MS-E2114 Investment Science Lecture 3: Term structure of interest rates

Transcription

1 MS-E2114 Investment Science Lecture 3: Term structure of interest rates A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science

2 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 2/40

3 This lecture We have determined the time value of money from fixed income securities (e.g., bonds) Prevailing interest rate implied by the yield to maturity (YTM) Yet YTM is not the same for all bonds YTM of long bonds tends to be higher than that of short ones This is not fully explained by different levels of default risk by issuers The yield curve shows YTM as a function of time to maturity 3/40

4 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 4/40

5 Euro area yield curves 5/40

6 Euro area yield curves 6/40

7 Euro area yield curves 7/40

8 Emergence of the term structure Why do the bond yields differ? Possible explanations: A. Expectation hypothesis Markets believe that interests will go up in the future B. Preference for liquidity Investors prefer to have their assets available for new prospects in the nearer future C. Market segmentation Specialization in trading short, medium or long bonds Little trading between segments arbitrages may exist inconsistent pricing 8/40

9 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 9/40

10 Spot rates Spot rate s t = rate of interest (p.a.) for a loan from the present to time t Principal and interest paid at time t Annualization with different compounding intervals gives the following growth factors Yearly: (1 + st ) t m periods per year (1 + s t /m) mt Continuous e st t 10/40

11 Determining the spot rates Method A: Bootstrapping Assume we know s 1 Implied by the YTM of a 1-year U.S. Treasury bill Consider a 2-year bond with annual coupon payment C and face value F Value of bond P = C + C + F 1 + s 1 (1 + s 2 ) 2 Solve for s2 to determine the spot rate for 2 years Continue similarly by considering 3, 4, 5,... year bonds to determine s 3, s 4, s 5,... 11/40

12 Determining the spot rates Method B: Replication Idea: Construct a portfolio which pays no net coupon payments Spot rate must be the IRR of this portfolio Example Bond Maturity (y) Face value Coupon rate Price ( e ) A % B % Consider the portfolio in which 10 units of B are bought and 8 units of A are sold The total annual coupon payments e received from B ( e 8%) and paid to A ( e 10%) cancel out At maturity year 10 there is a cash flow of e e = e Initial investment cost is e e = e (1 + s 10 ) 10 = s % 12/40

13 Determining the spot rates Spot rates derived from different bonds can be different from each other Estimate spot rates with many methods and bonds Averaging and other statistical methods can be used for improved accuracy 13/40

14 Time-dependent discounting factors Spot rates are different for different periods Discount rates are different, too PV of cash flow (x 0, x 1,..., x n ) is PV = x 0 + d 1 x 1 + d 2 x d n x n, where the discounts factors are d k = d k = 1 (1 + s k ) k yearly 1 (1 + s k /m) mk m times per year d k = e s k t continuous 14/40

15 Example: Spot rates What is the value of a 5-y bond with 8% coupon rate and face value 100 e? =1/( ) Time t s t d t x t PV Total /40

16 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 16/40

17 Forward rates Forward rate f ij = interest rate (p.a.) agreed upon today for a loan taken at time i and paid back at time j, i < j Spot rates imply corresponding forward rates 1) Investing at the 2-y spot rate gives the growth factor (1 + s 2 ) 2 2) Investing at the 1-y spot rate and then with forward rate f 12 gives the growth factor (1 + s 1 )(1 + f 12 ) Because there cannot be any arbitrage, these growth factors must be equal (1 + s 1 )(1 + f 12 ) = (1 + s 2 ) 2 17/40

18 Forward rates Thus we have f 12 = (1 + s 2) 2 (1 + s 1 ) 1 The forward rate f ij depends on the compounding convention: A) With yearly compounding (1 + s i ) i (1 + f ij ) j i = (1 + s j ) j ( ) (1 + sj ) j 1 j i f ij = (1 + s i ) i 1 18/40

19 Forward rates B) With compounding m times per year ( 1 + s ) mi ( i 1 + f ij m m f ij = m ( (1 + sj /m) j (1 + s i /m) i C) With continuous compounding ) m(j i) = (1 + s j ) mj ) 1 j i m e s i i e f ij (j i) = e s j j f ij = s jj s i i j i 19/40

20 Spot vs. forward rates The spot rate s i is a special case of a forward rate: s i = f 0i With n periods, there are n(n + 1)/2 forward rates and n spot rates For n = 3, the spot rates si and forward rates f ij are f 01 f 02 f 03 f 12 f 13 f 23 20/40

21 Example: Forward rates Given the spot rates below, determine the forward rates i i s i 3.00 % 3.40 % 3.60 % 3.70 % 1 f i,j = 1 + s j j i j s i i j f ij % 3.40 % 3.60 % 3.70 % 3.80 % % 3.90 % 3.93 % 4.00 % % 4.00 % 4.07 % % 4.10 % % 21/40

22 Forecasting spot rates with forward rates Let the current spot rates be s 1, s 2,... What will the spot rates s 1, s 2,... in a year s time? If the interest rates follow investors expectations, in one year the spot rates will equal the current forward rates! Hence s 1 = f 12, s 2 = f 13,... i Forecast for next years' j forward rate table f ij % 3.40 % 3.60 % 3.70 % 3.80 % % 3.90 % 3.93 % 4.00 % % 4.00 % 4.07 % % 4.10 % % 22/40

23 Forecasting spot rates with forward rates The updated table has one column/row less than the previous one Spot forecast from s 1, s 2,..., s k yields s 1, s 2,..., s k 1 Forecasts obtained for one period shorter than the length of the initial time series But even other forecasts could be made, e.g., one could assume that: The spot rates stay constant The spot rates increase by a constant/proportional amount Yet arbitrage opportunities must be eliminate We shall revisit these later on 23/40

24 Invariance theorem Theorem (Invariance theorem) Suppose that the interest rates evolve according to expectations dynamics and that interest is compounded annually. Then any sum of money invested at the interest rate for n years will grow by a factor of (1 + s n ) n regardless of the investment strategy on condition that all funds (including initial sum and accrued interest on investments) will remain fully invested. 24/40

25 Invariance theorem Proof. Consider the two period case n = 2 : A If you invest in a 2-year zero-coupon bond, the growth is (1 + s 2 ) 2 B If you invest twice in a row in 1-year zero coupon bonds, the growth is (1 + s 1 )(1 + s 1 ) = (1 + s 2) 2 By the expectation hypothesis s 1 = f 12 By the definition of forward rates, (1 + s 1 )(1 + f 12 ) = (1 + s 2 ) 2 The investments A and B have the same growth factors Note: Any other fixed income investment can be formed as a combination of these two strategies E.g, the value of the cash flow of a 2-year bond with coupon payments can be obtained by adding the present values of two investments (i) and (ii) whose future cash flows are: (i) the first coupon payment after 1 year (1 st year of type B) (ii) the second coupon and face value after 2 years (type A) This logic can be readily extended to any number of periods n 25/40

26 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 26/40

27 Short rates Short rate r k = is the forward rate for a single period rk = f k,k+1 Short rates can be derived from 1 + s 1 = 1 + f 01 = 1 + r 0 (1 + s 2 ) 2 = (1 + s 1 )(1 + f 12 ) = (1 + r 0 )(1 + r 1 ). (1 + s k ) k = (1 + r 0 )(1 + r 1 ) (1 + r k 1 ) 27/40

28 Spot vs short vs forward rates Consider the forward rate table The first row of forward rate table gives spot rates Spot rates define the forward rates on the other rows of the table Short rates are on the diagonal of the forward rate table i j f ij % 3.40 % 3.60 % 3.70 % 3.80 % % 3.90 % 3.93 % 4.00 % % 4.00 % 4.07 % % 4.10 % % 28/40

29 Running present value Present value of a cash flow can be computed as follows PV (k) = Cash flow discounted to k from periods k and later Set PV (n) = x n For k = n 1, n 2,..., 1, 0, calculate the backward recursion PV (k) = x k + where r k = short rate at time k PV (k + 1) 1 + r k, This running present value can be written alternatively by using the discount factor d k = 1/(1 + r k ) as PV (k) = x k + d k PV (k + 1) 29/40

30 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 30/40

31 Parallel shifts in spot rate curve The original spot rate curve is the middle curve This curve is shifted upward and downward by an amount λ to obtain the other curves It is possible to immunize a portfolio of fixed income securities against such shifts for small values of λ 31/40

32 Fisher-Weil duration Lecture 2: Duration measures sensitivity to interest rate changes Here we derive analogous results for changing spot rates s k s k + λ for all k, We first consider the case of continuous compounding Given the spot rate curve s t, t 0 t t n, the Fisher-Weil duration for the cash flow sequence, (x t0, x t1,..., x tn ) is D FW = 1 PV PV = n t i x ti e s t i t i, where i=1 n x ti e s t i t i, i=0 32/40

33 Fisher-Weil duration When s k s k + λ, we have dp(λ) dλ P(λ) = t x ti e (s t i +λ)t i i=0 = λ=0 n t i x ti e s t i t i i=0 n i=0 = t ix ti e s t i t i P(0) P(0) dp(0) dλ = D FW P(0) 1 P(0) dp(0) dλ = D FW 33/40

34 Quasimodified duration For periodic compounding, we get the quasi-modified duration D Q Let the spot rate in period k be s k and the cash flow stream be (x 0, x 1,..., x n ) (with periodical indexing) t ( P(λ) = x k 1 + s ) k + λ k m dp(λ) dλ k=0 = λ=0 n k=0 k ( m x k 1 + s k m ) k+1 Dividing by P(0), we get the definition for D Q as D Q = 1 n dp(0) P(0) dλ = k=0 k m x ( k 1 + s k m t k=0 x k (1 + s k +λ m ) k+1 ) k 34/40

35 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 35/40

36 Example: Immunization A firm has to pay 1 million e in 5 years It seeks protection from interest rate risk through immunization Both bonds A and B have face values of 100 e Bond Maturity (y) Coupon rate Price ( ) A 12 6 % B 5 10 % Bonds priced according to spot rates s 1 s 2 s 3 s 4 s 5 s s 7 s 8 s 9 s 10 s 11 s /40

37 Example: Immunization 37/40

38 Example: Immunization Based on spot rates, the quasi-modified durations can be computed as NPV D Q Liability L e 4.56 Bond A e 7.07 Bond B e 3.80 For immunization, one needs to buy z A bonds A and z B bonds B so that { P A z A + P B z B = PV L D A P A z A + D B P B z B = D L PV L z A = 2208, z B = 4745 rounded 38/40

39 Example: Immunization 39/40

40 Overview Term structure Spot rates Forward rates Short rates Term structure and duration Immunization 40/40