Central Bank of Egypt March 2014Rania Elsawy Bond Basics Rania Elsawy

Central Bank of Egypt March 2014Rania Elsawy A Bond is a debt instrument requiring the issuer (borrower of cash) to repay the lender for the amount borrowed plus interest over a specified period of time. 2. Type of Issuer: Federal Government & its agencies Municipals Corporations 1.Definition of a Bond 1.Definition of a Bond : I. Bond Basics

Central Bank of Egypt March 2014Rania Elsawy I. Bond Basics 3. Term to Maturity: The term to maturity is the number of years over which the issuer has promised to meet the conditions of the obligations. Bonds with a maturity between one and five years are considered short term, bonds with maturity between 5 and 12 years are viewed as intermediate term, long term bonds are those with a maturity of more than 12 years.

Central Bank of Egypt March 2014Rania Elsawy 4. Principal and Coupon Rate: The principal value (face value, par value) is the amount the issuer agrees to repay the bondholder at the maturity date. The coupon rate is the interest rate the issuer agrees to pay each year. The annual or semi annual amount of interest payment is called the coupon I. Bond Basics

Central Bank of Egypt March 2014Rania Elsawy Yield to maturity is the most popular measure of yield in bond markets It is the single discount rate that will make the present value of a bond’s cash flows equal to its market price P= CF 1 + CF 2 + CF 3 + … + CF n (1+r 1 ) 1 (1+r 2 ) 2 (1+r 3 ) 3 (1+r n ) n The YTM is found by determining the cash flows and looking for the interest rate by trial and error that will make the present value of the cash flows equal to the market price. I. Bond Yield 5. Yield to Maturity (YTM)

Central Bank of Egypt March 2014Rania Elsawy YTM considers not only coupon but also capital gain that the investor will gain if the bond is held to maturity It also considers reinvestment of coupons but 2 assumptions are made that could be unrealistic:  It assumes that the bond will be held to maturity  It assumes that all future coupons are reinvested at the same rate (YTM) The investor will realize the initially assumed yield only if the reinvestment rate actually equals the YTM used to discount the future cash flows  If the reinvestment rate is higher than YTM, the realized return will be higher and vice versa The true return of a bond cannot be known with certainty because the interest rate at which future coupons will be reinvested is unknown 6.Shortcomings of YTM: I. Bond Yield

Central Bank of Egypt March 2014Rania Elsawy I. Bond Basics 7.Price-Yield Relationship A fundamental property of a bond is that its price (present value of the cash flows) changes in the opposite direction from the change in the required yield. As the required yield increases, the present value of the cash flow decreases; hence the price decreases & vice versa

Central Bank of Egypt March 2014Rania Elsawy III. III. Risk of Bonds 1.Interest Rate Risk: The price of a bond will change in the opposite direction of the change in interest rates; as rates rise, prices will fall and vice versa. If an investor has to sell a bond prior to the maturity date at a higher interest rate that means the investor will realize a capital loss.

Central Bank of Egypt March 2014Rania Elsawy III. III. Risk of Bonds 2.Reinvestment Risk: The risk where the interest rate at which the interim cash flows can be reinvested will fall. Reinvestment risk is greater for longer holding periods or for bonds with large cash flows. Interest rate risk and reinvestment risk have offsetting effects.

Central Bank of Egypt March 2014Rania Elsawy III. III. Risk of Bonds 3.Credit Risk: The risk that the issuer of a bond will fail to satisfy the timely payment of interest and the amount borrowed. This type of risk is called default risk. 4.Exchange Rate Risk: Ex: from the perspective of a U.S investor, a non dollar denominated bond ( whose payments are in foreign currency) has unknown USD cash flows since they are dependant on the exchange rates.

Central Bank of Egypt March 2014Rania Elsawy III. III. Risk of Bonds 5.Liquidity Risk: Liquidity depends on the ease with which an issue can be traded near its value. The wider the bid ask spread the more the liquidity risk U.S. treasuries are the most liquid bonds in the world.

Central Bank of Egypt March 2014Rania Elsawy III. III. Risk of Bonds 6. Prepayment Risk: Arises from the fact that the issuer might call back the security at times of falling interest rates and therefore the investor will receive early payments. An investor must re- invest these prepayments at a lower new rate.

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics 1. Cash Flows of a Bond Ex: U.S. treasury Note 5 years Maturity paying coupons semi annually.

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics 2.Coupons: A coupon payment on a bond is a periodic interest payment that the bondholder receives during the time between when the bond is issued and when it matures. Ex: If the bond has a Face (Par) value of $1000 and a coupon rate of 4%. Therefore, the annual coupon = Face Value X Coupon Rate Annual coupon = 1000 X 4% Annual Coupon = $40 Semi Annual Coupon = $20

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics 3.Prices: Bond prices are expressed as a percentage of par value (100), which means that the price is always given in relation to 100. At Maturity, the investor receives par value. A bond price above 100 means the bond is trading above par, below 100 means its below par If Coupon rate > YTM therefore the bond will be trading above par If Coupon rate < YTM therefore the bond will be trading below par. IF Coupon rate = YTM therefore price of the bond is 100.

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Ex: A European government bond with a $1000 par value quoted at would have a market value of $1000 X 85.70/100= $857. Face value X Price = Market Value U.S government bonds are quoted in fractions EX: /8, , ¼ /8 = = ¼ = /8 = ?

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics 4.Value determination: The price of a bond is equal to the present value of its expected cash flows. Price = C [ 1- 1/(1+r) n ] + M r(1+ r ) n

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Ex: Consider a 20 year 10% coupon bond with a par value of $1000, the required yield on the bond is 11% Therefore, coupon = $50, number of period= 40 R= 0.055, M= $1000 Present Value of the coupons = $ Present Value of the maturity value (par)= $ The present value of the bond is equal to the sum of the two present values. $ $ = $919.77

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Suppose that instead of an 11% required yield, the required yield is 6.8% Therefore, the present value of the coupon payments= $ Present value of the par amount = $ Price or present value of the bond = $

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Ex: What is the price of a zero coupon bond that matures in 15 years if the maturity value is $1000 and the required yield is 9.4% Therefore, M=1000, R= 0.094/2 = 0.047, N= 2X15=30 Present value = M / (1+r) n Present Value= $252.12

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics 5.Accruals: When an investor purchases a bond between coupon payments, the investor (buyer) must compensate the seller of the bond for the coupon interest earned from the time of the last coupon payment to the settlement date of the bond This amount is called accrued interest The amount that the buyer pays the seller is the agreed upon price plus accrued interest, this is referred to as the dirty price Dirty Price= Clean Price + Accrued Interest

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Last coupon date Settlement date Next coupon date Coupon period Accrual period Trade date

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Coupon rate:3.625% (semiannual pay) Clean Price :99.5 Par value:100 Maturity date:04/30/07 Settlement date:05/12/05 Last coupon date:: 04/30/05 Next coupon date::10/31/05 Accrual period:04/30/ /12/05 (12 days) Coupon period:04/30/ /31/05 (184 days) Accrued Interest : (.03625/2) * 100 * (12/184) = Dirty Price : =

Central Bank of Egypt March 2014Rania Elsawy V. Bond Characteristics Coupon rate:4% (semiannual pay) Clean Price :98 Par value:100 Settlement date:01/Aug/06 Last coupon date:: 31/Mar/06 Next coupon date::30/Sept/06 Accrual period: Coupon period: Accrued Interest : Dirty Price :

Central Bank of Egypt March 2014Rania Elsawy The shape of the price/yield is a non-linear relationship that is referred to as convex A change in interest rates does not affect all bonds in the same way The illustration in the following slide shows the price/yield relationship of a straight (option-free bond) 1.The Duration / Convexity Approach: VI. Duration

Central Bank of Egypt March 2014Rania Elsawy VI. Duration 2.Price / Yield Relationship:

Central Bank of Egypt March 2014Rania Elsawy Price-yield relationship is negative, meaning that the price and yield move in opposite directions For small changes in interest rates, the upside and downside price moves are approximately equal in size For large changes in interest rates, the upside and downside will diverge from the duration line For option-free bonds, the upside price move will be greater than the downside price move for the same changes in rates The actual level of duration and convexity will vary with the terms of the bond As interest rates decrease, the duration of the bond will increase (and so will convexity) VI. Duration

Central Bank of Egypt March 2014Rania Elsawy Duration is a measure of the approximate price sensitivity of a bond to interest rate changes More specifically it is the approximate percentage change in price for 100 basis points change in rates Duration = Price if yields decline – Price if yields rise 2 (initial price) (change in yield in decimal) D = P - - P + 2P 0 Δy VI. Duration 3.Calculating Duration:

Central Bank of Egypt March 2014Rania Elsawy In calculating duration, it is necessary to shock interest rates (yields), up and down by the same number of basis points The below example shows how the size of the rate shock will affect duration For option-free bonds, duration estimates are not materially affected by the size of the rate shocks VI. Duration 4.Yield Shocks and Duration Calculation:

Central Bank of Egypt March 2014Rania Elsawy Assume a 5 year 5% par bond and Δy= 0.3% P- = P+ =98.71 D= P - +P + = – = P 0 Δy 200 x A duration of 4.33 years means that the price of the bond will change approximately by 4.33% for 100 basis point change in the yield. VI. Duration 5.Duration Calculation Example:

Central Bank of Egypt March 2014Rania Elsawy Duration can now be used to project how the price of a bond will change for a given change in interest rates. Approximate percentage price change = -Duration x Δy x 100 Continuing the previous example to calculate the expected new price from duration we will find that there is a minor discrepancy: VI. Duration

Central Bank of Egypt March 2014Rania Elsawy For the same 5 year 5% par bond: D= 4.33 P=100 Δy= 0.3% ΔP = - D x Δy x 100 = x x 100= P = = If we compare this to the previous example, we see that the price actually changed to VI. Duration

Central Bank of Egypt March 2014Rania Elsawy Approximate percentage price change = -Duration x Δy x 100 The key issue is that duration is a linear approximation of the sensitivity of the price of a fixed-income security to a change in its yield to maturity PROBLEM: This relationship is not linear Notice that: the error in the estimate gets larger, the larger the change in yield is. Also it is the larger the larger the convexity is The tangent line always underestimates what the new price will be 6.How Good Is Duration: VI. Duration

Central Bank of Egypt March 2014Rania Elsawy Duration is a linear approximation for a small change in yield The divergence from the straight line is called convexity Convexity is used to approximate the change in price that is not explained by duration C= P - + P + - 2P 0 2P 0 (Δy) 2 VII. Convexity 1.Convexity Adjustment:

Central Bank of Egypt March 2014Rania Elsawy Convexity is generally fairly small for modest yield changes. This means that for small changes a linear approximation is generally sufficient For large yield changes, however, it may be necessary to use a convexity adjustment Positive convexity is good – when the yield goes down, the price will increase by more than what Duration implies; when the yield goes up, the price will decrease less by what Duration implies. VII. Convexity

Central Bank of Egypt March 2014Rania Elsawy The chart below shows convexity and how the level of convexity will affect each bond: Duration combined with the convexity gives a better estimate of the bond’s price change to large changes in yield %ΔP = -D Δy + C Δy 2 VII. Convexity

Central Bank of Egypt March 2014Rania Elsawy Bond portfolios have duration and convexity just like the individual bonds It is calculated as the weighted average of the individual durations and convexities D Portfolio = w 1 D 1 + w 2 D 2 + … + w n D n C Portfolio = w 1 C 1 + w 2 C 2 + … + w n C n However, duration and convexity will not provide meaningful percentage value change estimates for portfolios unless the yield curve shifts in a parallel manner VIII. Duration & Convexity of Portfolios 1.Weighted Duration and Convexity:

Central Bank of Egypt March 2014Rania Elsawy Rate duration gives the sensitivity of the portfolio if yield changes for only one maturity and the yield for all other maturities remain unchanged To reduce the amount of work, it is calculated only for key maturities along the yield curve This measure is called key rate duration and can be used to measure yield curve risk in a portfolio The main point to remember here is:  Duration is used to measure % change in value if a parallel shift in the yield curve occurs  Key rate duration is used to project % change in value if yield for only one specific maturity on the yield curve shifts and the rest of the curve remains unchanged VIII. Duration & Convexity of Portfolios 2.Key Rate Duration:

Central Bank of Egypt March 2014Rania Elsawy Appendix

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds 1.Government Bond: Ex: U.S. Government Bonds which are regarded as the safest issues in the world giving its deep liquidity. 2.Corporate Bond: Corporate bonds are characterized by higher yields because there is a higher risk of a company defaulting than a government. The higher the quality of the company, the lower the interest rate the investor receives. Ex: Microsoft Bond, Toyota Bond

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds 3.Municipal Bonds: Sate and Local governments issue municipal bonds to raise funds. The returns of Municipals are non taxable, therefore, the yield on municipals are usually lower than that of a regular taxable bond.

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds 4.Zero Coupon Bonds: This is a bond that makes no coupon payments during the life of the bond and instead, pays the interest at the maturity. It is issued at a discount to par value. They are usually long term investments 10 years or more. Pension funds usually invest in zero coupon bonds Ex: Education Fund

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds 4.Treasury Inflation Protected Securities (TIPS) A Treasury security that protects investors from inflation where its par value rises with an inflation index like the CPI. TIPS are issued by U.S Government and its interest rate is fixed.

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds 5.Fixed Rate Bonds: It a bond with a fixed coupon rate 6.Floating Rate Bonds: It’s a bond that contains a variable coupon that is equal to a money market reference rate plus a specified spread. Libor is the most widely used MM reference rate for floaters. Example : Coupon= 6 Month Libor + 50 bps

Central Bank of Egypt March 2014Rania Elsawy I. Types of Bonds The issuer and the investor of the bond can limit their exposure to extreme fluctuations in the reference rate by placing upper and lower limits on the coupon rate. The upper limit is called a CAP, which puts a maximum on the interest rate paid by the issuer. The lower limit, on the other hand, is called a FLOOR which puts a minimum on the periodic interest payment received by the bondholders. When both limits are present simultaneously, the combination is called a COLLAR

Central Bank of Egypt March 2014Rania Elsawy Modified duration It assumes that changes in yield do not change the bond’s expected cash flow and thus is not a good measure for bonds with optionality Effective duration or option adjusted duration Duration measure that recognizes the fact the yield changes may change the expected cash flow Therefore it is used to measure duration if the bond has embedded options II. Duration Different Types of Duration:

Central Bank of Egypt March 2014Rania Elsawy Macaulay duration It is the weighted average time-to-maturity of the cash flows of a bond Macaulay Duration : ∑(n)(PV of Cash Flows) k x Bond Price Modified duration= Macaulay duration (1+Yield/k) k = number of periods per year n = number of periods until cash flow is paid II. Duration

Central Bank of Egypt March 2014Rania Elsawy Effective duration is the first derivative of the pricing function D E = - d P / d r P 0 Duration is some sort of weighted average maturity of the bond. Macaulay’s duration is measured in units of time Duration is the approximate percentage change in the price of a bond for 100 basis point change in interest rates. II. Duration Different Definitions of Duration:

Central Bank of Egypt March 2014Rania Elsawy Dollar duration is another measure of a bond’s price volatility or interest rate risk It’s the dollar change in a bond’s price for a given change in its yield Dollar Duration = |P 0 D E Δr | Price Value of a Basis Point (PVBP) is the dollar duration of a bond for a 1 basis point change in its yield PVBP= P 0 D E Dollar Duration and PVBP: II. Duration