Chapter 7 Introduction to the Measurement of Interest Rate Risk 1 Introduction ◼ In Week 1, we first discussed the interest rate risk in bond investment. We know that: Bond price and interest rate move in opposite directions In particular, a rise in the interest rates will lead to a decline in the bond value. This is known as the interest rate risk to bond investors ◼ The key in measuring the interest rate risk is the accuracy in estimating the value of the bond portfolio after an adverse interest rate move We can do so using either the full valuation approach or the duration/convexity approach 2 Full valuation approach ◼ This approach revalues a bond or bond portfolio for a given interest rate change scenario. It is also known as scenario analysis On the following two slides, we illustrate this approach using a portfolio of two bonds, bond A and bond B We consider two scenarios: in Scenario 1, there is a parallel increase in the yields on both bonds by 50 basis points ◼ in Scenario 2, there is a parallel decrease in the yields on both bonds by 100 basis points ◼ We can also do a similar analysis for a nonparallel shift in yields (see Exhibit 3 on p.159) 3 Full valuation approach ◼ Both bonds A and B are option free Bond A is 8% coupon payable semiannually, 5 years, $100 par, yields at 6%, and sells at $108.53 ◼ (N = 10; PMT = 4; FV = 100; I/Y = 3; CPT PV = -$108.53) Bond B is 5% coupon payable semiannually, 15 years, $100 par, yields at 7%, and sells at $81.61 ◼ (N = 30; PMT = 2.5; FV = 100; I/Y = 3.5; CPT PV = -$81.61) ◼ The value of the bonds and the value of the portfolio consisting of bonds A and B in each scenario are summarized in the table on the next slide … 4 Full valuation approach ◼ Scenario Yield change Current None 50 basis points () 1 100 basis points () 2 Bond A $108.53 $106.32 $113.13 Bond B $81.61 $77.71 $90.20 Porfolio $190.14 $184.03 $203.33 Portfolio value change None 3.21% () 6.94% () In Scenario 1 (i.e., interest rates rise), bond A declines by 2.04% [= ($106.32 - $108.53) / $108.53] in price, and bond B drops by 4.78% in price In Scenario 2 (i.e., interest rates drops), bond A increases by 4.24% in price, and bond B rises by 10.53% in price 5 Full valuation approach ◼ Bond B is a longer maturity bond and has a lower coupon rate: both of which mean higher interest rate risk The changes in bond B price are 4.78% and 10.53%, both higher than the corresponding changes in bond A price (2.04% and 4.24%, respectively) Recall from Week 1: ◼ Higher (lower) coupon rate means lower (higher) interest rate risk ◼ Longer (shorter) maturity (generally) means higher (lower) interest rate risk ◼ Higher (lower) yield means lower (higher) interest 6 rate risk Full valuation approach ◼ Advantages of this approach: It is more accurate than the duration/convexity approach (which is only for parallel yield curve changes, as we will see later in the chapter) ◼ Disadvantages of this approach: 1) Can be very complex and time consuming ◼ In our example, we only consider a portfolio of two option free bonds. Yet in reality bond portfolios usually consist of hundreds or even thousands of bonds, many of them have embedded options 7 Full valuation approach ◼ Disadvantages of this approach (continued): 2) Modeling risk: the model analyzing embedded options may produce wrong value because of the incorrect assumptions made in the model (see also p.15 and p.118) 3) Which scenarios to use? There might be tens of thousands of possible scenarios rather than just two! ◼ Stress testing: risk managers look at extreme scenarios (e.g., market crash, the entire industry collapses etc.) to assess the exposure of the portfolio to interest rate risk, especially for highly levered investors such as hedge funds (see also p. 118) 8 Price volatility of bonds ◼ Option free bonds Consider an option free, 8% coupon semiannual pay, 20 year bond currently trading at par ($100). The price yield relationship for this bond is shown as below along with another bond price curve (solid blue line) Price The price yield curve for an option free bond is convex toward the origin A $110.68 $100 $90.80 0 A bond with same duration, same yield but different convexity B 7% 8% Yield 9% 9 Price volatility of bonds ◼ Option free bonds: from the graph on the previous slide, we can see the following four properties of the price volatility of option free bonds Property 1: the price yield curve is downward sloping, but is not a straight line (compared to the straight line AB), so the percentage price change is not the same for all bonds Property 2: for small changes in yields, the percentage change in bond price for a given bond is roughly the same, no matter the yields increase or decrease (see the next slide for an example …) 10 Price volatility of bonds ◼ Option free bonds Property 2: ◼ For the bond that we are considering, suppose the yield rises by 10 basis points, then bond price drops to $99.02, a decline of 0.98% ( 1%) ◼ If the yield drops by 10 basis points, then bond price rises to $101, an increase of 1% 11 Price volatility of bonds ◼ Option free bonds Property 3: for large changes in yields, the percentage price change is not the same for yield increase as it is for yield decrease In our example, when yields rise by 1% (8% 9%), the bond price drops by 9.2% ($100 $90.80) ◼ In contrast, when yields drop by 1% (8% 7%), the bond price increases by 10.68% ($100 $110.68) ◼ 9.2% 10.68%! 12 Price volatility of bonds ◼ Option free bonds Property 4: for a given large change in yields, the percentage price increase is greater than the percentage price decrease ◼ In our example, for a given 1% (i.e., 100 basis points) change in yields, bond price may increase by 10.68%, or drop by 9.2%, depending on the sign of the yield change 10.68% > 9.2% ◼ This is known as the convexity (or positive convexity) of the price yield curve. This property is beneficial to bond investors (Why?) 13 Price volatility of bonds ◼ Callable or prepayable bonds The price yield curve for a callable or prepayable bond usually looks like: Price Call option value Call price Option free bond Callable bond 0 Negative y* convexity Yield Positive convexity 14 Price volatility of bonds ◼ Callable or prepayable bonds In the previous graph, y* is usually equal to the bond coupon rate When yield is higher than y*, the embedded option has almost no value, and the callable or prepayable bond is roughly equal to the corresponding option free bond ◼ When yield is lower than y*, the embedded option has a positive value to the issuer, the price appreciation potential of the callable bond is capped by the call price (known as price compression), and the callable bond is less valuable than the comparable option free bond ◼ 15 Price volatility of bonds ◼ Callable or prepayble bonds The region where yield is lower than y* is known as the negative convexity (or concave), as opposed to positive convexity where yield is higher than y* ◼ In negative convexity, for a given change in yield, the percentage price increase is less than the percentage price decrease (see the next slide for an illustration …) 16 Price volatility of bonds ◼ Callable or prepayable bonds Price P1 P0 P2 0 y 0 − y1 = y 2 − y 0 P1 − P0 P2 − P0 But | | P0 P0 y1 y0 y2 y* Negative convexity Yield Positive convexity 17 Price volatility of bonds ◼ Callable or prepayable bonds From the graph on Slide 16, the embedded call option value is the difference between the price of the corresponding option free bond and the price of the callable bond ◼ Recall from Chapter 2: Price of callable bond = price of option free bond – value of the embedded call option 18 Price volatility of bonds ◼ Putable bonds The value of an embedded put option is larger to the bondholders at higher yields and is smaller at lower yields (Can you see why?), opposite to the call option value ◼ Also typically the put price is set equal to par The price yield curve for a putable bond is shown on the next slide … ◼ In the graph, y* is usually equal to the bond coupon rate 19 Price volatility of bonds ◼ Putable bonds Price Putable bond Option free bond Put option value 0 y* Yield Positive convexity 20 Price volatility of bonds ◼ Putable bonds The embedded put option value is given by the sum between the price of the putable bond and the price of the comparable option free bond ◼ Recall from Week 1: Price of putable bond = price of option free bond + value of the embedded put option The price yield curve for a putable bond is flatter than the curve for the corresponding option free bond because embedded options tend to reduce the interest rate risk (see Week 1) 21 Duration/convexity approach ◼ Duration The formula for the approximate percentage price change for a 100 basis point change in yield is: price if yield declines - price if yield rises 2 (initial price) (change in yield in decimal) The above measure of interest rate risk is called duration 22 ◼ Duration/convexity approach Duration If we let y = change in yield in decimal ◼ V0 = initial bond price ◼ V- = bond price if yields decline by y ◼ V+ = bond price if yields increase by y ◼ then the duration formula becomes: V- − V+ 2 (V0 ) ( y) 23 ◼ Duration/convexity approach Duration Example: y = 0.0016 (i.e., 16 basis points), V0 = $1,020, V- = $1,022.50, and V+ = $1,019 $1,022.50 − $1,019 = 1.07 Duration = 2 $1,020 0.0016 ◼ Implication: the approximate price change when yield changes by 100 basis points is 1.07%; for a 50 basis point change in yield, price changes by roughly 0.535% (= 1.07% / 2) etc. 24 Duration/convexity approach ◼ Approximate percentage price change Given duration, we can approximate the percentage change in bond price using: approximat e percentage price change = - duration y * where y* is the yield change (in decimal) for which the estimated percentage price change is sought Note: do not confuse the y in the duration formula on Slide 25 with the y* in the above formula. y is the (typically small) change in yields used to estimate duration (we refer to y as “rate shock”). On the other hand, y* is the specific change in yields for which the approximate percentage price change is sought 25 Duration/convexity approach ◼ Approximate percentage price change Example: given a duration of 1.07, the approximate percentage price change for a 200 basis point decrease in yield is - duration y * = −1.07 (−0.02) = 0.0214 = 2.14% 26 Duration/convexity approach ◼ Modified duration is the duration measure in which it is assumed that yield changes do NOT alter the expected cash flows In the duration formula on Slide 25, the values of V- and V+ are calculated using the same cash flows as those used to compute the value of V0 Modified duration is appropriate for option free bonds such as noncallable Treasury securities etc. 27 Duration/convexity approach ◼ Effective duration (or option adjusted duration) is the duration measure that recognizes the fact that yield changes may alter the expected cash flows on a bond as well In the duration formula on Slide 25, the cash flows used to calculate the value of V- and V+ are different from those used to compute the value of V0 28 Duration/convexity approach ◼ Effective duration is more appropriate for bonds with embedded options (e.g., call, and put options and mortgage backed securities etc.) ◼ The difference between modified duration and effective duration for bonds with embedded options can be huge For example, for some mortgage backed securities, the modified duration could be 7 and the effective duration could be 20! 29 Duration/convexity approach ◼ Duration The sensitivity of a bond’s price (as a percentage of initial price) to a change in yield Modified duration Effective duration Assumes yield changes will not alter the expected cash flows Recognizes that yield changes may alter the expected cash flows 30 Duration/convexity approach ◼ Macaulay duration The price of an option free bond is given as follows: C C C + Par P =[ + + ... + ], 2 n 1 + y (1 + y) (1 + y) where P = price of the bond, C = semiannual coupon in $, y = one half of the YTM, n = number of semiannual periods (= number of maturity in years 2), and Par = face value of the bond in $ 31 Duration/convexity approach ◼ Macaulay duration (continued) Macaulay duration = C C (C + Par ) 1 + 2 + ... + n 2 (1 + y) (1 + y) (1 + y) n [ ]. P ➢ To understand Macaulay duration, think of the coupon bond as a package of zero coupon bonds. ➢ Macaulay duration is a weighted average of the maturities of the underlying zero coupon bonds, where the weight on each maturity is the present value of the corresponding zero coupon bond calculated using the coupon bond’s YTM (see the next slide for an illustration …) 32 Duration/convexity approach ◼ Macaulay duration (continued) Maturity 1 2 3 … n Present value of zero-coupon bond C / (1 + y) 2 C / (1 + y) 3 C / (1 + y) … (C + Par) / (1 + y)n 33 Duration/convexity approach ◼ Macaulay duration (continued) Note that Macaulay duration is measured in periods For zero coupon bonds, maturity measures the length of time that a bondholder has invested money. But for coupon bonds, maturity is an imperfect measure of this length of time because much of a coupon bond’s value comes from coupon payments made before maturity 34 Duration/convexity approach ◼ Macaulay duration (continued) Example: a 4-year $100 par 3% coupon payable annually option free Treasury bond yielding at 2.5% ◼ Bond price P = $101.88, C = $3, y = 2.5%, n = 4, and Par = $100 Macaulay duration = $3 $3 $3 $(3 + 100) 1 + 2 + 3 + 4 2 3 (1 + 2.5%) (1 + 2.5%) (1 + 2.5%) (1 + 2.5%) 4 [ ] $101.88 = 3.8305 35 Duration/convexity approach ◼ Macaulay duration (continued) Investors often refer to the ratio of Macaulay duration to (1+y) as modified duration: Macaulay duration Modified duration = 1+ y Example: for the bond on the previous slide, its modified duration is given by: [3.8305 / (1 + 2.5%)] = 3.7371 periods or 1.8686 years 36 Duration/convexity approach ◼ Portfolio duration A portfolio’s duration is the weighted average of the duration of the bonds in the portfolio, where the weight is the proportion of the portfolio that a bond comprises. That is: w1D1 + w 2 D 2 + ... + w K D K , where wi is the weight of bond i in the portfolio, i.e. wi = (price of bond i) /(market value of the portfolio), Di is duration of bond i, and there are a total of K bonds in the portfolio 37 Duration/convexity approach ◼ Portfolio duration Example: consider the following 4 bond portfolio Bond Price ($) Duration 1 1,156 4.50 2 978 10.70 3 915 0.52 4 1,362 3.67 Market value of the portfolio = $1,156 + $978 + $915 + $1,362 = $4,411 Portfolio duration = $1,156 $978 $915 $1,362 4.5 + 10.7 + 0.52 + 3.67 $4,411 $4,411 $4,411 $4,411 38 = 4.79 Duration/convexity approach ◼ Portfolio duration On the previous slide, a portfolio duration of 4.79 means that for a 100 basis point change in the yield on every bond, the portfolio value changes by roughly 4.79% ◼ Be careful: the yields on the four bonds must ALL change by 100 basis points for the above measure of portfolio duration to be useful (i.e., the yield curve must undergo a parallel shift). This is the critical assumption made! ◼ Yet in practice there is no reason to believe that there are only parallel shifts in yield curves Recall the yield curve risk in Week 1. This is indeed one limitation of duration 39 Duration/convexity approach ◼ Convexity adjustment The limitations of duration: let’s use the bond example on Slide 10 to illustrate ◼ Recall the bond is option free, 8% coupon payable semiannually, 20 years, currently trading at par ($100) When yield rises by 10 basis points, then bond price drops to $99.02, a decline of 0.98% ( 1%) If yield drops by 10 basis points, then bond price rises to $101, an increase of 1% $101 − $99.02 Duration = = 9.9 2 $100 0.0010 40 Duration/convexity approach ◼ Convexity adjustment A duration of 9.9 means that for a 100 basis point change in yield, the bond price changes by roughly 9.9%; or equivalently, for a 10 basis point change in yield, the absolute percentage price change is approximately 0.99% (= 9.9 0.001, see Slide 28 for the formula), which is close enough to the 1% actual change in bond price ◼ So duration works well for small changes in yields ◼ But what if the yield on bond changes by e.g., 250 basis points? 41 Duration/convexity approach ◼ Convexity adjustment When yield rises by 250 basis points, bond price drops to $79.27, a decline of 20.73% When yield drops by 250 basis points, bond price rises to $130.10, an increase of 30.10% ◼ But according to duration, the bond price should rise and drop equally at 24.75% (= 9.9 0.025) ◼ So duration underestimates the price increase (24.75% < 30.10%) and overestimates the price decrease (24.75% > 20.73%)! ◼ 42 Duration/convexity approach ◼ Convexity adjustment Moreover, duration incorrectly treats percentage price increase as equal to percentage price decrease for a given change in yield Yet we know from both Property 3 and Property 4 of price volatility of bonds that for a given large change in yield, the percentage price increase should be higher than the corresponding percentage price decrease due to the positive convexity of the price yield curve (see Slide 10) ◼ Duration is therefore only a first (linear) approximation for a small change in yield ◼ The approximation can be improved by using a second approximation called the “convexity adjustment” 43 ◼ Duration/convexity approach ◼ Convexity adjustment The formula for the convexity adjustment to the percentage price change is: C (y* ) 2 , V+ + V− − 2V0 where C = 2 V0 (y ) 2 Note: 1) the notations are explained on Slide 25; 2) y is different from y* (see Slide 28); 3) C is known as the convexity of the bond 44 Duration/convexity approach ◼ Convexity adjustment Example: we continue with the bond example. The convexity adjustment (in %) is calculated as: C (y* ) 2 V+ + V− − 2V0 2 =[ ] ( y ) * 2 V0 (y ) 2 =[ $99.0177 + $100.9970 − 2 $100 2 ] ( 0 . 025 ) = 4.59% 2 2 $100 (0.0010) In the above calculation, y = 0.0010 (i.e., 10 basis points), yet y* = 0.025 (i.e., 250 basis points) 45 Duration/convexity approach ◼ The approximate percentage price change based on both duration and convexity is found by adding these two estimates Example: we continue with our bond example on the previous slide When yield rises by 250 basis points, the estimated change using duration is -24.75%, the convexity adjustment is 4.59%, so the total estimated percentage price change = (-24.75%) + 4.59% = -20.16%, which is closer to the actual decrease of 20.73% than using duration alone (-24.75%) ◼ When yield drops by 250 basis points, the estimated change using duration is 24.75%, the convexity adjustment is 4.59%, so the total estimated percentage price change = 24.75% + 4.59% = 29.34%, which is closer to the actual increase of 30.10% than using duration alone (24.75%) ◼ 46 Duration/convexity approach For bonds exhibiting positive convexity in their price yield curve, the convexity adjustment is positive, as for option free bonds (e.g., the bond example on slides) ◼ For bonds in the negative convexity region of their price yield curve, the convexity adjustment is negative, as for callable or prepayable bonds ◼ 47 Duration/convexity approach ◼ Modified convexity adjustment assumes that when yield changes the cash flows used to compute convexity C won’t change ➢ It is appropriate for option free bonds ◼ Effective convexity adjustment (will see in Week 6) assumes that when yield changes the cash flows used to compute convexity C do change ➢ This adjustment is more appropriate for bonds with embedded options 48 Price value of a basis point ◼ Another measure of price volatility to quantify the interest rate risk is the price value of a basis point (PVBP, also known as the dollar value of an 01, DV01). That is: PVBP = |Initial price – price if yield changes by 1 basis point| Example : Initial price = $954, price if yield drops by 1 basis point = $954.8 PVBP =| $954 − $954.8 |= $0.8 49 Price value of a basis point In the PVBP formula, it does not matter if yield increases or decreases by 1 basis point, the resulted change to bond value (in absolute value) will be the same, according to Property 2 of price volatility of bonds ◼ The PVBP is related to duration. In fact, the PVBP is almost equal to the dollar price change for a 1 basis point change in yields estimated using duration ◼ 50 Yield value of a price change ◼ An alternative measure of the price volatility of a bond is the yield value of a price change This is estimated by first calculating the bond’s YTM if the bond’s price is decreased by, say, X dollars. Then the difference between the initial YTM and the new YTM is the yield value of an X dollar price change 51 Yield value of a price change ◼ The smaller the difference (in yields), the greater the dollar price volatility, since it would take a smaller change in yields to produce a price change of X dollars Example: An option free $100 par value 3 year maturity 5% coupon rate payable semiannually bond is selling at $97.2914 The bond’s initial YTM is 6% ◼ When the bond price drops by $1 to $96.2914, the bond’s new YTM is 6.377781%. So the bond’s yield value of a $1 price change is (6% - 6.377781% =) 0.377781% in absolute value 52 ◼ Convexity Formula ◼ The convexity (measured in periods squared) of a bond is defined as: Convexity = 1 C C (C + Par ) 1 2 + 2 3 + ... + n ( n + 1 ) 2P (1 + y ) 3 (1 + y ) 4 (1 + y ) n +2 ◼ To obtain the convexity in years squared, simply divide by m2 where m is the number of periods per year ◼ The percentage price change using duration and convexity is: P − Duration (y ) + Convexity (y ) 2 P 53 The importance of yield volatility ◼ The greater the expected yield volatility, the greater the interest rate risk for a given duration and current value of a bond portfolio Consequently, to measure the exposure of a bond portfolio to interest rate changes, it is necessary to measure yield volatility The measure of yield volatility can be estimated as the standard deviation of yield changes (as we will see in Week 5) 54 Value-at-risk (VaR) ◼ Value at risk (VaR) is defined as the maximum loss in the market value of a given portfolio that is expected to occur with a specified probability (see the next slide for a graphical illustration …) In finance VaR can be used for risk management, risk measurement, and computing regulatory capital etc. Back testing a VaR calculation methodology involves looking at how often exceptions (i.e., loss > VaR) occur 55 Value-at-risk (VaR) ◼ Assume a standard normal distribution VaR at an e.g. ,99% confidence level 56 Value at risk (VaR) Regulators base the capital they require banks to keep on VaR ◼ The market risk capital is k times the 10 day 99% VaR where k is at least 3.0 ◼ Under Basel II, capital for credit risk and operational risk is based on a one year 99.9% VaR ◼ 57 Value-at-risk (VaR) ◼ VaR asks: “What loss level is such that we are X% confident it will not be exceeded in N business days?” It captures an important aspect of risk in a single number It is easy to understand It asks the simple question: “How bad can things get?” 58 Value-at-risk (VaR) ◼ VaR example: a bond portfolio of $180 million, a portfolio volatility of 0.1407, then for a confidence level of 99% the portfolio VaR for a one day holding period is 1 VaR = $180,000,000 0.1407 2.33 = $3,732,094 250 There is only 1% probability that the portfolio losses will exceed $3,732,094 over one trading day Note: in the above calculations, we assume: 1) a standard normal distribution; and 2) there are 250 trading days in a year 59
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