The Classical Theory of Interest Duration and Volatility The Term Structure and YTM Explaining the Term Structure Allowing for the Risk of Default

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1 Bonds, Duration and the Term Structure of Interest Rates The Classical Theory of Interest Duration and Volatility The Term Structure and YTM Explaining the Term Structure Allowing for the Risk of Default

2 Credit Risk How to assess the risk of bonds? (1) Uncertainty of returns (subsequent changes in interest rates) (2) Risk of default (3) Early redemption

3 Valuing a Bond PV (1 C 1 r) 1 (1 C 2 r) ,000 (1 r) C N N

4 Valuing a Bond Example If today is October 2008, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2013 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%) Cash Flows Oct

5 Example continued Valuing a Bond If today is October 2008, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2013 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%) PV , $1,161.84

6 Price Bond Prices and Yields Year 9% Bond 1 Year 9% Bond Yield

7 Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher

8 Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher Nominal Interest Rate = The rate you actually pay when you borrow money

9 Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher Nominal Interest Rate = The rate you actually pay when you borrow money Real Interest Rate = The rate after taking inflation into account when you borrow money, as determined by supply and demand r Supply Real r Demand $ Qty

10 Bonds are risky We use Duration to measure the risk of bonds Duration is (1) an average life of the bond rates (2) a measure of risk (3) a sensitivity to interest

11 Duration Describing the life of a bond 3 bonds (IRR=8%) A: : 10, 10, 10, 110 B: 106.2: 1, 1, 1, 141 C: 106.6: 32.2, 32.2, 32.2, 32.2 Same maturity, different cash flows

12 Average Life Calculation of Duration Multiply each Time by the present value of the Cash Flow received at that Time and then sum. Divide result by total present value of the bond

13 The weighting system : 10, 10, 10, 110 So in the above bond, the Present value =106.1 (at 8%) The Present value of the first 10 is 10/1.08 = 9.3 Similarly the Present values of the second, third and fourth = 8.6, 7.9 and 80.9 The sum is

14 Calculation for A Cash Time = Time.PV =370.8 Duration of A =370.8/106.6 = Sim D B =3.95 and D C = so for B You wait longer to get your money back.

15 Formula D A = t. PresVal(C t )] PresVal(A) D = the Duration of the investment

16 Duration as risk Suppose interest rates rise soon after you buy investment A The new value will be the say 12% time Cash PV A = 93.9 PV B = 92.0 and PV C = 97.8

17 Implications The investment with the greater Duration is more volatile i.e. riskier with respect to changes in interest rates Calculated for most bonds in stock market In the form ( P/P) = -D r/(1+r) that is the proportionate change in price as a function of Duration

18 Duration and volatility The risk is that interest rates change => the current price will change. More risky if Current income is low, Duration is long, interest rates are low.

19 Terminal Values and Duration Duration suggests that the change in current price (to interest rate changes) is proportional to duration. What about future values?

20 Matching Assets and Liabilities Suppose we buy lease with expected cash flows of 10, 10, 10, 39 (V = 51.5) IRR = 10% Wealth if we re-invest income and sell To find the FV of cash flows we can use: FV(t) = PV(0) x (1+r) t

21 Future Value of cash flows. 10,10,10,39 PV at t=0 FV at t=2 FV at t=3 FV at t=4 2% % %

22 Terminal Wealth and Duration The terminal wealth is nearly constant when we sell at time = duration Pension funds calculate their liabilities Can minimise risk if D assets = D liabilities The matching strategy. Alternative is by cash matching Liabilities have long durations

23 Duration Calculation Calculation for equities For Gordon s growth model V=Div/(r-g) D = (1+r)/(r-g) e.g. if r = 10%, g=4%, D=18.3 But in practice g changes with k so that effectively D is lower

24 Revision questions 1. Calculate the Duration of the following: A lease of 5 years fixed rent 50 per sq ft. interest rate = 8% A bond with coupon 6%, 5 years to maturity and interest rates 8% A 5% zero-coupon bond with five years to maturity and interest rates 8%

25 Revision question 2. A 10 year lease has a rent review for the 6 th year. The current rent is 20 and the expected rent in the 6 th year is 30. The discount rate is 10%. What is the duration of the lease? Predict what would happen to the value of the lease if the interest rate suddenly changed to 6%? Make your assumptions clear

26 Revision question 3. Property is estimated to have a long duration and therefore should be very sensitive to interest rate changes. Why then do values remain unaffected by interest rate changes?

27 Questions continued 4. (Difficult) A company is liable to pay 10m in three years. It can invest in two types of projects. The first type (X) will pay 1m in two years, the second (Y) will pay 1m in four years. The company can invest in as many projects as it wishes. If the rate of interest is 10%,Explain how you can construct a portfolio that is the least risky combination of project which will match the liability in the face of interest rate uncertainty

28 Debt & Risk Example (Bond 1) Calculate the duration of our 6 7/8 % 4.9 % IRR Year CF PV@4.9% % of Total PV% x Year Duration 4.424

29 Debt & Risk Example (Bond 2) Given a 5 year, 9.0%, $1000 bond, with a 8.5% IRR, what is this bond s duration? Year CF PV@8.5% % of Total PV% x Year Duration= 4.249

30 example Spot/Forward rates 1000 = 1000 (1+R 3 ) 3 (1+f 1 )(1+f 2 )(1+f 3 ) Spot Forward rates

31 Spot/Forward rates Forward Rate Computations (1+ r n ) n = (1+ r 1 )(1+f 2 )(1+f 3 )...(1+f n )

32 Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = year zero treasury YTM = 9.660

33 Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = year zero treasury YTM = Answer FV of YTM 2 yr 1000 x ( ) 2 = yr 1000 x ( ) 3 = IRR of (FV & PV= ) = 11%

34 Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

35 Spot/Forward rates coupons paying bonds to derive rates Bond Value = C 1 + C 2 (1+r) (1+r) 2 Bond Value = C 1 + C 2 (1+R 1 ) (1+f 1 )(1+f 2 ) d1 = C 1 d2 = C 2 (1+R 1 ) (1+f 1 )(1+f 2 )

36 Spot/Forward rates example 8% 2 yr bond YTM = 9.43% 10% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = %= 1010 Step = 80d d > solve for d =100d d > insert d1 & solve for d2

37 Spot/Forward rates example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 =.8350 insert d2 and solve for d1 = d1 =.9150 Step 4 Insert d1 & d2 and Solve for f 1 & f = 1/(1+f 1 ).8350 = 1 / (1.0929)(1+f 2 ) f 1 = 9.29% f 2 = 9.58%

38 Term Structure YTM (r) s & Normal Year Spot Rate - The actual interest rate today (t=0) Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time. Future Rate - The spot rate that is expected in the future Yield To Maturity (YTM) - The IRR on an interest bearing instrument

39 Term Structure What Determines the Shape of the TS? 1 - Unbiased Expectations Theory 2 - Liquidity Premium Theory 3 - Market Segmentation Hypothesis

40 Expectations Theory The market prices bonds so that the return from buying 1-year bond followed by another 1-year bond = the return from investing in a 2-year bond. Implications: if the term structure is upward sloping, the market expects short-term interest rates to rise

41 Liquidity Theory Lenders don t like lending long. Therefore they will demand a risk premium for long bonds An upward sloping term structure implies that they are risk averse but not necessarily that interest rates will rise

42 Market Segmentation Theory Short term investors don t buy long term bonds. Neither do long term investors buy short term bonds. There are two different markets. The market is dominated by a clientele effect.

43 Yield To Maturity All interest bearing instruments are priced to fit the term structure This is accomplished by modifying the asset price The modified price creates a New Yield, which fits the Term Structure The new yield is called the Yield To Maturity (YTM)

44 Yield to Maturity Example A $1000 treasury bond expires in 5 years. It pays a coupon rate of 10.5%. If the market price of this bond is , what is the YTM?

45 Yield to Maturity Example A $1000 treasury bond expires in 5 years. It pays a coupon rate of 10.5%. If the market price of this bond is , what is the YTM? C0 C1 C2 C3 C4 C Calculate IRR = 8.5%

46 Default, Premiums & Ratings The risk of default changes the price of a bond and the YTM. Example We have a 9% 1 year bond. The built in price is $1000. But, there is a 20% chance the company will go into bankruptcy and not be able to pay. What is the bond s value? A:

47 Default, Premiums & Ratings Example We have a 9% 1 year bond. The built in price is $1000. But, there is a 20% chance the company will go into bankruptcy and not be able to pay. What is the bond s value? A: Bond Value Prob Value YTM = = $ % =expected CF

48 Default, Premiums & Ratings Conversely - If on top of default risk, investors require an additional 2 percent market risk premium, the price and YTM is as follows: Value YTM $ %.

49 Summary Bonds are interesting because they can be compared relatively easily. They reveal investors expectations They can be used to benchmark the price of money over different intervals Q: Can the same be done to Property leases??

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