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VALUATION OF FIXED INCOME SECURITIES

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## VALUATION OF FIXED INCOME SECURITIES

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**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**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**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 < r V > 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.**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**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**Risks Faced by a Bond Investor**• Default risk • Interest rate risk (price risk) • Reinvestment risk • Call risk • Inflation risk • Foreign exchange risk • Liquidity risk**Rating**Category Moody’s S&P ------------------------------------------ High Grade Aaa AAA Aa AA ------------------------------------------- Investment A A Grade Baa BBB ------------------------------------------- Speculative Ba BB B B ------------------------------------------- Default Caa CCC Ca CC C C D**Interest Rate Risk**Example: Two bond issues of ABC Co. N1=1 yr N2= 10 yrs r = 5% As term to maturity increases, value of the bond becomes more sensitive to movements in market interest rate.**Bond Value and Coupon RatesExample:Two issues of ABC Co.**n=20 yrs, r1=10%, r2=6% • Low coupon bonds are more sensitive to changes in market interest rates**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.**Term Structure of Interest Rates**Relationship between yield and time to maturity. Example: n=1 i=6% n=5 i=8% n=20 i=9% i Yield Curve Maturity**Possible Explanations of the Term Structure**1. Expectations Hypothesis 1 + in =[(1+ i1)(1+ 1i2)…….(1+n-1 in)]1/n Example: i2=8% i1=6% 1i2=? 1 + 0.08 = [(1+ 0.06)(1+ 1i2)]1/2 1i2 = 0.1004 or 10% 2. Liquidity Preference Hypothesis Slope of the yield curve is higher than specified in expectations hypothesis 3. Segmented Markets Hypothesis**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.**Duration Examplen=5 yrs, r=8%, i=10%**Bond Value = $92.41 Macaulay Duration = 4.28 years**Hedging Interest Rate Risk**$12 $12 $12 $12 $12 $12 $112 |___|____|____|____|____|...…..|___ | 0 1 2 3 4 5 9 10 V0=$84.94 when i=15% After i declines to 12%, V = $100 V when term to maturity is 4 years: V6 = $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%.**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)**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: 1 1000 909.09 0.133 0.133 2 2000 1652.89 0.242 0.484 3 2500 1878.29 0.275 0.825 4 2000 1366.03 0.200 0.800 5 1650 1024.52 0.150 0.750 2.990 Duration = 2.99 years**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)**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%**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**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**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**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 Vt+1 - Vt + rM HPY = -------------------- Vt**Approximate yield-to-maturity**Example V= $877.11 n=3 yrs r=8% M=$1000**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**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**- Riding the yield curve**• Investing in bonds assuming that the yield curve will not shift i A B Maturity Eg. 1 year bill i=6% V1 = $943.40 B 2 year zero coupon i=8% V2 = $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%**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**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