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1 Chapter 5 Bond Prices and Interest Rate Risk This Chapter … 2 Explain how interest rate movements affect the prices of assets and liabilities of investors and financial institutions. Time Value of Money 3 A dollar today is worth more than a dollar received at some future date Positive time preference for consumption Time Value of Money 4 Future Value: the value of a given amount of money invested today at a given point in the future at a given rate of interest The future value (FV) of a sum (PV) is: FV = PV (1+i)n where i is the periodic interest rate and n is the number of compounding periods. Time Value of Money 5 Example: $100 in savings for 5 years, and the bank pays 4% interest. Future value? Now assume interest is paid quarterly? The difference is because of compounding periods. Explain Time Value of Money 6 Present Value: the value today of a given sum of money to be received at given point in the future Risk-free interest rate Finding the present value is called discounting Time Value of Money 7 Present Value: Cont’d 1 PV = FV n (1 + i) Opportunity cost: Interest rate on the next best alternative investment Time Value of Money 8 Risk-free interest rate With risk present, a premium may be added to the risk-free rate. The higher the discount rate, the lower the present value. Bond Pricing 9 Focus on how investors and financial institutions price bonds An application of present value formula Bond Pricing 10 It is a contractual obligation of a borrower to make periodic cash payments to a lender over a given number of years Borrower issuer “ of bond” Lender holder “of bond” 11 1. 2. 3. Terms of the bond contract: Principle Coupon rate Term to maturity 12 Bond Pricing 13 Two types of cash flows: At maturity, holder receives principal (or face value or par value). Periodically before maturity, holder receives interest (coupon) payments determined by coupon rate, original interest rate promised as percentage of par on face of bond. c = C/F Term to maturity: timing of the cash flows Bond Pricing 14 Par value Coupon Rate Issued Matures $1,000 5% Today 30 years from today Scheduled Payments: $50/year interest for 30 years $1,000 par at end of year 30 Bond Pricing 15 Notes: It is assumed for most bonds that payment of both coupon and principle are made at maturity It can semiannually, quarterly Coupon rate and face value are fixed a similar bond: close substitute, same maturity and risk Coupon rate and interest rate may differ Bond Pricing 16 Bondholder thus owns right to a stream of cash flows: Ordinary annuity of interest payments and Future lump sum in return of par value, Discountable to a present value at any time while bond is outstanding. Bond Pricing 17 The value (price) of a bond is the present value of the future cash flows promised, discounted at the market rate of interest (the required rate of return on this risk class in today’s market) Bond Pricing 18 Bond Price Formula: C1 C2 CN + F N PB = + + ... 1 2 N (1 + i) (1 + i) (1 + i) Where PB = price of bond or present value of promised payments; Ct = coupon payment in period t, where t = 1, 2, 3,…, n; Fn = par value (principal amount) due at maturity; i = market interest rate (discount rate or market yield); and n = number of periods to maturity. Bond Pricing 19 Cash flows are assumed to flow at end of the period and to be reinvested at i. Bonds typically pay interest semiannually. Increasing i decreases price (PB); decreasing i increases price; thus bond prices and interest rates move inversely. Bond Pricing 20 If market rate equals coupon rate, bond trades at par. If coupon rate exceeds market rate, the bond trades above par—at a premium. If market rate exceeds coupon rate, bond trades below par—at a discount. Bond Pricing 21 Par value Coupon Rate Issued Matures $1,000 5% Today 30 years from today Scheduled Payments: $50/year interest for 30 years $1,000 par at end of year 30 Bond Pricing 22 Par value Coupon Rate Interest Rate Matures $1,000 8% 10% 3 years from today Bond Pricing 23 In your calculator: Enter the number of years over which the bond contract extends, n, the interest rate, i, the par value received at maturity, F (usually the FV key on your financial calculator), and the annual coupon payment amount Ct (usually the PMT key on your financial calculator) Bond Pricing 24 Price < face value Discount bond Price > face value Premium bond Price = face value Par bond Bond Pricing 25 Par bond: Par value Coupon Rate Interest Rate Matures $1,000 5% 5% 3 years from today Bond Pricing 26 a. b. What is the price if the market interest rate was: 8%? 2% ? Bond Pricing 27 Semiannual Compounding C1 m C2 m C N m+ F Nm PB = + + ... 1 2 Nm (1 + i) (1 + i) (1 + i) m m m: is the number of times coupon payments are made each year. m Bond Pricing 28 Par value Coupon Rate Interest Rate Matures $1,000 5% paid semi-annually 6% 3 years from today Bond Pricing 29 Zero Coupon Bonds: “have no coupon payments but promise a single payment at maturity” Examples: treasury bills, U.S savings bonds Issued at discount from par. Return on security is the difference between the purchase price and the face value. Bond Pricing 30 Zero Coupon Bond: F PB = mn i (1 + ) m Bond Pricing 31 Par value Interest Rate Matures $1,000 10% 10 years from today Recap.. 32 Already covered: 1. Time Value of Money 2. Bond Pricing We will cover: 3. Bond Yields 4. Important Bond Pricing Relationships 5. Interest rate and Duration Bond Yields 33 Yield: is the return on any investment Bonds Yield: Return on bond The yield should take into consideration the three sources of cash flow of a bond: 1. Coupon payment 2. Interest income from reinvesting coupon payment 3. Any capital gain or loss Bond Yields 34 1. 2. 3. Coupon rate : annual cash flow promised by the borrower to the lender Actual rate of return (for the lender), depends on risks: Credit or default risk Reinvestment risk Price risk Bond Yields 35 Important note: Reinvestment risk: change in market interest rate that might cause the lender to have to reinvest coupon payments at interest rates different from the interest rate at the time the bond was purchased. Price risk: change in interest rate that cause market value of the bond to change resulting in capital gains or losses (invers relationship) Bond Yields 36 1. 2. 3. 4. We will discuss four yield measures: Yield to Maturity Expected Yield Realized Yield Total Return Bond Yields 37 1. 2. 3. 4. We will discuss four yield measures: Yield to Maturity Expected Yield Realized Yield Total Return Bond Yields 38 Yield to Maturity: “Investor's expected yield if bond is held to maturity and all payments are reinvested at same yield” 1. 25 25 1,025 951.90 = + + ... 1 2 6 (1 + (i / 2) ) (1 + (i / 2) ) (1 + (i / 2) ) Normally determined by trail and error Bond Yields 39 1. Yield to Maturity (Cont’d): It is cumbersome and difficult method, but it can be solved using financial calculator. Calculation need the following input: 1. 2. 3. 4. Current bond price or PV number of periods over which the bond contract extends Coupon payments Bond face value Bond Yields 40 1. 1. 2. 3. Yield to Maturity (Cont’d): Important assumptions to calculate Yield to Maturity: Borrower makes all cash payments Interest rates does not change over the maturity Investors holds the bond to maturity Bond Yields 41 1. Yield to Maturity (Cont’d): the coupon payments are reinvested at a lower rate, the bondholder’s actual yield is less than the promised yield. Bond Yields 42 1. Yield to Maturity (Cont’d): Investor buys 5% percent coupon (semiannual payments) bond for $951.90; bond matures in 3 years. Solve the bond pricing equation for the interest rate (i) such that price paid for the bond equals PV of remaining payments due under the bond. Bond Yields 43 1. 2. 3. 4. We will discuss four yield measures: Yield to Maturity Expected Yield Realized Yield Total Return Bond Yields 44 2. Expected Yield “Predicted yield for a given holding period (same procedure as YTM, but for some holding period shorter than maturity)” Sell before maturity to know potential impact of interest rate change on returns on bonds investments Bond Yields 45 2. Expected Yield (Cont’d): Must forecast— Expected interest rate(s) Bond price at end of holding period Plug forecast results into bond pricing formula Bond Yields 46 2. Expected Yield (Cont’d): Given the future price, the investor can calculate the expected yield; that reflects the expected sale price Example: Book page 114 47 2. Expected Yield (Cont’d): Steps for solving for expected yield: - Find the expected future interest rate - Find the future price using the computed interest rate Bond Yields 48 1. 2. 3. 4. We will discuss four yield measures: Yield to Maturity Expected Yield Realized Yield Total Return Bond Yields 49 3. Realized Yield “The return earned on a bond given the cash flows actually received by the investor and assuming that the coupon payments are reinvested at the realized yield” It might differ from yield to maturity due to: a. change in the amount or timing of promised payments (e.g. default). b. change in market interest rates affecting premium or discount. Bond Yields 50 3. Realized Yield (Cont’d): Terminal price of a bond: the market price of the bond on the date you sell it. ( in other words, it is the price of remaining coupon payments and the final principle repayment) Realized yield is useful because it allows investors to evaluate the return on a bond ex-post (after the end of the holding period or investment horizon) Bond Yields 51 3. Realized Yield (Cont’d): Book example page 115: Investor pays $1,000 for 10-year 8% coupon bond; sells bond 3 years later for $902.63. Solve for i such that $1,000 (the original investment) equals PV of 2 annual payments of $80 followed by a 3rd annual payment of $902.63 (the actual cash flows this investor received). Bond Yields 52 3. Realized Yield (Cont’d): 80 80 902.63 1000 = + + ... 1 2 3 (1 + i) (1 + i) (1 + i) Solving either by trial and error or with a financial calculator results in a realized yield of 4.91%. Bond Yields 53 1. 2. 3. 4. We will discuss four yield measures: Yield to Maturity Expected Yield Realized Yield Total Return Bond Yields 54 4. Total Return: Both expected and realized yield calculation assumes that we will be able to reinvest coupon payments at the calculated yield. That’s not always true However, if we know or can explicitly predict the reinvestment rate, we will be able to calculate the Total Return on a bond. Bond Yields 55 4. Total Return (Cont’d): “It is the return we receive on a bond that considers capital gain or losses and changes in the reinvestment rate” To calculate the total return we need to determine two things: 1. 2. The terminal value of the bond The accumulated future value of all the coupon payments based on a known reinvestment rate Bond Yields 56 4. Total Return (Cont’d): Calculated using this formula: Initial Purchase Price = Terminal Value + Accumulated Future Value (1+i) Bond Yields 57 4. Total Return (Cont’d): We have learned how to calculate the terminal value when calculating expected and realized yield To calculate the accumulated future values using this formula: FVc = C[(1+i) -1] / i n Important Bond Pricing Relationships 58 3. There are three important relationship between bond prices and the change in the level of interest rates: Bond Prices and Yield Bond Price Volatility and Maturity Bond Price Volatility and Coupon Rate They apply to all fixed-income securities 1. 2. Important Bond Pricing Relationships 59 Bond Prices and Yield: They vary inversely (Negative Relationship): As the market rate of interest (yield) rises, a bond’s market price declines and vice-versa The negative relationship exists because the coupon rate ( interest on bond) is fixed when the bond is issued. Decreasing or increasing the bond’s price is the only way to adjust for changes in market interest rates 1. Important Bond Pricing Relationships 60 2. Bond Price Volatility and Maturity Bond Price Volatility: “the percentage change in bond prices for a given change in interest rates (yield) ” It is also a measure of how sensitive a bond’s price is to changes in yields. Long term bonds have greater price volatility than short term bonds, holding other bond characteristics constant. Important Bond Pricing Relationships 61 Important Bond Pricing Relationships 62 2. Bond Price Volatility and Maturity (Cont’d): Calculating Bond Price Volatility: Where Pt - Pt-1 %DPB = ´100 P t-1 change in price %∆PB = percentage Pt = new price in period t P t – 1 = bond’s price one period earlier Important Bond Pricing Relationships 63 3. Bond Price Volatility and Coupon Rate “ the lower a bond’s coupon rate, the greater the percentage price change (price volatility) for a given change in yield” Important Bond Pricing Relationships 64 Interest Rate Risk and Duration 65 1. 2. 3. This section will include the following topics: Interest Rate Risk Duration and bond properties Use of Durations: a. b. c. 4. To calculate bond price volatility As a measure of interest rate risk To measure and manage interest rate risk Duration of bond portfolios Interest Rate Risk and Duration 66 Interest Rate Risk “ Risk related to changes in the interest rates that causes a bond’s total return to differ from the promised yield or yield-to-maturity” - It includes two different but related risks: a. Price Risk b. Reinvestment Risk Interest Rate Risk and Duration 67 Interest Rate Risk: (Cont’d) Price risk: change in interest rate that cause market value of the bond to change resulting in capital gains or losses (inverse relationship) Interest Rate Risk and Duration 68 Interest Rate Risk: (Cont’d) Reinvestment risk: change in market interest rate that might cause the lender to have to reinvest coupon payments at interest rates different from the interest rate at the time the bond was purchased. The change in a bond’s total return caused by changing coupon reinvestment rate is what constitute reinvestment risk. Interest Rate Risk and Duration 69 Interest Rate Risk: (Cont’d) It is important to note that price risk and reinvestment risk partially offset each other. Understand this carefully: - When interest rate decline bond’s price increase result in capital gain but lower coupon reinvestment income Interest Rate Risk and Duration - - - Duration and Bond Properties It is important to evaluate the effect of interest rate risk on bond investments. Problem: bond price volatility varies directly with maturity and inversely with coupon rate. A good measure should take both into consideration. Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) Duration can be defined as : “ A measure of interest rate risk (bond price volatility) that considers both coupon rate and maturity” “ It is a weighted average of the number of years until each of the bond’s cash flow is received” Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) It can be computed using: n CFt * t t t 1 (1 i ) D n CFt t ( 1 i ) t 1 where: D = duration of the bond CFt = interest or principal payment at time t t = time period in which payment is made n = number of periods to maturity i = the yield to maturity (interest rate) Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) Suppose we have a bond with a 3-year term to maturity, an 8% coupon paid annually, and a market yield of 10%. Duration is: Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) If the yield increases to 15%: Interest Rate Risk and Duration Interest Rate Risk and Duration 1. Duration and Bond Properties: (Cont’d) Important properties of Duration: High coupon rates shorter duration - 2. When bonds have the same maturity because bond holder will receive more of the total cash flow earlier Long maturity High bond duration - maturity can be called term-to-maturity Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) 3. Bonds with a single payment (with or without coupon) duration = term to maturity - example: zero coupon bond Bonds with interim payments always have durations less than their final maturity Interest Rate Risk and Duration 78 Duration and Bond Properties: (Cont’d) 4. All other factors held constant, the higher the market interest rate shorter duration. Because in this case coupon reinvestment income accumulates faster. Interest Rate Risk and Duration Duration and Bond Properties: (Cont’d) 5. Direct relationship between duration and bond price volatility. Greater bond duration greater the percentage change in the bond’s price for a given change in interest rate. - Important for managers specifically Interest Rate Risk and Duration 80 Use Duration To calculate bond price volatility As a measure of interest rate risk To measure and manage interest rate risk Interest Rate Risk and Duration 81 - Use Duration to Calculate Bond Price Volatility “how to use duration to estimate the percentage change in the bond’s price” i %PB D 100 (1 i ) Interest Rate Risk and Duration 82 - - Use Duration to Calculate Bond Price Volatility: (Cont’d) It is important to note that this formula, work best for small changes in interest. Graph in the book page 125 To solve this issue, investors will add an adjusting factor to the formula called “convexity factor” Interest Rate Risk and Duration 83 Use Duration to Calculate Bond Price Volatility: (Cont’d) Interest Rate Risk and Duration 84 Use Duration to Calculate Bond Price Volatility: (Cont’d) To illustrate the preceding, work out this example: A $1,000, 3-year bond, has a coupon payment of 4%. It has a duration of 2.88 and market interest rates increased from 10% to 10.25%. Compute the bond price volatility. Interest Rate Risk and Duration 85 Use Duration to Calculate Bond Price Volatility: (Cont’d) Assume, that the rate actually increased from 10% to 12%, what would be the percentage change in the bond price? Now, find prices of each bond and find the percentage change, do you get the same answers? Interest Rate Risk and Duration 86 Using Duration as measure of interest rate risk We already studied that: 1. Long term bonds have higher interest rate risk 2. Low coupon bonds have more interest rate risk However, these properties cannot allow me to rank bonds of basis of riskiness. Interest Rate Risk and Duration 87 Using Duration as measure of interest rate risk: (Cont’d) For example: consider these bonds: a. 10 year, c=7% bond and, b. 8 year, c= 5% bond Which one is more risky in-terms of interest rate risk ? Interest Rate Risk and Duration 88 - - - Using Duration as measure of interest rate risk: (Cont’d) It is not possible to tell using the given information. Duration solves the problem. Low-duration bonds face low interest rate risk and vice-versa. Duration is relate to the price-yield profile steeper the price-yield profile greater duration greater exposure to interest rate risk Interest Rate Risk and Duration 89 Use Duration to Calculate Bond Price Volatility: (Cont’d) Back to the two bonds described earlier: Market interest rate is 5% Bond Coupon Rate Maturity A B 7% 5% 10 years 8 years Now weDuration can conclude that the to lower interest 7.71bond yearsthat is exposed 6.79 years rate risk is the one with the lower duration. Interest Rate Risk and Duration 90 - - Use Duration to measure and manage interest rate risk Duration is used as a tool for reducing or eliminating interest rate risk over a given holding period. There are three possible approaches to deal (or manage) interest rate risk: 1. 2. 3. A Zero Coupon Approach The Maturity-Matching Approach The Duration-Matching Approach Interest Rate Risk and Duration 91 Use Duration to measure and manage interest rate risk: (Cont’d) 1. A Zero Coupon Approach: - Price risk: Eliminated by holding the bond for the whole maturity - Reinvestment risk: Eliminate because no coupon payment - Return: the difference between purchase price and maturity value. Interest Rate Risk and Duration 92 Use Duration to measure and manage interest rate risk: (Cont’d) 2. The Maturity-Matching Approach: - It is a naïve alternative - Price risk: Eliminated by holding the bond for the whole maturity - Reinvestment risk: Still there - A Dramatic change in interest rates would have a grate effect Interest Rate Risk and Duration 93 Use Duration to measure and manage interest rate risk: (Cont’d) 3. The Duration-Matching Approach: - A surefire way to eliminate both risks - Duration of the bond equals the required holding period - By doing that, capital gains and losses caused by changes in interest rates are exactly offset by changes in reinvestment income Interest Rate Risk and Duration 94 - Duration of Bond Portfolio Investors rarely hold only one asset Bond portfolio, or bond mutual fund? Duration can help rank bond portfolios as it help us rank individual bonds in terms of exposure to interest rate risk 95 Duration of Bond Portfolio: (Cont’d) Federal bonds (15%) Shell bonds (50%) Bond Portfolio Google bonds (35%) Interest Rate Risk and Duration 96 Duration of Bond Portfolio: (Cont’d) Duration of a bond portfolio is a weighted average of the individual bond durations. The weight is according to the proportion of the portfolio accounted for each bond. Interest Rate Risk and Duration 97 Duration of Bond Portfolio: (Cont’d) Formula to calculate bond portfolio duration: i 1 Portfolio Duration wi Di n where: wi = proportion of bond i in portfolio and Di = duration of bond i. Interest Rate Risk and Duration 98 Duration of Bond Portfolio: (Cont’d) Example: Suppose a bond portfolio contains four bonds, A,B,C, and D. Bond A has duration of 15.7 years and makes up 20 percent of the portfolio. Bond B has duration of 22.3 years and makes up 40 percent of the protfolio. Bond C has duration of 10.2 years and makes up 15 percent of the portfolio. Finally, Bond D has duration of 7.6 years and makes up 25 percent of the portfolio. Calculate the duration of the portfolio. Interest Rate Risk and Duration 99 Duration of Bond Portfolio: (Cont’d) - Managers can change the duration of their bond portfolio by changing the proportions of bonds in the portfolio. 100 End of Chapter 5