Quantitative Finance - Fixed Income securities

Transcription

1 Quantitative Finance - Fixed Income securities Lecture 2 October 21, 2014

2 Outline 1 Risk Associated with Fixed Income Products 2 The Yield Curve - Revisit 3 Fixed Income Products

3 Risks Associated The return obtained from a fixed income security from the day it is purchased to the day it is sold: 1 the market value of the security when it is eventually sold 2 the cash flows received from the security over the time period that it is held, plus any additional income from reinvestment of the cash flow We can define the risk in any security as a measure of the impact of any market factors on the return characteristics of the security

4 The different types of risk that an investor in fixed income securities is exposed to are Market or Interest rate risk Reinvestment Risk Timing, or Call, risk Credit risk Liquidity risk Volatility Risk

5 Market or Interest rate Risk The price of a typical fixed income security moves in the opposite direction of the change in interest rates For an investor who plans to hold the security till maturity, this may not be of any concern; however the investor who may have to sell the security before maturity, an increase in interest rates will mean the realization of a capital loss To control interest-rate risk, it is necessary to quantify it. The most commonly used measure of interest-rate risk is duration

6 Reinvestment Risk The cash flows received from a security are usually (or are assumed to be) reinvested The additional income from such reinvestment, sometimes called interest-on-interest, depends on the prevailing interest- rate levels at the time of reinvestment, as well as on the reinvestment strategy. The variability in the returns from reinvestment from a given strategy due to changes in market rates is called reinvestment risk

7 Timing or Call Risk Bonds may contain a provision that allows the issuer to "call", all or part of the issue before the maturity date, the issuer usually retains this right to refinance the bond in the future if market interest rates decline below the coupon rate. There are three disadvantages to investors: First, the cash-flow pattern of a callable bond is not known with certainty. Second, because the issuer may call the bonds when interest rates have dropped, the investor is exposed to reinvestment risk.

8 Finally, the capital appreciation potential of a bond will be reduced because the price of a callable bond may not rise much above the price at which the issuer may call the bond. In mortgage backed securities - Prepayment risk

9 Credit Risk The credit risk of a bond includes 1 The risk that the issuer will default on its obligation (default risk) 2 The risk that the bond s value will decline and/or the bond s price performance will be worse than that of other bonds against which the investor is compared (rating agencies may lower the credit ratings) The first is called default risk whereas the second is called downgrade risk.

10 Liquidity Risk Liquidity risk is the risk that the investor will have to sell a bond below its true value where the true value is indicated by a recent transaction liquidity risk is the risk that a given security or asset cannot be traded quickly enough in the market to prevent a loss The primary measure of liquidity is the size of the spread between the bid price and the ask price quoted by a dealer. The wider the bid-ask spread, the greater is the liquidity risk

11 Volatility Risk The risk that a change in volatility will adversely affect the price of a security is called volatility risk.

12 The yield curve The most commonly occurring yield curve is the yield to maturity yield curve. The yield curve is a line graph that plots the relationship between yields to maturity and time to maturity for bonds of the same asset class and credit quality.

13 From figure 2.2 note the yield spread differential between German and Italian bonds. Yield curves are usually upward sloping Although both the bonds are denominated in euros and, according to the European Central Bank (ECB) are viewed as equivalent for collateral purposes (implying identical asymptotically: credit quality), the higher yield thefor longer Italian government thebonds maturity, proves that the market the views them as higher credit risk compared to German government bonds. higher the yield Yield % Negative Positive Humped Years to maturity Fig 2.1 Yield to maturity yield curves Figure 2.2 Bloomberg page IYC showing three government bond yield curves as at 2 December 2005

14 this could be due to two reasons 1 First, it may be that the market is anticipating a rise in the risk-free rate. If investors hold off investing now, they may receive a better rate in the future 2 Longer maturities entail greater risks for the investor - the economy faces more uncertainties in the distant future than in the near term

15 A humped curve results when short-term and long-term yields are equal and medium-term yields are higher than those of the short-term and long-term Under unusual circumstances, long-term investors will settle for lower yields now if they think the economy will slow or even decline in the future. The New York Federal Reserve regards it as a valuable forecasting tool in predicting recessions two to six quarters ahead. Historically, the yield curve has become inverted 12 to 18 months before a recession.

16 When does the slope of the yield curve change? A sharply upward sloping, or steep yield curve, has often preceded an economic upturn. Investors demand more yield as maturity extends if they expect rapid economic the Federal Reserve will often raise interest rates to fight inflation. growth because of the associated risks of higher economy inflation began to recover andfrom higher the recession interest of rates, whichfigure can 2 both hurt bond returns. Historically, the slope of the yield curve has been a good leading indicator of economic activity. Because the curve can summarize where investors think interest rates are headed in the future, it can indicate their expectations for the economy. A sharply upward sloping, or steep yield curve, has often preceded an economic upturn. The assumption behind a steep yield curve is interest rates will begin to rise significantly in the future. Investors demand more yield as maturity extends if they expect rapid economic growth because of the associated risks of higher inflation and higher interest rates, which can both hurt bond returns. When inflation is rising, Figure 2 below shows the steep U.S. Treasury yield curve in early 1992 as the U.S. Yield (%) Steep Yield Curve: April 30, m 2Y 5Y 10Y 30Y

17 Yield Curve A flat yield curve frequently signals an economic slowdown. A flat yield curve is unusual and typically indicates a transition to either an upward or downward slope. The flat U.S. Treasury yield curve in Figure below signaled an economic slowdown prior to the recession of September 2004 Figure 3 9 Flat Yield Curve: December 31, 1989 Yield (%) 8 7 3m 2Y 5Y 10Y 30Y

18 Fixed Income Products

19 Caps A caplet with reset date T and settlement date T + δ pays the holder the difference between a simple market rate F (T, T + δ) (e.g. LIBOR) and the strike rate κ. Its cash flow at time T + δ is δ (F(T, T + δ) κ) + A cap is a strip of caplets. It thus consists of

20 a number of future dates T 0 < T 1 < < T n with T i T i 1 = δ (T n is the maturity of the cap), a cap rate κ Cash flows take place at the dates T 1,..., T n. At T i the holder of the cap receives δ (F (T i 1, T i ) κ) + i = 1,..., n Let t T 0 then we write cpl (t, T i 1, T i ) i = 1,..., n

21 for the time t price of the i-th caplet with reset date T i 1 and settlement date T i, and cp(t) = n cpl (t, T i 1, T i ) i=1 for the time t price of the cap. A cap gives the holder a protection against rising interest rates. It guarantees that the interest to be paid on a floating rate loan never exceeds the predetermined cap rate κ.

22 It can be shown (Assignment no. 3) that the cash flow (above) at time T i is the equivalent to (1 + δκ) times the cash flow at date T i 1 of a put option on a T i -bond with strike price 1/(1 + δκ) and maturity T i 1, that is, ( ) 1 (1 + δκ) 1 + δκ Z (T i 1, T i ) This is an important fact because many interest rate models have explicit formulae for bond option values, which means that caps can be priced very easily in those models.

23 Floor A floor is the converse to a cap. It protects against low rates. A floor is a strip of floorlets, the cash flow of which is - with the same notation as above - at time T i δ (κ F (T i 1, T i )) + write Fll(t; T i 1, T i ) for the price of i-th floorlet and Fl(t) = n Fll(t; T i 1, T i ) i=1

24 It is a common market practice to price caps and floors using Black s formula. Let t < T 0 then Black s formula for the value of ith-caplet is Cpl(t; T i 1, T i ) = δz (t, T i )(F (t; T i 1, T i )N (d 1 (i; t)) κn (d 2 (i; t))) where d 1,2 (i; t) = log(f(t;t i 1,T i ) κ ) ± 1 2 σ(t)2 (T i 1 t) σ(t)(t i 1 t)