Lecture on Duration and Interest Rate Risk 1 (Learning objectives at the end)

Transcription

1 Bo Sjö (updated formulas 0a and 0b) Lecture on Duration and Interest Rate Risk (Learning objectives at the end) Introduction In bond trading, bond portfolio management (debt management) movements in the interest rates are the key factor. he most important tool, that investors have to calculate how the value of bonds and bond portfolios change with the interest rate, is called duration. Duration is simply answering the question how much will the price of a bond change if the interest rate change (and thereby the yield) a bit. Mathematically we talk about the derivate of the price with respect the interest rate. Duration is used to construct bond portfolios with a specific duration (=maturity). You can set up a bond portfolio that pays you a given amount at a specific date in the future that matches a required payment you must do on that date. his is referred to as immunisation. You can also use duration to calculate how much you stand to loose given one or several bad days on the market. hereby, you can get idea how big reserves you need to keep for a rainy day he concept of duration comes up after ) learning about what a bond is, ) how to value a bond, 3) understanding the link between interest rates and bond price, 4) learning about the terms structure, 5) how bonds with different maturities are related, and 6) theories about the terms structure.. Interest Rate Risk and Duration hese notes deal with the concept of duration and how to measure interest rate risk. If you buy a zero coupon bond with face value $000 and a maturity of 0 years today, you will get the nominal value of $000 paid out to you after 0 years. If I buy a 0 year bond that pays coupons, you will get the amount of $000 back quicker because you can reinvest the coupons payments and collect additional interest on these coupons up until the maturity of the bond. he higher the interest rate the quicker you can cash in $000. hus, the maturity of a bond is an ambiguous concept. If some bonds pay coupons, others do not, and the YM is different, the maturity of bonds are not directly comparable from a given investment horizon. Duration is a measure of the price sensitivity of a bond with respect to the interest rate. It is also a measure of the maturity of a bond in terms of the average lifetime of a bond. Or, by taking the reinvestment of the coupons into account when do we get the nominal value of the bond back in our hands. he value of financial assets and liabilities changes with the interest rate. When the interest rate goes up the value of the bond goes down. Interest rate risk is the risk associated with changes in the market values of a fixed income instrument following (unexpected) changes in the interest rate. he risk applies both to assets and to liabilities; it affects income streams, liquidity and solvency. his is my informal lecture note memo on duration.

2 Interest rate risk Since most institutions hold fixed income securities among assets and liabilities, interest rate risk is the net change in the value of all fixed income securities held by firm. hus, interest rate risk originates from mismatched maturities and cash flows of fixed income assets and liabilities. Look at the following balance sheet of a financial institution, able. Economic Balance Sheet Market Values Assets Loans (A) Liabilities Debt (L) Owner's Equity (E) As the interest rate changes, the value of the outstanding loans and the debt changes and effects in turn the value of owner's equity, A - L = E. he problem of interest rate changes comes from the fact that the change in value of assets and liabilities will not be the same if the interest rate where to change by say one basis point (0.000). Depending on the type of assets and liabilities the net effect of an increase in market interest rates can be positive or negative on the net position. Bond value and Duration Duration measures the sensitivity of market determined values (prices) with respect to changes in the interest rate changes. Duration is used mainly for two things. he first is to measure interest rate risk, and given the measure decide on hedging the risk. he second way, to use the duration measure, is to immunize the value of a fixed income portfolio from unexpected changes in the interest rates. he general definition of duration is that it measures the average lifetime of a security. he lifetime of a zero coupon bond is equal to its time to maturity. he average lifetime of a coupon-paying bond is shorter because the coupon payments can hypothetically be reinvested to yield additional income up until the maturity. he intuition here is that the money paid for the bond is "repaid" faster for a coupon bond than for a zero-coupon bond with the same maturity. Consider the valuation formula, here for an asset that pays fixed interest (coupons) and a face value: 0 C C C + FV = , () ( + r) ( + r) ( + r) where the discount rate r, is the required return, or alternative investment income for a comparable investment.. he value at time zero (today) is 0, we will assume, since we use YM in the formula, that the value is identical to the price on the market. (Different authors use different symbols. Instead of you can find B, sometimes V etc.) In the following we will assume that the price is given, and we solve for the yield to maturity from the market price. hough bonds are the typical fixed income instruments, it is important to remember that we talk about all types of bonds, including all types of fixed income securities, bills, notes, certificates of deposits (CDs), commercial papers, loans, deposits etc. he valuation principle is always the same. In the context of interest rate risk and duration, we often talk about the interest rate rather than the yield (to maturity) because the YM is a function of the prevailing interest rates in the economy, and our final aim is to analyse how the market value of fixed income assets varies with changes with "the interest rate of the economy". Furthermore, the interest rate, or the yield, is the nominal interest rate,

3 meaning that expectations about inflation, the expected business cycle and monetary policy play an important part. o find the change in the price following a change in the interest rate take the derivative of 0 with respect to y, d/dy. he price of the bond and the YM is related as, 0 C C C + FV Ct = = ( + y) t= t FV +. () A US government bond pays semi-annual coupons and the formula becomes, = C FV t 0 + t t= ( + y / ) ( + y / ). (3) In the following we will use annual coupon payments, until we start with practical calculations. Derivation of Duration (advanced) o find the price change given a change in YM, take the derivative of 0 with respect to a change in y. hat is we look the change in value given a small change in the yield, d/dy. his is a tricky derivative to find, so rather than taking the derivative with respect to y, we approximate and take the derivative with respect to instead. If we talk about a small change in y or in, will not matter that much for the effect on. he derivative of C = C( + y) with respect to (+y) is -(+y) -. General, the derivative of (+y) -n is given as -n (+y) -n-. Applying this to standard value formula above gives, d dy C C ( C + FV ) (4) For calculation purposes, we can break out -/(+y), d dy C C = + ( C + FV ) , (5) which simplifies to d = t CF t DF t +, (6) dy ( y) t= where CF t is the cash flow at time t and DF t the discount factor for time t. 3

4 If both sides of this expression are multiplied with (/ 0 ) we get the percentage change in the price as d / = dy 0 C C = ( y) + + ( + y) t CF DF t t ( + y) t= 0 ( C + FV ) (7) where CF t is the cash flow at time t, DF t is the discount factor at time t, and 0 is the price at time t=0, today. In Equation (7) the expression within the bracket, multiplied with (/ 0 ) is referred to as Macaulay duration (D), C C D = + ( + y) CF DF t t = t t = 0 ( C + FV ) , (8) Duration he duration measure tells how long it will take you (in years) to get the nominal value if you keep reinvesting coupons at the present YM, D = CF DF t t t t =. 0 he duration measure will also tell you about the interest rate sensitivity of the bond value to changes in the interest rate. he greater the distance between the time to maturity of the bond and duration, the more sensitive is the bond to changes in the interest rate. It follows, that the change in the price of the bond following a change in the interest rates (=yield change) is, d dy = (( + y) ) D. (9) he ratio of duration to (+y) is referred to as modified duration (MD) MD = D ( + y) (0) Modified duration simplifies the calculations that follows and provides a form of standardization of the duration value in the sense that the duration of bonds with different yields Calculate rice changes Given these results a change in the relative price is given as, 4

5 y = D ( + y) or = MD y () where represent an observable change in discrete time. he change in vale (, in dollar terms) is given by, = D y ( + y) or = MD y () Of course, the same expressions can be, and is often, written with modified duration. Modified duration can be used to find the price change (in per cent) as dp = MD y (3) p Convexity he duration measure is an approximation. he curve that links bond prices with changes in yields is convex. It is possible to improve the formula with the following adjustment, 3 = MD y Convexity (4). he Duration of a ortfolio ( y) o calculate the duration of a fixed income portfolio, start by calculating the duration of each individual security in the portfolio. he duration of the portfolio is then given by weighting the individual duration measures together using market values. Suppose there are N securities in the portfolio, the market value of the portfolio is then MV = MV + MV MV N, where MV i is the market value of security i. Let the duration of asset i be D i. he duration of the portfolio is then given by D = D (MV /MV ) + D (MV /MV ) D N (MV N /MV ). (4) Calculating the duration of a fixed income portfolio is easy. Start by calculating the duration of the individual instruments, and then weight them together using weights based on their market values. 3. Calculating changes in the value of assets and liabilities We know that the value of assets is the present value of the cash flows associated with the asset until maturity. (he value of a liability follows the same principle). Up until now, we used continuous time, indicated with d, dy etc. We cannot observe changes in continuous time. hus, when we move actual changes in prices, we move to small observable changes in discrete time, that is changes between to points in time. 3 For more on Duration and convexity, incl. the convexity formula, see textbooks in Investments such as Bodie, Kane and Marcus. 5

6 With the help of the duration formula, we can estimate the price change following a change in the interest rate. For various reasons, developed below, the formula is in practice only an approximation. Setting d 0 = 0, we get that the percentage change in the price, is given by the following expression, y 0 = D 0, (5) or y 0 / 0 = D, (6) which simplifies to 0 0 / = MD y, (7) where MD = ( / ) D is modified duration, and the change in YM is expressed in decimals. Notice that though duration (D) is typically measured in years the yield y is expressed on the same time interval as the frequency of the coupon payments. For a bond paying annual coupons, YM (y) is just the yearly YM, or the yield per annum. For a security paying semi-annual interest, the annual YM (y) must be divided by two, for quarterly payments divided the annual YM with 4, etc. Notice that y is expressed in decimals, not in percent. When analysing interest rates and yield changes, it is common to talk about changes in terms changes in basis points. One basis point ( bp) change is 0.0% (or 0.000). Example Consider a bond, with face value $,000, is selling for $964.54, with an YM of 0%, duration of.8853 years, and semi-annual coupon payments. What is the predicted change in the price of the bond, given a one basis point (0.0%) increase per 6 months in the level of the interest rate? If interest rates go up, the price of the bond will fall. he predicted percentage drop in the price is given by - D [ y/(+y)] = [0.000 / ( )] = or -0.08%. Notice that 0% and semi-annual coupons give us ( +y/). he price declines to $ ( ) = $ In this situation, a one basis point increase has a relative small impact on the price. Alternatively, use the formula = -D [ y/(+ y)]. he change in the price is, $ [0.000 / ( )] = $ = he new, lower price, is predicted to be $ $0.736 = $ (as before). Conclusion: If we know the duration we can calculate the expected price change following an expected change in market interest rates, here understood as a parallel shift in the yield curve. We assume that all spot interest rates that exist at one point in time changes up or down uniformly over time, and that yield curve is flat. If we expect the interest rate to go up, we should reallocate our bond portfolio in such a way that we decrease the duration of the portfolio. Higher interest rates will lower the price of bonds. he effect of the interest rate, and decline of value, is reduced by shifting to lower duration bonds. In this way I can use the concept of duration for active bond portfolio management. If we expect the interest rate to go down, bond prices will go up. In this situation, the maximum gain is found by swapping low duration bonds in the portfolio to high duration bonds. 6

7 Furthermore, if we want to hedge the value of the bond portfolio at a given value in the future, against yield changes, we should construct a portfolio with a duration that matches the future date. In this way it is possible to lock in a future nominal value. (his is discussed below, exact matching or immunization) 4. Computing Duration- practical tips Duration is simple to calculate, as long as you do things in the right order. If you study the formula, calculate the nominator first, and then divided it by the price of the security. It is recommended that you set up the following type of matrix (and fill in the cells): ime to payment in years Cash flow - Coupon ayment CF t Discount Factor Discounted Cash Flow (V) CF t DF t Weighted Discounted Cash Flow CF t DF t t DF t t C /(+r) /(+r) t C + FV /(+r) - - Sum =.0000 CF t DF t t Duration =D = ( V t ) / 0 Modified duration = MD = D/(+r) Another way of simplifying the calculation is to write the duration formula in the following two steps, C ( + r) = wt, 0 and duration is given by D = t t= w t t. In this case set up the following matrix, and fill in the following matrix: ime to payment (in years) t i Cash Flow ayment (C + FV) CF Discounted payment CF/( + r) Weight [CF(+r)]/ 0 = t t t CF FV w t he sum of this he sum of this column = 0 column =.0 Duration = D Modified Duration = MD = D/(+r) Weight time w t t he sum of this column = D Given that the left most column, in both the matrixes above, is in years, duration is also measured in years. I the left most column is,, 3,..., in say quarters, duration is also measured in quarters, but can be expressed in years if it is dived by 4, the number of quarters per year. Notice that the discount 7

8 rate must correspond to the time interval of the coupon payments, i.e. interest rate per six months for semi-annual coupon payments. 6. he Duration of Net Worth and Reserves he calculation of net worth is a combined calculation of assets minus liabilities. Calculate the duration of assets and liabilities separately, and then weight them together. Assume that an FI has assets (A) and liabilities (L) consisting of fixed income securities. From the balance sheet we have, A = L + E, (8) where E is equity. Equity is given by E = A - L. (9) It follows that the change in net worth is given by E = A - L. he threat to any financial intermediary is that changes in the interest rates will not only make them loose money due to a falling spread between lending and borrowing rates, but also that it might lead to insolvency. (Assets - Liabilities does not cover the owner's equity) We can use the duration formula for a portfolio to calculate the duration of the portfolios of assets and liabilities, respectively. his gives D A and D L. From these we can calculate D E, or rather set up the formula for how E is changing when the interest is changing. A change in the level of interest rates will affect the markets values of assets and liabilities, and therefore also the net worth of the FI. he threat to the FI is that the interest change can make net worth become negative. he change in net worth is given by E = A - L. o analyse this risk we need the duration for assets and liabilities, respectively. hese are given by the duration formulas: A/A = -D A [ r / ( + r)] or A = -D A A [ r / ( + r)], L/L = -D L [ r / ( + r)] or L = -D L L [ r / ( + r)]. (0a) (0b) hese formulas show the change in the value of total assets and total liabilities. It follows that the change in net worth is E = -[D A A - D L L] [ r / ( + r)]. () he formula can be written as E = -[D A - k D L ] A [ r / ( + r)], () where k = L/A, the leverage ratio of the FI. he change in the value of the FI originates from three components. he first is the duration gap: -[D A - k D L ]. he second is the size of the FI, measured by the market value of its assets. he third is change in the interest rate. he bigger the change in the interest rate the bigger is the change in net worth. 8

9 An FI can manage interest rate risk, by changing the duration of its assets and liabilities, or by changing k. Example: Suppose we have a FI with the following balance sheet: Assets Liabilities $00m $90m $0m Where the total value of the assets are $00m. Duration: D A = 6 years, D L = 4 years. k = 0.9. Suppose the interest rate is 8% and increases by %, what is the effect on the net worth? he answer is given by E = - (D A - k D L ) A [0.0/( + r)] = - ( ) 00 [(0.0/(.08)] = -$.. he new balance sheet becomes Assets Liabilities $94.44m $86.66m $7.78m he net worth declines, as expected, but is not eroding net worth. It is easy to see why banks might run into trouble if capital market liberalisation is not accompanied by a tight monetary policy that controls inflation. A huge variance in the interest rate can have large consequences for the banking sector. + It is possible to talk about duration of any asset, say stocks. As long as you can calculate d/dr you get the duration, but always as an exact value as in the calculations above. Learning Objectives: Explain what duration is? What is it measuring? What can you do with duration? What does immunization mean? Why is not maturity matching enough to hedge against interest rate risk? If you want to fix the maturity of a bond portfolio, how do you do it? Know the formula and apply it, to a single bond and to portfolios of bonds. Calculate the change in the value following a change in the interest rate, both for a single bond as well as for a portfolio (Asset value, and liability value).he relation and demand for stripped bonds, how is that related to duration? How does interest rate changes affect E? What are the variables one must take into account? What is convexity in the context of bonds and duration? How do you deal with it? Explain the link between Duration and VaR models, and Bank reserves. 9

10 References Cox,, J.E. Ingersoll and S.A. Ross (985) A heory of the erms Structure of Interest Rates, Econometrica 53, Heath, D., R.A. Jarrow and A. Morton (99) Bond ricing and the erms Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica 60, Hull, John C. (003) Options, Futures and Other Derivatives, 5 ed., earson education, rentice Hall, New Jersey. Jarrow, Robert and Stuart urnbull (000) Derivative Securities, ed, South-Western College ublishing, homson Learning. Saunders, Anthony and Marcia Millon Cornett (008) Financial Institutions Management A Risk Management Approach, McGraw-Hill. Vasicek, O. (977) An Equilibrium Characterization of the erm Structure, Journal of Financial Economics 5,