Chapter 4 Interest Rate Measurement and Behavior Chapter 5 The Risk and Term Structure of Interest Rates
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1 Chapter 4 Interest Rate Measurement and Behavior Chapter 5 The Risk and Term Structure of Interest Rates Fisher Effect (risk-free rate) Interest rate has 2 components: (1) real rate (2) inflation premium I = r+ IP LEVEL OF INTEREST RATES STRUCTURE OF INTEREST RATES Term Structure of Interest Rates - defines the relationship between maturity & annualized yield, holding other factors such as risk, taxes, etc., constant. Graphic presentation is the yield curve. Curve shifts and twists through time. Four Basic Shapes: Positive Yield Curve: upward sloping Negative Yield Curve: downward sloping Flat Yield Curve Humped Yield Curve Level of Interest Rates: Loanable Funds Theory -- market interest rate is determined by the factors that control the supply of and demand for loanable funds. Demand Factors Household: Y increases then installment debt increases R increases then installment debt falls
2 Business: NPV = - I n CFt + t (1+i ) T =1 if i decrease _ NPV (?) Government: interest inelastic 1980: debt/gnp =25% 1987: debt/gnp=42% Foreign Foreign interest rates vs. U.S. rates Supply Households - largest suppliers Very steep slope for supply. Why? What happens if expect higher inflation? Savers -
3 Liquidity Preference Theory - Market rate of interest is determined by demand/supply of money balances. Demand of Money Transaction Precautionary Speculative + f(y) + f(y) - f(r) Supply of Money Fed basically determines supply Few uncontrollable factors (1) banks lending (2) public's preference for cash Letting D=S then solve for interest rate: r = f _ (,, ) y m p e
4 THEORIES ON SHAPE OF YIELD CURVE Unbiased Expectations - shape of yield curve is determined solely by current & expected future short-term interest rates. A 2 year security should offer a return that is similar to the anticipated return from investing in 2 consecutive one-year securities. 2 (1+ t R2 ) = (1+ t R1 )(1+ t+1 r1 ) 3 (1+ t R3 ) = (1+ t R1 )(1+ t+1 r1 )(1+ t+2 r1 ) t+1r 1 / one year interest rate that is anticipated as of time t+1. t+2r 1 / one year interest rate that is anticipated as of time t+2. R 1 } R 2 } Known annualized rate on 1 year security, 2 year security, 3 year security. R 3 }
5 Eg.. t 2 R =.10 if. t 1 R =.08 2 (1+.1 ) =(1.08)(1+ t+1r1 ) then 2 (1.1 ). t+1r1= Expect interest rates to rise: investor (saver) wants s-t security borrowers want L-T security Risk neutral Borrowers are the suppliers of securities (IOU's) therefore s-t demand > supply then increase in Price which results in a decrease in yield and we have an upward sloping curve. Liquidity Premium Investor becomes risk averse prefer s-t security over L-T security. Inducement to buy L-T must add premium. Segmented Markets Unlike first 2 theories which treat market as a whole - this theory treats market as if made up of segments. Investors/borrowers choose security with maturities that satisfy their forecasted cash needs. Therefore, need rather than expectations of s-t rates determine where a person invests. Demand/Supply within segments drive rates. rate Upward Pension Life Ins. Thrifts Bank Time
6 Downward Bank Thrifts Pension Demand for L-T increases or D>S for S-T Prices increases for L-T and yield for L-T decreases Preferred Habitat Combines Expectations & Segmented Investors may have a preferred investment horizon (habitat) but their expectations on int. rates move them within or even out of their habitat. Which Theory is Correct? Research by Meiselman on interest rate expectations found they strongly influence on the term structure-- they have looked at how accurate forward rates are--not very accurate--therefore other factors also influence interest rates Kessel found that liquidity premium caused forward rates to have a positive bias Other research on liquidity premium have found that the size of the premium varies inversely with interest rate levels and yet others have found the opposite to be true Elliot and Echols have examined the segmented market idea--they found discontinuities in the yield maturity relationship--may be due to supply and demand of segments All of the theories have some evidence on their validity Integrating the theories Example: 1. Borrowers and investors expect rates to rise they invest (expectation theory) 2. Borrowers need long term funds while investors prefer short-term (segmentation theory) 3. Investors prefer more liquidity than less (liquidity premium theory) In this example, all theories suggest an upward sloping curve ---domination of the views would determine the slope--ie. If #1 was dominate but the expectation was for declining rates then might have a slightly downward sloping curve
7 Other factors that affect yield on Sec. (1) Default risk (2) Liquidity (3) Tax Status (4) Term-to-Maturity (5) Special Provisions Call Convertibility Yield on Sec r = R / + DP + LP + Special Provision
8 INTEREST RATES/BOND PRICES Appropriate price of a sec. is the PV of remaining cash flows PV considers (1) size of cash flow (2) timing of cash flow (3) required rate of interest. which includes adj. for risk Price of m Ct Bond = t (1+i ) t =1 Two Cash Streams Annuity (interest) PV a i 1-(1+ = A [ m i m ) -nm ] Single Sum S PV s = i (1+ m ) nm Semiannually therefore need to adjust interest rate and period.
9 As interest rates rise, bond prices lower, but this is not 1-1 relationship. Coupon Rate - YTM Relationship Coupon Rate > YTM Ψ sell at premium Coupon Rate < YTM Ψ sell at discount Coupon Rate = YTM Ψ sell at par or face value Coupon Rate = Interest in dollars Par Value Current Yield = Interest in dollars Price of Bond YTM= Capital gain (loss) + income from interest payments YTM (Exact) Bond C1 Price= (1+YTM C2 + 1 ) (1+YTM Cm + Par ) (1+YTM ) m Solve for YTM YTM (approximate) = YTM app Par - Price Annual Interest dollars + number of yrs maturity Price + Par 2 YTM (Exact) Bond C1 Price= (1+YTM C2 + 1 ) (1+YTM Cm + Par ) (1+YTM ) m Solve for YTC Rate Maturity Bid Ask Bid Chg Yld 81/4s * May 105:25 106: * * May Bond matures on 2005 but is callable starting in 2000 ** Yld For callable bonds the YTM is calculated in one of two ways a. to first call date when the asked price is above par b. to maturity date when the asked price is equal to or below par
10 YTC 1. Substitute the Call Price for Par 2. Substitute the number of years to first call for maturity years Effective Yields: Realized r Yield = (1+ ) m mn - 1 r Continuous Compounding = e - 1 e Point of interest: Difference between nominal rate and add-on and discount. Add-On 36 months, r = 10% $5000 borrowed monthly payment: Interest Payment: 1) 5000 x.10 x 3 = $1500 2) ( )/36 = Actual Interest Rate 5000 = [PVIFA x,36 ] = x x = 1.5%/month Ψ Annual 18% Discount 36 months Interest $1500 Pay $5000 Receive $3500 Payment 5000 = $ = (PVIFA x, 36 ) 25.2 = x x = 2%/month 24% Annual
11 Valuation of Zero Coupoin Securities: Price = Maturity Value ( PVIF %,n) If the security is sold on a discount then the price is equal to the PV of a lump sum. Money market instruments sold on a discount basis are sensitive to changes in interest rates but not to the same degree as bonds with coupon payments. Treasury Bill Yields P Discount Yield = [ ] n 100 Equivalent Yield 365 = [ n P P ] i.e. Discount yield on bill 8.6% with 80 days till maturity..086 = Price = 4.5 [ Price ] 4.5 [Price] = P = Price.086 = 80 Price Equivalent Yield = 8.6%.0188 = Price Price [Price] = 100 Price = 98.15
12 Factors Affecting Bond Prices Strong economic growth tends to place upward pressure on interest rates Weak economic growth tends to place downward pressure on interest rates Money supply increase (demand not affected) downward pressure on rates Money supply increase ( demand increases for funds) upward pressure--inflation Oil prices have major impact upon prices drop in oil prices--lower interest rates Weaker dollar ( everything else constant) increase inflationary expectations ( prices of imports increase); thus increase in rates Default risk Exchange Rates and Foreign Bonds Risk that currency will depreciate and more than offset any coupon rate advantage when the interest payments are converted into US dollars International Diversification Reduce exchange rate risk by diversifying among foreign securities denominated in various foreign currencies. Diversify internationally to reduce exposure to interest rate risk and default risk. By investing in many countries avoid only one country=s economy and interest rates to affect the bond portfolio. Measurement of timing of Cash Flows (1) Term to Maturity - Number of years to final payment - Ignores interim cash flows - Ignores Time Value Duration (2) Weighted Average Term to Maturity - Computes the proportion of each individual payment as a percentage of all payments and makes this proportion the weight for the year the payment is made. CF1 CF 2 CFm WATM = (1)+ (2)+...+ (m) TCF TCF TCF Cf t = the cash flow in year t m = maturity TCF = Total Cash Flow e.g. 10 year 4% bond TCF = WATM = (2)+ (3)+...+ (9)+ (10)= 8.71 yrs Considers timing of all cash flow - Does not consider time value of the flows
13 DURATION Duration: Weighted Average number of years until an initial cash investment is recovered with the wight expressed as the relative present value of each payment of interest and principle. In order for a bond to be protected from the changes in interest rates after purchase the price risk and coupon reinvestment must offset each other. Duration is the time period at which the price risk and coupon reinvestment risk of a bond are of equal magnitude but opposite in direction. Assume a 8% coupon on 1000 Face Value bond with 2 years to maturity and a YTM of 20%. Duration Calculation (1) (2) (3) (4) (5) Periods Coupon 1 (1+i) n where i=10% 2 x 3 Unweighted PV 1 x 4 Weighted PV 1 $ $ $ $ Duration Calculation: = 3.75 semi -annual periods Present Value Interest Factor, is obtained by using the formula 1 where n is the number of compound periods. (1+i) n P P = - D (1+r) (1+r) If you know a bond's duration you know how much its price will change as its yield changes. D: duration; P: initial price; r: initial YTM P 100%= r( P - D (1+r) )100%
14 Ex. Yield on 8% 5 year bond selling at par has duration of 4.31 years rates go to 82% Price decline will be of its initial price % [.005( )] = % 1.08 Yield Differential Yields vary according to factors such as default risk, marketability, & tax status. Bond Risks (1) Interest Rate Risk Factors influencing change in interest rate: a) Money Supply Growth Rate b) Impact of Oil Prices c) Dollar Value d) Event Risk e) Default Risk
15 15
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