Fixed-Income I. Alan Gu. What s a bond? An obligation. To make a fixed amount of periodic payments In exchange for money lent

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1 Fixed-Income I Alan Gu What s a bond? An obligation To make a fixed amount of periodic payments In exchange for money lent 2 1

2 Quick Definitions Principal = What one borrows Interest = What one pays to borrow Maturity = How long one borrows the $$ for 3 Example: Anand Sharma borrows $100 from me at 6% per annum for 1 year to buy a Movado. Principal (What Anand owes me): $100 Interest (What Anand pays to borrow to buy his dress): 6.00%. Maturity (How long before Anand pays me back the $100): 1 year In fixed-income, we use the term coupon and interest interchangeably Think of bonds as loans 4 2

3 $1,200 Dollar Value vs. Present Value 9% 10-Year Note, Priced at 6% YTM $1,000 $800 $600 Dollar Value Present Value $400 $200 $ Periods in 1/2 Years Before the Formulas Think of bonds as individual cash flows For instance, think of a 30-year bond w/ semi-annual coupon payments as 60 individual cashflows. You get paid interest every 6 months, and at the end of 30 years, you get paid you principal back 6 3

4 Formulas Where: P = Price of the Bond C = Coupon Payment Z = Discount Rate M = Principal Amount 7 Sensitivity to Rates As discount/interest rates fall, bonds rally As discount/interest rates rise, bonds sell off In the extreme, if interest rates fall to 0%, then the present value of future payments is equal to the actual payments that you should be expected receive 8 4

5 Duration-Convexity Relationship 9%, 10-Year Note Priced at 6% YTM $1,200 $1,000 $800 $600 $400 Bond w. Convexity Bond w /o Convexity $200 $0 0.00% 0.60% 1.20% 1.80% 2.40% 3.00% 3.60% 4.20% 4.80% 5.40% 6.00% 6.60% 7.20% 7.80% 8.40% 9.00% 9.60% 10.20% 10.80% 11.40% 12.00% Levels of Interest Rates Bond Math Modified Duration (First Derivative): The percentage change in the price of the bond with respect to the change in the level of interest rates (market rates). Convexity (Second Derivative): The rate of change of duration. Yield to Maturity (YTM/IRR): The rate that sets the present value of coupon and principal payments to the current price. YTM: The nominal return one expects to get should one hold the bond to its maturity. YTM is also the discount rate. 10 5

Macaulay Duration/Modified Duration 11 Calculating

948 6.161 6.161 2 6.50 0.898 5.840 11.680 3 6.50 0.852 5.

867 6 6.50 0.725 4.714 28.285 7 6.50 0.687 4.468 31.

6 Macaulay Duration/Modified Duration 11 Calculating Duration Maturity 10/15/06 Settlement 10/15/97 Coupon(C) 6.50 Yield(r) 5.50 t C t 1/(1+r) t C t /(1+r) t C t *t/(1+r) t Total: Duration = / =

7 Duration Contd. Maturity Yields Coupon Duration Duration Duration 13 Duration.. Examples Maturity 4% Yield 6% Yield 8% Yield Coupon (Years) Price Mod Dur Price Mod Dur Price Mod Dur 4.00% % % % % % % % %

8 Popular Conventions In reality, YTM has little relevance. Cash flows are rather individually discounted off of the spot curve (zero coupon curve interpolated bootstrapping ) BPV (DV01): Basis point value or Dollar Value of 1 Basis Point E.g. how much in dollar terms does my bond move in value for every 1 basis point (0.01%) change in rates (market rates) 15 Formulas Where: P = Price of the Bond C = Coupon Payment Z = Discount Rate M = Principal Amount 16 8

9 17 Assumptions/Flaws of Duration Future cash flows do not change with the change in interest rates We will see why the above is a big problem when using duration to value mortgages Yield curve movements are parallel 18 9

10 Why is Convexity Valuable? Convexity represents the second derivative of the price/yield relationship of a bond. It measures the rate of change in duration. Convexity = CURVATURE As rates fall, a more convex bond will rally MORE than a less convex bond. As rates rise, a more convex bond will sell off LESS than a less convex bond. 19 $1,200 Duration-Convexity Relationship 9%, 10-Year Note Priced at 6% YTM $1,000 $800 RED LINE: No Convexity/Curvature, the Price-Yield Relationship is linear $600 $400 Bond w. Convexity Bond w /o Convexity $200 $0 0.00% 0.60% 1.20% 1.80% 2.40% 3.00% 3.60% 4.20% 4.80% 5.40% 6.00% 6.60% 7.20% 7.80% 8.40% 9.00% 9.60% 10.20% 10.80% 11.40% 12.00% Levels of Interest Rates 10

11 Why is Convexity Valuable? Therefore, all else equal, more convex bonds will trade richer (lower yield) than a less convex bond. You can make bets on realized volatility of interest rate using a non-convex and a convex bond. Euro$$ futures and Swaps. You can make lots of $$$ if traders misprice the convexity in a swap vs. Euro$$s Shorting Euro$$ futures vs. long swaps is a doubly convex position due to MTM 21 Using Duration and Convexity We can use duration and convexity to estimate the change in the price of the bond. a = initial yield, F(a) = price of bond at initial yield x = initial yield + change in yield, F(x) = price of bond at yield x x a = change in yield, F(x) F(a) = change in price of the bond f (a) = first derivative at point a f (a) = second derivative at point a 22 11

12 Introduction to Treasuries Interest payments are made on a semi-annual basis Most liquid and actively traded issues are the on-theruns (most recently issued) 2 s 5 s, 10 s and 30 s They trade special versus other bonds in that their repo costs (financing costs) are lower than the general collateral rate 23 Introduction to Strips STRIPS = Separate trading of interest and principle We can trade the individual interest and principal payments as zero coupon bonds We can also reconstitute different interest (c-strip) and principal (p-strip) and trade them as a coupon bond That is, we can bring different c-strip and p-strip together and create a new bond 24 12

13 Arbitrage 101 The laws of arbitrage ensures that the price of the C and P Strips must be equal to the price of the Bond as a whole. Otherwise 25 Treasury Yield Curve Graphical description of the most currently issued 2- year, 5-year, 10-year and 30-year securities with their respective yield to maturities on the Y-axis and their maturities on the X-axis. Many securities are priced off of the Treasury Yield curve. (Swaps) Treasuries are the BENCHMARK curve for the fixedincome markets much like the S&P 500. Many long term interest rates mortgage rates (fixed) are linked to the long end of the yield curve 26 13

29 Trading the Yield Curve (The Super-Basics) Blending Macro views on growth & inflation, with quantitative measures Steepeners (bull-bear) Flatteners (bull-bear) Carry Trade bet against forwards

15 29 Trading the Yield Curve (The Super-Basics) Blending Macro views on growth & inflation, with quantitative measures Steepeners (bull-bear) Flatteners (bull-bear) Carry Trade bet against forwards Butterflies: butterfly. 2 s 10 s are cheap relative to 5 s -- Buy 2 s 10 s (Wings) and sell 5 s (Belly) Condors: butterfly Every Wall Street bank uses quantitative models to break down when the fly is cheap/rich. (PCA) 30 15

16 How Do I Make $$ Macro: Macroeconomic views on growth, inflation, employment and the Federal Reserve Micro: Understanding market flows and the linkage between Treasuries and other segments of Fixed- Income mortgages in particular Quantitative: Building systematic trading models to trade different fixed-income instruments.. E.g. Euro$$ options vs. Swaptions, etc. Quantitative: Using models to analyze the richcheapness of different segments of the yield curve Quantitative: Using quantitative measures to model cash bonds and futures. 31 Important Stuff Fixed-Income and mortgage-backed securities in particular, are intimidating because of the jargon. It s the market to trade if you understand macroeconomics and are good at math/statistics. Treasuries are only one small part of the market. Some securities are incredibly exotic and complex. Example: Digital IO A structured mortgage-backed derivative which only receives interest payments from a pool of mortgages. It pays an initial coupon of 3%, but if LIBOR > 7.00%, the coupon jumps to 25%. This is a super-leveraged bet on prepayments

National University of Singapore Dept. of Finance and Accounting. FIN 3120A: Topics in Finance: Fixed Income Securities Lecturer: Anand Srinivasan

National University of Singapore Dept. of Finance and Accounting FIN 3120A: Topics in Finance: Fixed Income Securities Lecturer: Anand Srinivasan Course Description: This course covers major topics in