# Chapter Three Understanding Interest Rates Slide 3–3 Present Value  Four Types of Credit Instruments 1.Simple Loan – all principal & interest due at.

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Chapter Three Understanding Interest Rates

Slide 3–3 Present Value  Four Types of Credit Instruments 1.Simple Loan – all principal & interest due at maturity 2.Blended Payment Loan 3.Coupon Bond 4.Discount Bond  Concept of Present Value  A dollar today is worth more than a dollar to be received in the future.  Yield to maturity  The interest rate that equates today's market value with the present value of all future payments  Equivalent to the IRR (Internal Rate of Return)

Slide 3–4 Yield to Maturity: Loans 1.Simple Loan Example: You borrow \$1,000 today. Repayment of \$1,469.33 is due in 5 years. What is the yield to maturity on the loan? Solution: Solve for i in the formula

Slide 3–5 Yield to Maturity: Loans 2.Blended Payment Loan Example:You borrow \$10,000. You will repay the loan in 60 monthly payments of \$212.47. What is the nominal annual yield to maturity for the loan? Solution: You can either use trial & error or use a financial calculator. We will opt for the calculator. 10,000PV 0FV 212.47 +/-PMT 60N i0.833326% per month or 10% per year

Slide 3–6 Yield to Maturity: Bonds 2.Coupon Bonds Example:A 10 year, \$1,000 face value bond with a 6% annual pay coupon is currently selling for \$929.76. What is the bond’s yield to maturity? Solution: You can either use trial & error or use a financial calculator. We will opt for the calculator. 1,000FV 60PMT 929.76 +/-PV 10N i7%

Slide 3–8 Bond Page of the Newspaper

Slide 3–9 Bond Page of the Newspaper

Slide 3–10 Relationship Between Price and Yield to Maturity Three interesting facts in Table 1 1.When the bond sells at par, yield equals coupon rate 2.Price and yield are negatively related 3.Yield is greater than the coupon rate when the bond’s price is below par value 4.Yield is less than the coupon rate when the bond’s price is above par value

Slide 3–11 Decomposing a Bond’s Yield to Maturity  The yield to maturity on a bond can always be decomposed into:  Current yield  Capital gain or loss  For example, assume that we are holding the 10 year bond reviewed earlier. It has a \$1,000 face value, a 6% annual pay coupon and it is currently selling for \$929.76, providing a 7% YTM.  We want to decompose the YTM into its constituent parts

Slide 3–12 Decomposing Yield to Maturity  Step #1: Calculate the Current Yield  Step #2: Calculate the Capital Gain or Loss  First, calculate the market price of the bond, assuming one year has passed but all else remains equal. 1,000FV 60PMT 9N 7i PV\$934.85

Slide 3–13 Decomposing Yield to Maturity  Then calculate the capital gain  Now add up the current yield plus the capital gain to obtain the YTM Where: P 1 = Price at end of period P 0 = Price at start of period

Slide 3–14 Yield to Maturity: Bonds 4.Discount Bonds Example: A 91 day, \$100,000 face value Treasury bill is currently selling for \$98,600. What is the bond’s annualized yield to maturity? Solution: Where: Price = current market price Face = face value of T bill B = Annual basis (365 in Canada) n = days to maturity

Slide 3–15 Clean & Dirty Bond Prices  The usual formula for calculating the market price of a bond is:  This formula assumes that the sale transaction is taking place on a coupon payment date, which is unrealistic for most market transactions  Therefore, have to calculate two prices:  Clean price – the quoted price  Dirty price – the clean price plus accrued interest since the last coupon payment date

Slide 3–16  Graphically, it appears like this: Clean & Dirty Bond Prices +

Slide 3–17  Example: Assume that it is March 1, 2006 and we are holding a \$1,000 face value bond with a 6% YTM and a 7% coupon paid semi-annually. The bond matures December 31, 2010. Coupon payment dates are December 31 and June 30. Calculate the clean & the dirty price of the bond. Clean & Dirty Bond Prices Dec 31, 2006 Dec 31, 2007 Dec 31, 2008 Dec 31, 2009 Dec 31, 2010 June 30 Today – March 1, 2006 The clean price is the PV of all future cash flows not yet received, as of the transaction date. \$35 \$1035

Slide 3–18 Clean & Dirty Bond Prices Calculating the Clean Price Calculating the Dirty Price

Slide 3–19 Day Count Conventions  In the capital markets, there are a number of ways that days between dates are computed for interest rate calculations. Many of these conventions were developed before the wide spread introduction of computers. The historical rationale for many of these calculations was to simplify the math involved in performing normally complex financial calculations. And as in most industries with a long history, many of these conventions have stayed with us despite considerable advances in computers and computational methods.  The day count basis indicates the manner by which the days in a month and the days in a year are to be counted. The notation utilized to indicate the day count basis is (days in month)/(days in year).  The five basic day count basis are the following:  Actual/360  Actual/365  Actual/Actual  30/360  30/360 European

Slide 3–20 Day Count Conventions: Actual/360  This calculates the actual number of days between two dates and assumes the year has 360 days. Many money market calculations with less than a year to maturity use this day count basis.  For example, a \$1 Million six month CD issued on 4/15/2006 and maturing on 10/15/2006, with an 8% coupon would pay an interest payment of:  Actual days between 4/15/2006 to 10/15/2006 = 183 days  Interest = 0.08 x 1,000,000 x (183/360) = \$40,666.67

Slide 3–21 Day Count Conventions: Actual/365  This calculates the actual number of days between two dates and assumes the year has 365 days.  Using an Actual/365 day count basis, a \$1 Million six month CD issued on 4/15/2006 and maturing on 10/15/2006, with an 8% coupon would pay an interest payment of:  Actual days between 4/15/2006 to 10/15/2006 = 183 days  Interest = 0.08 x 1,000,000 x (183/365) = \$40,109.59

Slide 3–22 Day Count Conventions: Actual/Actual  This day count basis calculates the actual number of days between two dates and assumes the year has either 365 or 366 days depending on whether the year is a leap year. More accurately, if the range of the date calculation includes February 29 (the leap day), the divisor is 366, otherwise the divisor is 365.  Using our CD example, the interest payment would be:  Actual days between 4/15/2006 to 10/15/2006 = 183 days  Interest = 0.08 x 1,000,000 x (183/365) = \$40,109.59  Notice that even if 2006 were a leap year, the denominator used for this calculation would be 365 because February 29 does not fall into the date range of the calculation. If the issue date was before February 29 and the year were a leap year, the divisor would have been 366 instead.

Slide 3–23  This day count convention assumes that each month has 30 days and the total number of days in the year is 360 (12 months x 30 days per month). There are adjustments for February and months with 31 days.  The formula for the 30/360 day calculation is as follows:  Assume Date 1 is of the form M1/D1/Y1 and  Date 2 is of the form M2/D2/Y2.  Let Date 2 be later than Date 1.  Then: If D1 = 31, change D1 to 30  If D2 = 31 and D1 = 30, change D2 to 30  Days between dates = (Y2-Y1) x 360 + (M2-M1) x 30 + (D2-D1) Day Count Conventions: 30/360

Slide 3–24  The 30/360 day count basis is different outside of the United States. The Europeans further simplified this calculation as follows.  Assume Date 1 is of the form M1/D1/Y1 and  Date 2 is of the form M2/D2/Y2.  Let Date 2 be later than Date 1.  Then: If D1 = 31, change D1 to 30  If D2 = 31, change D2 to 30  Days between dates = (Y2-Y1) x 360 + (M2-M1) x 30 + (D2-D1) Day Count Conventions: 30/360 European

Slide 3–25 Distinction Between Real and Nominal Interest Rates  Real interest rate 1.Interest rate that is adjusted for expected changes in the price level 2.Real interest rate more accurately reflects true cost of borrowing 3.When real rate is low, greater incentives to borrow and less to lend = the expected rate of inflation

Slide 3–26 Distinction Between Real and Nominal Interest Rates (cont.) Fisher Equation: developed by Irving Fisher (1867 – 1947) Cross product term When inflation is low, use the approximation. When inflation is high, use the exact method

Slide 3–27 Distinction Between Real and Nominal Interest Rates (cont.) Real RateInflation Rate Nominal Rate After-tax Nominal After-tax Real Rate 5%0%5%2.5% 5% 10%5%0% 5%10%15%7.5%-2.5% Nominal rate = Real rate + Inflation rate Assumptions: 50% marginal tax rate

Slide 3–28 Inflation as a Form of Taxation  Any tax is a transfer of purchasing power from A to B  The inflation tax is a transfer of purchasing power from creditors to debtors  The largest debtor is the federal government  The government issues a long term bond with a fixed coupon  During the life of the bond, inflation rises  The bundle of goods & services able to be purchased at the maturity of the bond is less than that purchased at the bonds issue date

Slide 3–29 U.S. Real and Nominal Interest Rates Figure 3: Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2001 Sample of current rates and indexes http://www.martincapital.com/charts.htm

Slide 3–30 Real Return Bonds  Issued by the Government of Canada  Principal is grossed up every 6 months, based on the change in the CPI.  The coupon is paid based on the grossed-up face value  Protects the holder against unexpected inflation

Slide 3–31 Key Facts about the Relationship Between Rates and Returns

Slide 3–32 Burton Malkiel’s Five Bond Theorems 1.Bond prices move inversely to bond yields 2.Long bonds have greater price volatility than short bonds 3.For a given change in yield, price volatility increases but at a decreasing rate as term to maturity increases 4.The capital gain due to a fall in market yields is always larger than the capital loss due to a rise in market yields 5.High coupon bonds have less price volatility than low coupon bonds

Slide 3–33 Reinvestment Rate Risk  Question: You buy a \$1,000, three year bond with a 6% coupon at a price of \$1,027.23 to yield 5% to maturity. If you hold the bond to maturity, are you assured of realizing a yield of 5% on your bond investment?

Slide 3–34 Reinvestment Rate Risk  Answer: Contrary to popular thought, you are not assured of earning a realized yield of 5%, unless you can reinvest each of the coupons at 5%. Assume that shortly after you purchase the bond, market rates drop to 4%. What is your realized yield if you hold to maturity? 0123 1,027.2360 1,060.00 62.40 64.90 1187.30 Now solve for the yield required for \$1,027.23 to grow to \$1,187.30 in three years (see next page for solution).

Slide 3–35 Reinvestment Rate Risk 1,187.30FV 0PMT 1027.23 +/-PV 3N i4.9456% Or, using Algebra

Slide 3–36 Reinvestment Risk 1.All coupon bonds are subject to reinvestment rate risk 2.The higher the coupon, the greater the reinvestment rate risk 3.The only bonds not subject to reinvestment rate risk are zero coupon or strip bonds

Slide 3–37 Duration  Three factors affect the price volatility of a bond. These are:  Term to maturity  Size of coupon  General level of interest rates  Duration captures all three factors in one number.  Duration was introduced by Frederick Macaulay in 1938

Slide 3–38 Two Definitions of Duration 1.Duration is the approximate percentage change in the price of a bond, given a 1% change in market yields 2.Duration is that point in time when the capital gain or loss due to a change in the YTM is exactly offset by the change in the reinvestment rate on the coupons.

Slide 3–39 Calculating Duration i =10%, 10-Year 10% Coupon Bond

Calculating Duration i = 20%, 10-Year 10% Coupon Bond Copyright © 2004 Pearson Education Canada Inc. Slide 3–40

Slide 3–41 Formula for Duration  Where:  Dur = Macaulay’s Duration  t = the number of time periods  C = the cash flow at time period t  i = the bond’s initial yield to maturity

Slide 3–42 Formula for Duration Numerator 1.First calculates the PV of each cash flow 2.Then multiplies the PV by the number of time periods until the cash flow occurs Denominator 1.Is equal to the current market price of the bond

Slide 3–43 What are we Measuring with Duration? YTM Price P0P0 YTM 0 True price-yield relationship Price-Yield relationship as measured by duration

Slide 3–44 Duration  Duration is a linear relationship that is attempting to measure something that is not linear  This leads to errors in measurement  As the size of the interest rate shock grows, so does the size of the error  Duration will always overstate the capital loss and understate the capital gain  Can correct the duration error using convexity

Slide 3–45 Modified Duration  Modified duration reduces some of the measurement error  To calculate the dollar & the percentage change in the price of a bond using modified duration

Slide 3–46 Using Modified Duration  Assume that you are holding a three year, \$1,000 bond with a 7% coupon and a YTM of 8%.  Calculate the bond’s duration  Calculate the dollar change in the price of the bond using modified duration, assuming the YTM rises by 20 basis points  Calculate the percentage in the price of the bond using modified duration, assuming the YTM rises by 20 basis points

Slide 3–47 Duration  Duration of a coupon bond is always less than maturity  Duration of a zero coupon bond is equal to duration  Duration will rise as maturity increases  Duration rises as the coupon decreases  Duration rises as the YTM decreases

Slide 3–48 Duration of a Portfolio  Assume that you have a portfolio consisting of 4 assets, as shown on the right side of the page: AssetDuration (D i ) Proportion of Portfolio (X i ) #11.510% #23.025% #35.035% #48.030%

Slide 3–49 Duration & Immunization - Banks  A Bank can immunize its Balance Sheet against changes in interest rates by matching the duration (rather than the maturity) of its assets & liabilities. Why might it want to do this?  To calculate the duration of the assets and liabilities, first calculate the duration of each instrument. Then multiply the duration of each instrument by its market value weight (See formula on next page).

Slide 3–50 Duration of the Assets & Liabilities  Duration of the assets & liabilities Where: D A = Duration of the assets D L = Duration of the liabilities X A1 = The proportion of the 1 st asset D 1 A = The duration of the 1 st asset

Slide 3–51 Calculating the Change in Market Value of the Assets & Liabilities When Rates Change  We use the duration formula to calculate the change in the market value of the assets & liabilities due to a change in market interest rates (remember modified duration) But what we really want to know is, what is the change in equity due to a change in interest rates?

Slide 3–52 Calculating the Change in Equity Where: k = a measure of leverage

Slide 3–53 Understanding the Formula  Can now decompose the effect of a change in market interest rates on the bank’s equity into three separate effects:  the leverage adjusted duration gap (D A - D L k)  Is measured in years  Reflects the exposure of the B/S to interest rate shocks  Size of the bank (reflected by the size of A)  The size of the interest rate shock  The change in equity captures the exposure of the bank to an interest rate shock

Slide 3–54 Convexity  Duration is an accurate measure of price sensitivity for small changes in interest rates. It is not accurate for large changes. R Price True Price/Yield relationship Duration Error R1R1 R2R2 P1P1 P2P2 P3P3 When rates rise from R 1 to R 2, duration suggests that price will fall from P 1 to P 2, but in fact, the true fall in price is only to P 3. The distance from P 2 to P 3 is the error due to duration.

Slide 3–55 Convexity  Duration  Overstates the decline in price due to a rise in interest rates  Understates the increase in price due to a fall in interest rates  Convexity is used to correct for this pricing inaccuracy  Convexity is:  desirable for an asset  undesirable for a liability

Slide 3–56 Calculating Price Changes, Including Convexity  To calculate the percentage change in price, use the following formula: Where: CX = convexity And where CX is calculated as follows:

Slide 3–57 Using Convexity  Example: To calculate the convexity of a \$1,000, 6 year, 8% coupon, 8% YTM Eurobond, first calculate the capital loss and the capital gain from a 1 basis point change in interest rates. Then multiply by a scaling factor of 10 to the 8th power:

Slide 3–58 Calculating the Change in Price  The percentage change in price is now calculated as follows (assuming a 2% increase in interest rates, from 8% to 10%):

Slide 3–59 Convexity: Things to Remember  Convexity increases with maturity  Convexity increases as coupon size decreases  When duration is the same, zero coupon bonds are less convex than coupon bonds

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