 5 years bond 10 years bond y Yield to Maturity and Bond Price

6. Yield to Maturity and Bond Price • Every things else are equal, the sensitivity of the bond price movement to movements inyields increases as: • The yields are lower • The time to maturity is longer • The coupon rate is lower • Duration • Duration is a measure of the sensitivity of the bond price movement to movement in yields. • Duration is measured as the weighted maturity of each payment, where the weights are proportional to the present value of the cash flow.

7. Thus, • where D* is modified duration and D*P is also known as the dollar duration. • This sensitivity is sometimes expressed in dollar value of a basis point:

8. Numerical Example A bond with $1,000 notional value and 8% coupon and 3 years to maturity is traded with 10% yield to maturity. Bond’s Payments

9. The modified Duration is: This implies that increasing of 1% in the yield will cause to:

10. Spot and Forward Rates • The yield curve is the relationship between the yield to and the time to maturity. • The yield described by the spot rates, ST , which are derived from zero-coupon bond prices with different maturity. • Prices of zero-coupon bonds with different maturity

12. Forward Rates • Forward rates, Ft,T are the rate on investment that start at a future date t to time T. • Example • An investor who wishes to invest for 2 years has two alternatives: • Buying two years bond with a spot rate S2 • Buying one year bond with a spot rate S1, and roll over the investment by entering to forward contract to buy in the next year a one year bond with a forward rate F1,2.

13. S1 S3 S3 F1,2 F2,3 F1,3 1 2 3 0

14. Forward Rates • Since the two portfolios must have the same payoff, we can infer F1,2 form: • and in general:

16. Default Risk Premium – Spreads Over Treasuries • Corporate bonds have an additional risk factor over government bonds - the risk of default. • Default Risk – firm’s failure to pay the coupon payment and/or the par value at maturity. • It causes the yields for corporate bonds to exceed those for Treasury bonds – the difference known as the spread over Treasury • The higher the risk of default, the lower the firm’s bond rating, the lower the bond’s market price, and the higher its yield.

17. Bond Rating

18. Default Risk Premium – Spreads Over Treasuries • For a given maturity, the lower the bonds rating, the higher its yield to maturity DRP

19. Fixed-Income Risk • Fixed income risk arises from potential movement in the level of bond yields. • The Fixed income risk can be measured either as return volatility or yield volatility:

20. Fixed-Income Portfolio Risk • The major problem with individual bonds is that there may not be sufficient history to measure their risk. • Therefore, we model the movement in each corporate bond yield by: • A movement in Treasury zero-coupon rates with a closest maturity - zj • A movement in the DRP of the credit rating class to which it belong - sk. • The remaining component, ei, which assumed to be independent across the bonds.

21. The movement in the bond price is: • DVBP is the total dollar value of a basis point for the associated risk factor. Specific bond z+s+e BBB z+s z Treasury 3M 10Y 20Y 5Y

22. Summing across the portfolio and collecting terms across the common risk factors: • Thus, a portfolio may consist of N=100 corporate bonds, but we can summarize the yield risk only with j=5 government bonds.

23. The total variance:

24. Numerical Example Portfolio Composition Change in Basis Points at Time t

25. Equity Portfolio Risk • The different market risk can be measured by the volatility of the major indexes. DAX HSI S&P500 Nikkei 225

26. Equity Portfolio Risk • The diagonal model is a statistical decomposition of the return of the stock i into a market-wide return and a residual which called the specific risk.

27. Equity Portfolio Risk • The diagonal model assumes that all specific risks are uncorrelated. • Thus, any correlation between two stocks must come from the joint effect market. • Therefore, with a large portfolio the specific risk should cancel each other, and the only remaining risk is the general market risk.

28. Equity Portfolio Risk The portfolio variance is: Suppose, is equally weighted and the residual variance are the same for all stocks:

29. Factor Model • The one factor model may miss common industry effects. • Adding factors, such as industry factors to the model improves the precision of the individual stock return, and decreases the error term. • The factors X are assumed to be independent

30. Currency Risk • Currency risk arise from potential movement in the value of foreign currencies. • Currency risk includes currency specific volatility and correlations across currencies, and devaluation risk. It arises in the following environments: • A pure currency float • A currency devaluation • A change in the exchange rate regime • Currency risk is also related to the interest rate risk – Often, interest rate are raised in effort to stem the depreciation of the local currency.

31. Exchange Rates Volatility Against the USD