CH 11

Figure 11.4 Immunization

At 8% the values are equal and the obligation is fully funded by the bond -as the interest rates change, the change in value of both the asset and obligation are equal, so the obligation remains fully funded -for greater changes in interest rates, the present value curves diverge, resulting in a small surplus at rates other than 8% -claim is valid for only SMALL changes in the interest rate, because as bond yields change so does duration

Convexity

Curvature of price−yield relationship of bond -Why Do Investors Like Convexity? -- More convexity = greater price increases, smaller price decreases when interest rates fluctuate by larger amounts --Bonds with greater curvature gain more in price when yields fall then when they lose when yields rise EX PG 344

Active Bond Management: Strategies (6)

1. Substitution Swap 2. Intermarket Swap 3. Rate Anticipation Swap 4. Pure yield pickup swap 5. Tax Swap 6. Horizon Analysis

Active Bond Management -Substitution Swap

Exchange of one bond for bond with similar attributes and better price -motivated by the belief that the market has temporarily mispriced the two bonds, with a discrepancy representing a profit opportunity -EX swap variable rate for fixed rate

Interest Rate Sensitivity -If you believe interest rate will increase during the next couple of years, what characteristics would you prefer a bond to have?

If market yields increase, bond prices will decrease -You should prefer bonds that have the following characteristics (have less sensitivity) --Short-term --High coupon --High yield, if possible

Passive Bond Management -Cash Flow Matching

Matching cash flows from fixed-income portfolio with those of obligation, perhaps with a zero-coupon -automatically immunizes portfolio from interest rate risk because the CF from the bond and the obligation offset each other -may not be able to find a matching zero coupon

Immunization—Example Situation 1—YTM remains at 8% for 5 years:

PVB= n=4 (7-5=2x2=4), I/Y=8/2=4, PMT=70, FV=1000 CPT PV= 1,108.90 nc= 4x2=8 140(coupon PMT yearly) x 5=700 CPT I/Y= 8.16/2= 4.08

Active Bond Management -Tax Swap

Swapping two similar bonds to receive tax benefit

Table 11.5 Market Value Balance Sheets

The duration matching strategy has ensured that both assets and liabilities react equally to interest rate fluctuations -originally 8% but the decrease or increase in assets/ liabilities is the same for 7% and 9%

Therefore, immunization is a passive strategy only in the sense that...

it does not involve attempts to identify undervalued securities - managers must still proactively update and monitor their positions

Figure 11.1 Change in Bond Prices as a Function of Change in Yield to Maturity

- Bond A and B are the same besides YTM. A is flatter than B because B is more sensitive to change in YTM than A because B has longer maturity - Bond B and C are the same besides coupon rate and C's lower coupon has a steeper line because its more sensitive than B -Bond C and D are the same besides YTM, C is flatter because it is less sensitive to changes in market rates

Immunization—Example - Consider three $1,000 face value bonds whose characteristics and provisions are identical except with respect to their terms to maturity. Each bond has a coupon interest rate equal to 14 percent, and interest is paid semiannually. Assume the investment horizon for the investor is five (5) years, and the simple return per year currently is 8 percent.

- Maturity match is A because we need 5 years - Duration match is B because duration is 5 years - EAR= (1+y)^n -1 n= # compounding periods in year

Interest Rate Risk -Macaulay's Duration(D)

- Measures effective bond maturity -the time it takes to recapture the initial investment and the expected return on the investment - Weighted average of the times until each payment, with weights proportional to the present value of payment - t= period t

Annual Coupon Bond Duration -TABLE 11.3 Duration of annual coupon bonds (initial yield to maturity = 6%)

- Perpetuity= 1.06/.06 and the duration will remain the same - Maturity 1= almost zero-coupon bond - Maturity 10= the higher the coupon the lower the duration - Duration gets higher for longer maturities -duration decreases as coupon rate increases

Figure 11.2 Duration as Function of Maturity

- Zero coupon bond maturity= duration - The lower YTM (6% vs 15%) the greater the duration - 3% coupon has a higher duration than the 15% coupon -duration increases with maturity -the 6% YTM has a higher duration than the 15% YTM

What Determines Duration?

- Zero-coupon bond's duration is its time to maturity - Time/yield to maturity constant, bond's duration and interest-rate sensitivity higher when coupon price lower(if coupon is lower it takes longer to get original investment back) - Coupon rate constant, bond's duration and interest-rate sensitivity generally increase with time to maturity; duration always increases with maturity for bonds at or above par

Figure 11.5 Bond Price Convexity

- where there is 0 change, the difference between actual and approximation gets bigger and bigger - the greater the change in YTM the duration approximation is less accurate -small changes have more accurate duration -duration approximation always understates the value of the bond: it underestimates the increase in bond price when yield falls and overestimates the decline in price when yields rise

Interest Rate Risk & Duration—Example Consider a bond with a 10 percent coupon rate and 10 years remaining until maturity. Interest is paid semiannually. If the YTM increases from 12 percent to 13 percent, by approximately how much will the price of this bond change? -%change P0 -Modified Duration

-DUR x [(1+ y1)- (1+y0)]/(1+y0) -Modified DUR= DUR/(1+y0) market value will decline by 5.633%

Interest rate risk -Interest Rate Sensitivity 1. Bond price.... 2. Increase in bond's YTM.....

1. Bond prices and yields are inversely related -as yields increase, prices fall 2. Increase in bond's yield to maturity results in smaller price change than yield decrease of equal magnitude EX: n=10, I/Y=4, PMT= 1,000 x .05=50, FV=1000 CPT PV=1081.11 4% has a greater change than 8%

2 sources of potential value in active bond management

1. Interest rate forecasting: anticipating movements across the entire spectrum of the fixed income market -If interest rate declines are forecast, increase portfolio duration -if interest rate increases are forecast, decrease duration 2. Identification of relative mispricing within the fixed-income market

Duration is a key concept in bond portfolio management for three reasons

1. it is a simple summary measure of the effective average maturity if the portfolio 2. it is an essential tool in immunizing portfolios from interest rate risk 3. it is a measure of the interest rate sensitivity of a bond portfolio

Convexity Graph

3 bonds with different terms to maturity -the greater maturity the greater convexity aka curvature -if interest rates were to decline, then you'd want the 30 years bond since its value is worth more -if interest rates were to increase, want 5 years bond since it curves a little but is a flatter line and lower decrease in value of this bond relative to the 15 or 30

Interest rate risk -Interest Rate Sensitivity 3. Longer-term bond prices.... 4. As maturity increases, sensitivity of bond prices...

3. Long-term bond prices more sensitive to interest rate changes than short-term bonds 4. As maturity increases, sensitivity of bond prices to changes in yields increases at decreasing rate (less than proportional to bond maturity) EX: n=20, I/Y=8%, PMT=50, FV=1000 CPT PV= 705.46 change I/Y=6% then PV= 885.30 -20 year bond has a greater change than 5 year

Interest rate risk -Interest Rate Sensitivity 5: interest rate risk is _____ to bonds coupon rate

5.Interest rate risk is inversely related to bond's coupon rate; low-coupon bonds are more sensitive to interest rate changes. EX: n=10,I/Y=8, PMT=1000 x .05= 50, FV=1000 CPT PV= 798 then I/Y=6 CPT PV=926.4

Interest rate risk -Interest Rate Sensitivity 6: Sensitivity of bond's price to yield change is....

6. Sensitivity of bond's price-to-yield change is inversely related to current yield to maturity Bonds with higher yields, are less sensitive to changes in market yields than bond with lower yields EX: n=10, I/Y= 4, PMT=1000 x .05=50, FV=1000 CPT PV=1081.11 With 10% the change is the smallest

Interest rate risk -Lecture Example Interest Rate Sensitivity 5

n= 10, I/Y=8, PMT=1000 x .10=100, FV=1000 CPT PV=1134.20 then I/Y=6 CPT PV= 1294.4

Immunization Example Situation 2—YTM increases to 10% from 8% immediately after purchase and remains at that level for 5 years:

Bond B Price in Year 5 = n=4 (7-5=2x2=4), I/Y= 10/2=5, PMT=70, FV=1000, CPT PV= 1,070.92 Coupon= 10 x $70= 700 Reinvestment income= FVA: n=5 x 2=10, I/Y= 10/2=5, PV=0, PMT=70, CPT FV=880.45-700= 180.45 EARB= N=10, PVB=-1,316.89, PMT=0, FV= 1,951.37 CPT I/Y= 4.01 EAR= (1.0401)^2-1= 8.18%

What Determines Duration? - Zero-coupon bond's duration is its time to maturity EXAMPLE

Face value is the only payment PV of CF= 1000/ (1.07)^5

Table 11.4 Terminal Value of Bond Portfolio after Five Years

For a horizon equal to the portfolio's duration, price risk and reinvestment risk are offsetting -when the interest rates rise, there is a capital loss for investors, however by reinvesting the interest the income will grow and the loss will be offset -since assets and liability duration is equal, the company is immunized against interest rate fluctuations

Active Bond Management -Horizon Analysis

Forecast of bond returns based largely on prediction of yield curve at end of investment horizon -want to know whether rates will increase (dont want big changes) or decrease (values change the greatest) EXPG 347

Interest Rate Risk & Duration—Example -%change in P0 using financial calculator

HPR=End-beg/beg

Figure 11.3 Growth of Invested Funds

Immunizing a lump sum in 5 years -find duration for 5 years -if interest rates do increase, then duration is greater than 5 years and would receive a greater amount ---this increase in rates initially is a capital loss (decrease in price), but the loss is offset by the faster growth rate of the reinvested funds -if interest rates decreases, then duration is less

Active Bond Management -Fixed-Income Investment Strategy

Key features 1. Firms respect market prices -believes only minor mispricing can be detected 2. To have value, information cannot already be reflected in prices 3. Interest rate movements extremely hard to predict

Active Bond Management -Pure Yield Pickup Swap

Moving to higher yield bonds, usually with longer maturities -exchange of a shorter duration bond for a longer duration bond -pursued not in response to perceived mispricing but as a means of increasing return by holding higher yielding longer maturity bonds -may take on poor credit for higher yields

Passive Bond Management -Deduction Strategy

Multi-period cash flow matching -manager selects either zero coupon or coupon bonds with total CFs that match a series of obligations -advantage: it is the once and for all approach to eliminating interest rate risk. Once the CFs are matched there is no need for rebalancing --difficult to match

Duration Example Maturity = 5 years Face value = $1,000 Coupon rate = 6% YTM = 12%

N = 5; I/Y = 12; PMT = 60; FV = 1,000 PV = -783.71 Year 5 includes the face value and interest PVCF= CFn/ (1+r)^n Duration of 4.386 is the amount of time to recapture the investment of $783.71 and earn return

Duration Example Maturity = 5 years Face value = $1,000 Coupon rate = 6% YTM = 9%

N = 5; I/Y = 9; PMT = 60; FV = 1,000 PV = -883.31 Duration is longer due to lower market rate (higher PV to be repaid)

Duration Example 2 From Lecture Video r=10%

PV of CF3= 2,000/ (1.10)^3= 1,502.63 Weight= 1,502.63/ 18,763.70 = .080 Duration= .08 x 3= .24 This example has greater cash flows at the beginning

Duration Example 1 From Lecture Video r=10%

PV of CFs= CFn/ (1+r)^n Wn=PV of CFn/total PV of CFs Duration= year x weight This example has greater cash flows at the end

Passive bond management -Rebalancing

Realigning proportions of assets in portfolio as needed -without rebalancing the durations will become unmatched -As interest rates and asset durations continually change, managers must adjust the portfolio to realign its duration with the duration of the obligation -even if interest rates do not change, asset durations will change solely because of the passage of time

Immunization—Example Comparison of Situation 1 and Situation 2:

Situation 1 YTM =8% Situation 2 went to 10% immediately from 8% hence why the prices go down because interest rates went up - Bond B is immunized and if done perfectly would have no change at all and keeps ROR where it should be hence only a $2 change

Passive bond management -Immunization

Strategy to shield net worth from interest rate movements -Funds should match their interest rate exposure of assets and liabilities so that the value of assets will track the value of the liabilities whether rates fall or rise -meet obligation's despite interest rate movements

Active Bond Management -Rate Anticipation Swap

Switch made in response to forecasts of interest rate changes -exchange of bonds with different maturities -if you believe rates will increase, want shorter terms bonds -if rates will fall, swap into bonds with longer duration

Active Bond Management -Intermarket Swap

Switching from one segment of bond market to another -pursued when an investor believes the yield spread between two sectors of the bond market is temporarily out of line -EX: corporate to treasury

What Determines Duration? Other factors constant,...

duration and interest rate sensitivity of coupon bond are higher when bond's yield to maturity is lower - at lower yields the more distant payments have relatively greater present values and thereby account for a greater share of the bond's total value (greater weights) -Duration of a perpetuity = (1+y)/y

Interest Rate Risk & Duration—Example Consider a bond with a 10 percent coupon rate and 10 years remaining until maturity. Interest is paid semiannually. If the YTM increases from 12 percent to 13 percent, by approximately how much will the price of this bond change? -Duration calculation

y=6% n=20 m=2