Forward Rates, Duration, and Interest Rate Swaps

1. Fin 603 Week 5 Forward Rates, Duration, and Interest Rate Swaps

2. Some Quick Notes • Class slides to appear sooner and be more stable • Barring unforeseen circumstances, slides will appear by midnight the day before they are first presented • Changes to slides once they are posted should be minimal (minor typos, etc.)—any major changes will be noted on WebCT • Professor Miller will not be holding office hours on Thursday, October 20 Professor Ross Miller • Fall 2005

3. Looking Ahead to Week 6 • Quick review of the course to date • Go over Week 5 problems • Preview of the two class projects • More on bonds, including topics from BKM Chapters 14 – 16 that have not yet been covered Professor Ross Miller • Fall 2005

4. The Following Week: The Mid-Term Exam • Place and Time • BA 229 (the usual classroom) • Special time: 6:15PM – 8:15PM (with an additional 20 “grace minutes”) • Dates: Thursday, October 20 • Format • 10 True-False (3 points each) • 2 Problems each with 5 questions (10 questions in total worth 7 points each) • Partial credit is possible throughout the exam Professor Ross Miller • Fall 2005

5. Parameters for Open-Book Exam • Allowed • Anything in print form, including textbooks, class slides, notes, etc. • Calculators (including financial, scientific, and graphing); however, all problems can be solved with a cheap, minimal calculator • Prohibited • Any device remotely capable of communication including cell phones, PDAs, and computers Professor Ross Miller • Fall 2005

6. 100 Total Points Possible for an Exam, Which are then Converted to GPA points • Points convert to GPA equivalent number according to the following formula: GPA equivalent = 4.33 – (100 – Test score)/15 • For example, • 95 = 4.00 = A 88 = 3.53 = lowest A- • 83 = 3.20 = lowest B+ 73 = 2.52 = lowest B- • Grades (in rare circumstances) may be “curved up,” they will never be “curved down” Professor Ross Miller • Fall 2005

7. What the Mid-Term Exam Covers • New material covered in class from Weeks 1 through 5 and in the Week 6 in-class problem review • New material introduced in Week 6, while useful for understand the previous material, is not explicitly covered on the mid-term • The main focus is what is covered in class and a basic understanding of the world’s current financial situation (exchange rates, yields, the Fed, etc.) • Although you should try doing Week 5 problems yourself before looking at the answers, the answers are already posted here Professor Ross Miller • Fall 2005

8. The Treasury Yield Curve Professor Ross Miller • Fall 2005

9. Four Main Yield Curve Shapes • Upward-sloping • The situation that occurs most of the now (including currently) • Provides a higher interest rate to lenders willing to assume the risk and reduced liquidity that comes from lending for a longer period of time • Downward-sloping (also known as inverted) • Usually the result of prolong Fed tightening • Often the forerunner of a recession • Flat (a fleeting moment between the above two) • Humped (so rare that few worry about it) Professor Ross Miller • Fall 2005

10. Assorted Yield Curves (a) Upward sloping (b) Inverted or downward sloping (c) Flat Yield to Maturity (a) (c) (b) Time to Maturity Professor Ross Miller • Fall 2005

11. The Yield Curve Is Dynamic • One component of this is predictable as the yield curve, as we shall soon see, contains information about the expected shape of future yield curves • Much of the movements, however, are unpredictable • View the animated history of the yield curve Professor Ross Miller • Fall 2005

12. Why Sweat the Yield Curve? • It contains within it a forecast of future interest rates and provides the prices of forward rate agreements (FRAs) and interest rate swaps • The bond yield curve that is typically given in newspapers is not the one used by professionals because it is complicated by the coupon payments of the bonds • Professionals use the pure yield curve based on zero-coupon securities • It is a reminder that despite the fact that most of financial analysis assumes all cash flows from an investment can be discounted using the same interest rate, doing so it sometimes unwise Professor Ross Miller • Fall 2005

13. What Hides Inside the (Pure) Yield Curve • Estimates of forward interest rates are there for the taking • An example of the basic idea: If you know the yield out 2 years (y2) and the yield out one year (y1), you can reverse engineering the what the one-year yield should be one year from now (this is an implied forward rate) • The same idea works between any two points in time Professor Ross Miller • Fall 2005

14. A Note on the Approach Taken by BKM • BKM introduce the idea of a short interest rate (itself a confusing term) • As BKM note repeatedly, these rates do not really exist in reality, they are purely theoretical constructs that act just like the forward rates that can be implied from the pure yield curve • If they help you understand things, that is fine; otherwise, feel free to ignore them Professor Ross Miller • Fall 2005

15. Consider This Situation from BKM • 1-year yield is 8.000% • 2-year yield is 8.995% • So what is the implied forward rate for 1 year starting 1 year into the future • If interest rate math were user-friendly, a rate of 9.990% for the second year would then make the 2-year average rate be exactly 8.995% • Instead, we have to remember that (1+yn)n is the appropriate way to discount cash flows Professor Ross Miller • Fall 2005

16. Computing the Implied 1-Year Forward Rate • $1,000 invested in the 2-year 8.995% zero-coupon bond gives:$1,000(1+y2)2 = $1,000(1.08995)2 = $1,187.99 • $1,000 invested ion the 1-year 8.000% zero-coupon bonds gives:$1,000(1+y1)1 = $1,000(1.08) = $1,080.00 • Turning the $1,080 into $1,187.99 requires a one-year forward yield of:f2 = $1,000(1+y2)2 / $1,000(1+y1)1 – 1 = $1,187.99/$1,080.00 – 1 = 10.00% Professor Ross Miller • Fall 2005

17. The General Formula for Computing Implied 1-year Forward Rates Professor Ross Miller • Fall 2005

18. Theories of the Term Structure of Interest Rates • (Unbiased) Expectations Hypothesis: At a given point in time, the yield curve reflects the market’s current expectations of future short-term rates • Liquidity Premium Theory: Investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value • Market Segmentation Theory: Investors have specific maturity preferences and will generally demand a higher maturity premium Professor Ross Miller • Fall 2005

19. The (Unbiased) Expectations Hypothesis and Implied Forward Rates • Under this hypothesis, the forward rates implied from the yield curve are an unbiased (accurate on average) estimate of what interest rates will be in the future • Even without this hypothesis, the implied forward rates from the curves should be very close to the actual forward rates that prevail in the market • If they were not, profitable arbitrage could occur • This is very similar to interest-rate parity Professor Ross Miller • Fall 2005

20. A Closer Look at f2 • In the example, the implied 1-year forward rate 1 year from now, f2, was 10% • If we were a financial institution, we would like to have a way to “lock in” a 10% one-year fixed rate on money we borrow for 1 year beginning 1 year from today (Note: This is a liability to the bank) • If the agreement involves a loan of $1,000,000, we would pay $1,100,000 a year later Professor Ross Miller • Fall 2005

21. What if a Year from Now (When the Loan Begins) the Interest Rate Rises to 11% • Then, instead of paying $1,100,000 back on the loan, we would have to pay $1,110,000 back, which is $10,000 more • We could, however, be “made whole” (get back to the original 10% rate) if someone gave us $9,009, which is $10,000/1.11 • At the end of the year, this $9,009 (at 11% interest) would become $10,000—making up for the extra $10,000 in interest we would need to pay out Professor Ross Miller • Fall 2005

22. What if a Year from Now (When the Loan Begins) the Interest Rate Drops to 9% • Then, instead of paying $1,100,000 back on the loan, we would have to pay $1,090,000 back, which is $10,000 less • Now to get back to the 10% rate we would pay $9,174, which is $10,000/1.09 • At the end of the year, this $9,174 (at 9% interest) would become $10,000—offsetting the extra $10,000 in interest we made when rates dropped Professor Ross Miller • Fall 2005

23. Forward Rate Agreement (FRA) • The financial arrangement described in the previous slides constitutes a forward rate agreement or FRA • It has the effect of freezing a future interest rate at a fixed level determined by the implied forward rate • The “twist” is that it is settled in cash rather than by making the two parties to the agreement actually transact at the fixed rate Professor Ross Miller • Fall 2005

24. Eurodollar Futures (traded on the CME) • A popular exchange-traded alternative to FRAs • Very similar because most FRAs are quoted in terms of Eurodollars • Like FRAs, Eurodollar futures are settled in cash rather than through the actual physical delivery of a 3-month Eurodollar • Because it is a futures contract, cash flows related to posting and maintaining margin take place prior to the “delivery” date Professor Ross Miller • Fall 2005

25. The PV of Fixed-Rate Securities Vary With Interest Rates • Obviously this is the case for a bond that pays a fixed coupon rate—rates go up and its PV goes down and vice versa • Somewhat less obviously this is the case for a bank that enters into an FRA that commits it to borrow money at a fixed rate • When rates go up, this commitment becomes less of a liability to the bank (someone else would be happy to pay the bank to be committed to borrow at 10%, for example, when the going rate is 11% Professor Ross Miller • Fall 2005

26. What About a Floating-Rate Alternative? • Suppose that the bank agrees to borrow $1,000,000 for a year one year from today; however, it will not commit to a rate now. Instead, it will pay whatever the prevailing rate is in a year • What would someone be willing to pay (or need to pay) to relieve the bank of this liability? Professor Ross Miller • Fall 2005

27. Duration: A Measure of Interest Rate Sensitivity • The duration of an investment is a quantitative indication of its interest rate sensitivity that is measured in years • Higher duration means a greater sensitivity (in percentage terms) to changes in interest rates • Doubling duration means doubling the interest rate sensitivity • A bond with a duration of 6 years will drop twice as much (as a percent of its price) as a bond with a duration of 3 years when interest rates increase by 1 basis point (0.01%) Professor Ross Miller • Fall 2005

28. Duration Example • Consider a Treasury bond (or note) with the following properties • One year left to maturity • $1,000 face value • 8% coupon rate (paid semiannually) • Its yield (to maturity) is 10% • Cash flow from this bond • $40 (4% of $1,000) paid in ½ year • $1,040 (4% of $1,000 plus the $1,000 face value) paid in 1 year Professor Ross Miller • Fall 2005

29. Duration Example Timelines Professor Ross Miller • Fall 2005

30. Computing the Present Value of the Bond • Value of $40 in ½ year = $40/(1.05) = $38.10 • Value of $1,040 in 1 year = $1,040/(1.05)2 = $943.31 • Total value = $38.10 + $943.31 = $981.41 Professor Ross Miller • Fall 2005

31. Computing Duration • Duration is the cash-flow weighted time to maturity Duration = (38.10)(0.5) + (943.31)(1.0) 981.41 = 0.9806 years • Notice that if the first cash flow were eliminated, making the total value of the bond $943.31, the duration of the bond would simply be 1 year Professor Ross Miller • Fall 2005

32. Features of Duration • Duration and Coupon Interest • The higher the coupon payment, the lower is a bond’s duration • Duration and Yield to Maturity • Duration decreases as yield to maturity increases • Duration and Maturity • Duration increases with the maturity of a bond but at a decreasing rate Professor Ross Miller • Fall 2005

33. Relationship Between Maturity and Duration on a Coupon Bond Professor Ross Miller • Fall 2005

34. Duration, Zero-Coupon Bonds, and Bond Portfolios • For a zero-coupon bond, duration equals maturity • The duration of a portfolio of bonds is the value-weighted average of the durations of the individual bonds • A portfolio with $1,000 in a bond with a duration of 2 years and $2,000 in a bond with a duration of 4 years has a duration of [2(1,000)+4(2,000)]/(3,000) = 3.33 years Professor Ross Miller • Fall 2005

35. Stocks Also Have Duration • The typical stock goes down on average when interest rates go up • Unlike bonds, however, they sometimes go up with interest rates (usually when the economy is getting stronger) • Why should this happen? The value of a stock is determined by (uncertain) discounted cash flows • Stock duration is usually high (20 years and up) and ignored by most finance people Professor Ross Miller • Fall 2005

36. Modified Duration • Duration works out nicely because an N-year zero-coupon bond has a duration of N years • Duration has one problem, as a measure of interest-rate sensitivity it is calibrated to changes in the discount factor (1+r) • To calibrate duration to interest rates directly (r), one needs to compute modified duration • The problem with modified duration is that the duration of an N-year zero-coupon bond is now less than N years Professor Ross Miller • Fall 2005

37. A Big Problem with Both Durations • Duration only applies to very small changes in the discount factor (as the price of the bond changes, so does duration) • The duration of zero-coupon bonds; however, only changes with time, not price • The measure of how much duration changes is known as convexity, which we will cover in great depth next time Professor Ross Miller • Fall 2005

38. Financial Institutions and Other CorporationsOften Desire Immunity from Interest Rates • The concept of duration can be extended to a company’s balance sheet • The ideal situation for most companies is total immunity—no balance sheet exposure to changes in interest rate • This can be achieved (in theory) by matching asset duration with liability duration • Note: Because shareholder equity is greater than zero, assets will exceed liabilities Professor Ross Miller • Fall 2005

39. The Duration Gap • A formal measurement of how much interest rate risk a company has on its balance sheet • The formula for the duration gap is: Professor Ross Miller • Fall 2005

40. Condition Where Duration Gap is 0 Dassets x Total assets = Dliabilities x Total liabilities Professor Ross Miller • Fall 2005

41. Quick Duration Gap Example • Assets of $1,000 with a duration of 4.2 year • Liabilities of $800 with a duration of 4.4 years (Note: Shareholders’ equity must be $200) Duration Gap = 4.2 – ($800/$1,000) x 4.4 = 0.68 years When interest rates go up, this company’s assets will drop faster than its liabilities, eating into its equity Professor Ross Miller • Fall 2005

42. More About Duration Gaps • If the duration gap is positive • Same situation as holding a bond with a positive duration • Increase in interest rates hurts earnings • Decrease in interest rates helps earnings • Hence, the FI is betting that rates will drop • If the duration gap is negative • Increase in interest rates helps earnings • Decrease in interest rates hurts earnings • Hence, the FI is betting that rates will raise Professor Ross Miller • Fall 2005

43. How to Reduce the Gap • One solution: Sell some of the bank’s loansand hold “cash” instead • A better solution: Enter into a contract that hedges the interest rate risk • While both the forward and futures contracts described earlier can do this, a variant on them known as an interest rate swap is the preferred method Professor Ross Miller • Fall 2005

44. Interest Rate Swaps • Designed as an exchange of fixed-rate interest payments (high duration) for floating-rate interest payments (low duration) • A swap enables one to change one’s level of interest rate risk without disrupting the underlying portfolio or the balance sheet • Most swaps have a value of zero at the time of creation • The value of a swap changes with changes in interest rates • Popular with bankers, corporate treasurers, and portfolio managers who need to manage interest rate risk Professor Ross Miller • Fall 2005

45. Why Interest Rate Swaps Affect Duration • The fixed side of the swap has the standard positive duration of a fixed coupon bond—for short-term bonds this duration is a bit less than the bond’s maturity as we saw earlier • The floating side of the swap has only a slightly positive duration because the interest rate is reset frequently—if the interest rate were continuously reset, the floating side would have zero duration • The net duration is then almost that of the fixed-rate side Professor Ross Miller • Fall 2005

46. Swaps from the Financial Institution’s Perspective • Interest rate swaps affect the duration of the financial institution’s liabilities • The buyer of a swap (the one who pays the fixed rate) increases the duration of the liabilities, which decreases the value of the duration gap • The seller of a swap (the one who pays the floating rate) decreases the duration of the liabilities, which increases the value of the duration gap Professor Ross Miller • Fall 2005

47. An Important Point: The Assets Themselves Are Not Swapped • An interest rate swap is equivalent to a series (or bundle) of FRAs • Both parties to the swap continue to own the “swapped” assets • Only the cash flows are swapped • Each period during the swap, the difference in cash flows between the two assets is computed and the difference is paid Professor Ross Miller • Fall 2005

48. Swaps Are the Prime Determinant of U.S. and Most Global Interest Rates • Swaps are the easiest way for large financial institutions to adjust their duration • Large swap transactions by influential market participants affect the entire yield curve • Hence, the influence of swaps on interest rates is similar to what we will see is the effect of S&P Index futures on the stock market Professor Ross Miller • Fall 2005

49. A Useful Comparison • At the short end of the yield curve, the management of cash/reserves (liquidity) by financial institutions drives the market • At the longer end, the management of interest rate risk (duration) by financial institutions drives the market Professor Ross Miller • Fall 2005

50. Some Swap Details • The standard type of interest rate swap is the fixed for floating rate swap • One party makes a fixed interest rate payment to another party making a floating interest rate payment • Only the net payment (difference check), if any, is made when the swap is entered into • The firm paying the fixed rate is the swap buyer • The firm paying the floating rate is the swap seller Professor Ross Miller • Fall 2005