Class #6, Chap 9 1

 Purpose: to understand what duration is, how to calculate it and how to use it.  Toolbox: Bond Pricing Review  Duration  Concept  Interpretation  Calculation  Examples 2

3  Bond Pricing Review  Zero coupon bond with the YTM  Coupon bond with the YTM  Coupon bond with Yield Curve  Yield Curve and YTM

Price a zero coupon bond with 10 years left to maturity, face value of $1000 and YTM of 5% 4 Step 1: Find coupon payments & draw cash flows Coupon Payments = $1,000*0 = $0 Step 2: discount cash flows

Price a 4 year coupon bond with face value of 1000 and an annual coupon of 7% if the yield to maturity is 13% 5 Step 1: Find coupon payments & draw cash flows Coupon Payments = $1,000*.07 = $70 Step 2: discount cash flows

6 Yield curve gives the market rate for a pure discount bond at each maturity The market price of a coupon bond incorporates several different rates for different time horizons (1 year money, 2 year money …. ) Every bond has its own yield curve – treasury bonds, Ford bonds, GM bonds The yield curve will change every day Current Yield Curve 4.3% 3.1% 1.1%

Example: Find the price of a three year bond that pays an annual coupon of 8%. The following rates are taken from the current yield curve. Face value is $1,000 7 TermRate 6 months1.2% 1 year2.1% 2 years2.5% 3 years2.7%

We used two different methods what is the difference? Method 1: we pulled rates off the yield curve Method 2: we used one constant rate – lets value the bond above with a constant rate of 2.676% (Yield to Maturity) 8 What does this yield curve look like?

9  The yield to maturity (YTM) is basically a weighted average of rates off the yield curve  It is the constant rate, over the full maturity, that gives you the market price of the bond  When you use the YTM you assume that the yield curve is flat!

DURATION  Concept  Calculation  Using duration 10

Concept of Duration 11

 What does duration do?  It measures the sensitivity of the asset price to changes in interests rates  How is that going to help us?  We have been trying to measure interest rate risk ▪ The movement in asset prices in response to a change in interest rates  Repricing gap gave us a rough measure but had several problems  Duration improves upon some of these short falls  Advantages of Duration  It is a market based measure so it takes into account current values  It considers the current time to maturity rather than the defined term  Disadvantages of Duration  It requires more information to calculate 12

 Definition of Duration:  Duration is the present value cash flow weighted average time to maturity of a loan/bond  What the ?  Duration tells us, in terms of present value, the average timing of cash flows from a loan or bond. (i.e. on average, when do we receive the value of our bond)  Not much help? 1. What is duration – work on explaining the definition 2. Why is it important – how does it help measure interest rate sensitivity 13 Lets split this discussion into two parts: This is what we are after. I want you to see that duration tells us: on average when do we receive the value of our bond/loan

Duration is the present value cash flow weighted average time to maturity of a loan/bond 14  First thing we want to realize is that any bond can be thought of as a portfolio of zero coupon bonds  Consider a 6 year coupon bond that pays an annual coupon of 4% and has a $1,000 face value

Duration is the present value cash flow weighted average time to maturity of a loan/bond We can think of the coupon bond as a portfolio of 6 zero coupon bonds. The average maturity Average TTM =

Duration is the present value cash flow weighted average time to maturity of a loan/bond Average TTM = 3.5 yrs  Do you think that this tells you when, on average, you receive the value of you payments?  Duration is going to depend on two things: 1. The timing of payments 2. The amount of payments – in terms of present value Before or after 3.5 years?Why? Take away: We can calculate the average time to maturity of a bond but that does not always tell us when, on average, we receive the full value of payments

PV = 27.32PV = PV = Duration is the present value cash flow weighted average time to maturity of a loan/bond  For this part, let’s just start with how much of the bond value we receive at each point in time. Assume YTM = 10%  On average, when do you think we receive the full bond value?  How can we adjust the average to account for this?  What do we use for weights? Duration is the present value cash flow weighted average time to maturity of a loan/bond PV = PV = PV = weighted average Present value of cash flows Somewhere around year 5 or 6 is a good guess Take away: On average we receive the full bond value close to the largest PV(payment). So we need to weight by PV(CFs) Which pmts is most/ least valuable?

Duration is the present value cash flow weighted average time to maturity of a loan/bond  On average we will receive the full value of our payments 5.35 years from today. Duration = 5.35 yrs

1. What is Duration – Definition 2. Why is it important – how does it measure interest rate sensitivity 19

Duration is important because it tells us the interest rate sensitivity of a bond!!! But how?  It turns out that the weighted average time to maturity (duration) gives us the maturity of the equivalent zero coupon bond.  This “equivalent” zero coupon bond will have the same interest rate sensitivity as the coupon bond  Example: consider two bonds: Bond 2: Bond 1: Duration = 5.35

Example: Price both bonds with YTM = 10% then again with YTM = 10.5% and compare the price sensitivity Bond 1: Duration = 5.35 Sensitivity Bond 2: Sensitivity The two bonds have the same interest rate sensitivity

20,728,012 What if we change the principal amount? What does this get us? …. Lets see Bond 1: Bond 2: Which bond is more interest rate sensitive? What is the duration of this bond? Which bond is more interest rate sensitive? Is it harder to see?

What does this get us? …. Lets see Bond 2: Bond 1: $40 Which bond is more interest rate sensitive?

What does this get us? …. Lets see Bond 2: $40 Which bond is more interest rate sensitive? Bond 1: $90 Duration = 5.96 Duration = 6.95

25 Bond 1 Face value Time to Maturity Coupon Rate Compounding Bond 2 Beginning of Period YTM End of Period YTM Bond 1 Interest Rate Sensitivity Bond 2 Interest Rate Sensitivity

Conclusion:  Duration tells us:  With respect to interest rate sensitivity, this bond will behave like a zero coupon bond with D years to maturity (where D is the bond duration/maturity)  We know that the zero coupon bond with longer maturity is more interest rate sensitive  Therefore, we also know that a bond with longer duration is more interest rate sensitive 26

Calculating Duration 27

Step 1: draw out the cash flows Step 2: take the present value of all the cash flows Step 3: calculate weights Step 4: calculate the weighted average time to maturity (duration) 28

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value Step #1: draw out the cash flows

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value Step #2: Take the present value of cash flows

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value Step #3: Calculate weights First thing we need to do is sum the present values What is this number?

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value Step #3: Calculate weights What do the weights mean? They are the percentages of the present value of all cash flows that occur on that time period Example: 3.61% of the present value of all cash flows is received at year 1.5 Why do we take the present value? We are trying to compare the relative importance of different cash flows so we need to compare them at the same point in time – which is more valuable to an investor $1 today or $1.10 in one year if the one year interest rate is 10%?

Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM, and $1,000 face value Step #4: Calculate the duration (present value cash flow weighted average time to maturity) 33 weights What are the ts? What are these? Years Macaulay Duration

 To this point, we know that duration is a measure of interest rate sensitivity  It turns out that duration is also the interest rate elasticity of a security (bond) price  What more does that tell us?  Because it is an elasticity, we can use it to determine how much the bond price will move in response to a change in interest rates  Elasticity Equation 34

We can rewrite the duration equation: This gives us an equation for calculating the percent change in the bond price (return) due to a change in the interest rate 35

 We have been working with a two year 8% coupon treasury bond with 12% YTM and a price of $  Suppose the interest rate decreased to.115 what would you expect the percent change in the price to be? 1. Calculate the percent change in the bond price: 2. With Duration: 36 Not exact but pretty close

 Duration gives us a way to measure the sensitivity of an asset price to changes in the interest rate  Duration also gives us a way to calculate the magnitude of the percent change in price in response to a change in interest rates  Alternative forms of duration  Bond traders developed more convenient ways to write duration  Modified duration (MoD)  Dollar duration 37

 Macaulay Duration: (D)  Modified Duration (MoD)  Dollar Duration 38 MoD allows you to calculate the %change in the bond price just by multiplying by the change in interest rate $D dollar duration allows you to calculate the change in bond price from the change in the interest rate

Example: calculate the duration, modified duration and dollar duration for a bond with: face value = 1000; annual coupon; coupon rate = 3%; YTM = 9%; and four years to maturity 39

Examples: Suppose the YTM = 9% i) Find the percent change in the bond price if YTM increases from 9% to 14% the duration is 3.8 years ii) The percent change in the bond price if the YTM increases to 11% given MoD = iii) The raw change in the bond price if the YTM decrease to 8.5% if $D =

 How would things change if the bond had semiannual coupons?  The bond pricing would change as we have already seen  Modified duration would also change: 41 For semiannual coupons we have

Different durations:  Macaulay Duration = D  Modified duration(MoD)  Dollar duration = (MoD)(bond price) 42 D = Macaulay duration R = the yield to maturity k = compounding periods

 We learned the meaning of duration (concept)  How to calculate duration (D, MoD, $D)  How to use duration to calculate the expected change (%change in price) 43