Chapter ThreeInterest Rates and Security Valuation

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Various Interest Rate MeasuresCoupon rate Determines the periodic cash flow received by the bond investor (and paid by the bond issuer)Required rate of return (r)rates used by individual market participants to calculate present values (PV)Expected rate of return or E(r)rates participants would earn by buying securities at current market prices (P)The required return and the expected return are the same if market prices reflect present valuesRealized rate of return ( r )rate actually earned on investments

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Required Rate of ReturnThe present value (PV) of a security is determined using the required rate of return (r) as the discount rate

CFt = expected cash flow in period t (t = 1, , n)n = number of periods in the investment horizon

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Expected Rate of ReturnThe current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate

CFt = cash flow in period t (t = 1, , n)n = number of periods in the investment horizon

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Realized Rate of ReturnThe realized rate of return ( r ) is the discount rate that just equates the actual purchase price ( ) to the present value of the realized cash flows (RCFt) t (t = 1, , n)

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Bond ValuationThe present value of a semi-annual bond (Vb) can be written as:

M = the maturity (or par value or face value) of the bond, usually $1,000INT = the annual interest (or coupon) paymentT = the number of years until the bond maturesr = the annual interest rate (often called yield to maturity (ytm))

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Bond ValuationA premium (discount) bond has a coupon rate (INT) greater (less) than the required rate of return (r) and the price of the bond (Vb) is greater (less) than the face or par valuePar bond: if INT = r, then Vb = ParPremium bond: if INT > r; then Vb > ParDiscount bond: if INT < r, then Vb < Par

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Equity Valuation: Constant Dividend ModelThe present value of a stock (Pt) assuming zero growth in dividends can be written as:

D = dividend paid at end of every yearPt = the stocks price at the end of year trs = the interest rate used to discount future cash flows

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Equity Valuation: Constant Growth ModelThe present value of a stock (Pt) assuming constant growth in dividends can be written as:

D0 = current value of dividendsDt = value of dividends at time t = 1, 2, , g = the constant dividend growth rate

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Expected Return on EquityThe return on a stock with zero dividend growth, if purchased at current price P0, can be written as:

The return on a stock with constant dividend growth, if purchased at price P0, can be written as:

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Equity Valuation: Supernormal or Non-Constant Growth ModelThree steps:Find the PV of each dividend during the supernormal growth periodFind the price of the stock at the end of the supernormal growth period (using the constant growth model)Discount all the values from steps 1 and 2 and add them together.

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As Interest Rates Fall, Bond Prices IncreaseInterest Rate

Bond Value

12%10%8%874.501,0001,152.47

FV100010001000PMT100/2100/2100/2N12 x 212 x 212 x 2I/Yr12/210/28/2PV874.501000.001152.47

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As Time to Maturity Increases, Price Sensitivity Increases but at a Decreasing Rate

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Interest Rate Sensitivity is Lower for Higher Coupon Rate Bonds

Required return10%12%7.0%$1,240.88 $1,401.46 7.5%$1,195.56 $1,352.01 8.0%$1,152.47 $1,304.94 8.5%$1,111.48 $1,260.12 9.0%$1,072.48 $1,217.43 9.5%$1,035.35 $1,176.76 10.0%$1,000.00 $1,137.99 10.5%$966.33 $1,101.02 11.0%$934.24 $1,065.76 11.5%$903.66 $1,032.11 12.0%$874.50 $1,000.00 12.5%$846.68 $969.34 13.0%$820.14 $940.05 13.5%$794.80 $912.06 14.0%$770.61 $885.31

$ Price 7% to 14%$470.26 $516.15

% Price 7% to 14%-37.9%-36.8%

Chart1

1240.87551404521401.4591900754

1195.56030135261352.0085424346

1152.4696314141304.9392628279

1111.48054799941260.1212786654

1072.47739183021217.4321754905

1035.35134412961176.7567206479

10001137.9864179435

966.32696959971101.0190912008

934.24150523761065.7584947624

903.65815263441032.1139491219

874.49642472241000

846.6804964694969.3360992939

820.1389193172940.0463064391

794.8043539232912.0590088242

770.6133199854885.3066599927

Coupon rate of 10%

Coupon rate of 12%

Required return

Bond Price

Bond Prices at Various Interest Rates for a 10% and a 12% semi-annual coupon bond with par value of $1000 and 12 years to maturity

Sheet1

Bond Price SensitivityChange in Interest Rate from 8% to 10%

FV1000Time to maturity1213141516

PMT50Abs value of change in bond price22.06%24.12%25.77%27.11%28.20%

N20

Required returnFair pricePrice change% price change

8%$1,135.90

10%1,000.00-$135.90-11.96%

12%885.30(114.70)-11.47%

FV1000

PMT50

N24

Required returnFair pricePrice change% price change

8%$1,152.47

10%1,000.00-$152.47-13.23%

12%874.50(125.50)-12.55%

FV1000

PMT50

N28

Required returnFair pricePrice change% price change

8%$1,166.63

10%1,000.00-$166.63-14.28%

12%865.94(134.06)-13.41%

FV1000

PMT50

N32

Required returnFair pricePrice change% price change

8%$1,178.74

10%1,000.00-$178.74-15.16%

12%859.16(140.84)-14.08%

FV1000

PMT50

N36

Required returnFair pricePrice change% price change

8%$1,189.08

10%1,000.00-$189.08-15.90%

12%853.79(146.21)-14.62%

Sheet1

Abs value of change in bond price

Time to Maturity

Percent change in bond price

Absolute value of change in bond price as required return changes from 8% to 12%(10% coupon, $1000 par value)

Sheet2

FV1000

N12

Coupon rate of 10%Coupon rate of 12%

Required return10%12%

7.0%$1,240.88$1,401.46

7.5%$1,195.56$1,352.01

8.0%$1,152.47$1,304.94

8.5%$1,111.48$1,260.12

9.0%$1,072.48$1,217.43

9.5%$1,035.35$1,176.76

10.0%$1,000.00$1,137.99

10.5%$966.33$1,101.02

11.0%$934.24$1,065.76

11.5%$903.66$1,032.11

12.0%$874.50$1,000.00

12.5%$846.68$969.34

13.0%$820.14$940.05

13.5%$794.80$912.06

14.0%$770.61$885.31

Sheet2

Coupon rate of 10%

Coupon rate of 12%

Required return

Bond Price

Bond Prices at Various Interest Rates for a 10% and a 12% semi-annual coupon bond with par value of $1000 and 12 years to maturity

Sheet3

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DurationDuration is the weighted-average time to maturity (in years) on a financial securityDuration measures the sensitivity of a fixed-income securitys price to small interest rate changes

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DurationDuration (Dur) for a fixed-income security that pays interest annually can be written as:

P0= Current price of the securityt = 1 to T, the period in which a cash flow is receivedN = the number of years to maturityCFt = cash flow received at end of period tr = yield to maturity or required rate of returnPVt = present value of cash flow received at end of period t

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9% Coupon, 4 year maturity annual payment bond with a 8% YTM Calculating DurationDuration = 3.54 years

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9% Coupon, 4 year maturity semi-annual payment bond with a 8% YTM Calculating DurationDuration = 3.46 years

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DurationDuration and coupon interestthe higher the coupon payment, the lower the bonds durationDuration and yield to maturitythe higher the yield to maturity, the lower the bonds durationDuration and maturityduration increases with maturity but at a decreasing rate

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Economic Meaning of DurationGiven a small change in interest rates, the estimated percentage change in a semi-annual coupon bonds price is approximately:

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Duration Based Prediction Errors

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ConvexityConvexity is good because it provides partial insurance against interest rate movementsConvexity (CX) measures the change in slope of the price-yield curve around interest rate level rConvexity incorporates the curvature of the price-yield curve into the estimated percentage price change of a bond given an interest rate change:

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ConvexityExample: Consider an 8% semi-annual coupon bond with 6 years to maturity and an 8% rate of return. What is the change in price if rates rise by 2%?The current price is $1000 because the coupon rate equals the rate of return. Calculate what the price will be if the rate of return is 0.01% higher and 0.01% lower?PMT = 80, N = 6, FV =1000If I/Yr = 8.01 then PV = 999.53785If I/Yr = 7.99 then PV = 1000.46243

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ConvexityExample: Consider an 8% semi-annual coupon bond with 6 years to maturity and an 8% rate of return. What is the change in price if rates rise by 2%?Calculate the convexity

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ConvexityExample: Consider an 8% semi-annual coupon bond with 6 years to maturity and an 8% rate of return. What is the change in price if rates rise by 2%?Calculate the change in price

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ImmunizationIn order to eliminate risk of an obligation due in N years:Buy a zero coupon bond that matures in N yearsBuy a coupon bond with a duration of N years

Example: Horizon is 5 years but you buy a 6 year bond with a duration of 5 years.Bond pays 8% annual coupon or $80 a year for 5 years.What happens if rates are 8%? 7%? 9%?

****************Note that if the security makes semiannual or monthly payments then the cash flow, the interest rate and the number of periods must be adjusted to reflect the payment frequency.***We just worked a problem with a 9% coupon thats just like the example in Table 3-7 with an 8% coupon. Our answer was 3.46 years with the 9% coupon while theirs was 3.42 years with the 10% coupon.**Duration is an accurate predictor of price changes only for very small interest rate changes. For day to day fluctuations duration works quite well but when interest rates move significantly, such as when the Fed makes an announcement of a rate change, the predicted pricing errors can become significant. The prediction errors arise because bond prices are not linear with respect to interest rates. ****So a 2% change in interest rates will cause an 8.69% change in the bond price rather than a 9.25% change when convexity is not considered.

True answer is that if rates rise 2% then bond price will fall by 8.71% so this is quite close.FV = 1000, PMT = 80, N =6, I/Yr= 10 causes a PV of $912.89 and 912.89/1000 1 = 8.71%*After showing them the calculations, remind them that this only works because the duration of this bond matches the investment horizon.