1. 1 VALUATION OF FIXED INCOME SECURITIES Bond: A debt instrument with periodic payments of interest and repayment of principal at maturity rM rM rM rM rM rM rM+M |___|____|____|____|____|...…..|___ | 0 1 2 3 4 5 n-1 n r: coupon interest rate M: maturity (par value) n: term to maturity

2. 2 Bond Valuation V= rM(PVIF) i,1 +rM(PVIF) i,2 +……… rM(PVIF) i,n + M(PVIF) i,n i: market rate of interest Coupon payments (rM) can be regarded as an annuity, V= rM(PVIFA) i,n + M(PVIF) i,n or (1+i) n -1 1 V = rM ------------- + M ------------ (1+i) n (1+i) n

3. 3 Bond Valuation example n=10 years, coupon rate: 8% M= $1,000 Market rate : 10% $80 $80 $80 $80 $80 $80 $180 |___|____|____|____|____|...…..|___ | 0 1 2 3 4 5 9 10 V= $80x(PVIFA) 10%,10 + $1,000x(PVIF) 10%,10 = $877.11 If i > r V < M(discount) i M(premium) i = r V = M(par) Yield-to-maturity: the rate of return on a bond In the example, the YTM is 10%. A bond’s YTM is the market rate of interest for that risk group and maturity.

4. 4 Valuation Between Interest Payment Dates V: invoice price of the bond c: days until first payment g: number of days between two payment periods P= quoted price = V - accrued interest Accrued Interest = rM (g-c)/g

5. 5 Valuation Example Eg. N=5 years,semiannual coupon r=8%, i=10%, first payment 2 months from today. V= Invoice Price = $953.29 Accrued Interest = 40 x (4/6) = $26.67 Quoted price = $926.62

6. 6 Risks Faced by a Bond Investor Default risk Interest rate risk (price risk) Reinvestment risk Call risk Inflation risk Foreign exchange risk Liquidity risk

7. 7 Rating CategoryMoody’sS&P ------------------------------------------ High Grade AaaAAA Aa AA ------------------------------------------- Investment A A Grade BaaBBB ------------------------------------------- Speculative Ba BB B B ------------------------------------------- Default CaaCCC Ca CC C C D

8. 8 Interest Rate Risk Example: Two bond issues of ABC Co. N 1 =1 yr N 2 = 10 yrs r = 5% As term to maturity increases, value of the bond becomes more sensitive to movements in market interest rate.

9. 9 Bond Value and Coupon Rates Example:Two issues of ABC Co. n=20 yrs, r 1 =10%, r 2 =6% Low coupon bonds are more sensitive to changes in market interest rates

10. 10 Value of a Bond in Time Example: Market rate stays at 10%, values of two bonds with coupon rates of 8% and 12% as the term to maturity approaches: Assuming that interest rates remain the same, bond value approaches to par over time as term to maturity shortens.

11. 11 Term Structure of Interest Rates Relationship between yield and time to maturity. Example:n=1 i=6% n=5i=8% n=20i=9% Maturity i Yield Curve

12. 12 Possible Explanations of the Term Structure 1. Expectations Hypothesis 1 + i n =[(1+ i 1 )(1+ 1 i 2 )…….(1+ n-1 i n )] 1/n Example: i 2 =8%i 1 =6% 1 i 2 =? 1 + 0.08 = [(1+ 0.06)(1+ 1 i 2 )] 1/2 1 i 2 = 0.1004 or 10% 2. Liquidity Preference Hypothesis Slope of the yield curve is higher than specified in expectations hypothesis 3. Segmented Markets Hypothesis

13. 13 Duration Volatility in bond price is directly proportional to term to maturity but inversely proportional to coupon payments. Duration of a bond is a measure that incorporates both factors that affect volatility.

14. 14 Duration Example n=5 yrs, r=8%, i=10% Bond Value = $92.41 Macaulay Duration = 4.28 years

15. 15 Hedging Interest Rate Risk $12 $12 $12 $12 $12 $12 $112 |___|____|____|____|____|...…..|___ | 0 1 2 3 4 5 9 10 V 0 =$84.94 when i=15% After i declines to 12%, V = $100 V when term to maturity is 4 years: V 6 = $100 Future value of the first 6 coupon payments reinvested at 12%: 12 x PVIFA 12%,6 = $97.38 Total savings = $100 + $97.38 = $197.38 $84.94 in 6 years grows to $197.38 Annual growth of 15%.

16. 16 Immunization Example $1,000 $2,000 $2,500 $2,000 $1650 |_____|______|______|______|______| 0 1 2 3 4 5 Total Premiums = Assets = $6,830.82 Market rate = 10% Flat yield curve Strategy 1: Invest in 1-yr bills with 10% interest 6830.82 -> 7513.90 (1000.00) 6513.90 --> 7165.29 (2000.00) 5165.29 --> 5681.82 (2500.00) 3181.82 ->3500 (2000) 1500 ->1650 (1650)

17. 17 Immunization Example (Cont’d) However, if interest rates fall, assets will be short of liabilities Strategy 2: Invest in 3-yr zero coupon bonds yielding 10% Duration of Liabilities: 11000 909.090.1330.133 220001652.890.2420.484 325001878.290.2750.825 420001366.030.2000.800 516501024.520.1500.750 2.990 Duration = 2.99 years

18. 18 Immunization Example (Cont’d ) Market rate 10%, V = $6,830.82 M = $9,091.82 Duration = 3 years If interest rates fall from 10% to 8%, V= $9,091.82 x PVIF 8%,3 = $7,217.38 7217.38 ->7794.77 (1000.00) 6794.77 ->7338.35 (2000.00) 5338.35->5765.42 (2500.00) 3265.42->3526.66 (2000.00) 1526.66->1650 (1650)

19. 19 Modified Duration D MD = ----------- (1 + i) In the example above, MD = 4.28/1.10 = 3.89 Approximate Change in V = -MD x Change in yield Example: If the yield decreases from 10% to 8% % Change in V= -4.28 x (-2) = 8.56% In fact when i=10% V = $92.41 i=8% V = $100 increase 8.21%

20. 20 Convexity Price-Yield Relationship V Yield The shape of the curve depends on the coupon rate and term to maturity High coupon + Short term -----> Linear Low coupon + Long term ------> Convex

21. 21 Convexity (Cont’d ) Higher convexity means that when interest rates go up, bond value declines slowly; but when rates decline, increase in bond price is large Therefore high convexity is a desirable feature. Factors that increase convexity: * Low coupon * Long term to maturity * Low yield

22. 22 Convexity (Cont’d ) Convexity = [1/(1.10) 2 ][2247.41][1/92.42] = 20.10 Appox. Change in V = -MD x  i + K x (  i) 2

23. 23 Alternative Measures of Yield Current Yield = rM / V Yield-to-maturity –Bond is held until maturity –All coupon and principal repayments are made on time –Bond is not called before maturity –Coupon payments are reinvested at yield-to-maturity Yield-to-call Holding period yield V t+1 - V t + rM HPY = -------------------- V t

24. 24 Approximate yield-to- maturity Example V= $877.11 n=3 yrs r=8% M=$1000

25. 25 Bond Investment Strategies I. Passive Strategies Investing $100 in 1925 T-bill Deposits Stock Market AAA Corporate Bonds Gold Inflation Passive Strategies are better when: Interest rate risk is low, and Inflation is low and stable

26. 26 II. Active Strategies Strategies based on maturity structure –Maturity matching - duration –Spreading the maturity –Investing only in short term bills and long term bonds Strategies based on forecasting interest rate movements –Interest rate fluctuations Buy when rates are high, sell when low Increase duration if higher rates are forecast, reduce duration otherwise

27. 27 - Riding the yield curve Investing in bonds assuming that the yield curve will not shift i Maturity B A Eg. 1 year bill i=6% V 1 = $943.40 B 2 year zero coupon i=8% V 2 = $857.34 A Buy the 2-year bond at $857.34, sell it next year at $943.40 HPY = (943.40 - 857.34) / 857.34 = 10.04%

28. 28 Strategies based on lack of market efficiency Junk bonds Bond swaps –Yield swap : same coupon, rating, maturity and industry, different yield –Exchange swap: same rating, maturity, industry, yield, different coupon. Exchange current yield for capital gains –Tax swap: Selling a bond to realize a loss, and replacing it with a similar bond –Swapping bonds with different tax status: eg. AAA corporate bond vs. municipal bond

29. 29 Strategies based on lack of market efficiency (cont’d) Possible shortcomings of bond swaps: –time to execute the swap –taxes –transaction costs –risk level of bonds Portfolio rebalancing: adjusting the bond portfolio for the changes in market conditions