VALUATION OF FIXED INTEREST SECURITIES FOCUS Bond valuation Yield measures Yield maturity relationship Effect of reinvestment on realised return Calculating forward rates from the spot rate curve Forward rate agreements Yield curve theories Bond price volatility Duration Convexity Duration and convexity of callable bonds

VALUATION OF FIXED INTEREST SECURITIES Bond valuation Recall how we did it previously  Two stage calculation OR  Can work PV out with calculator Make the PMT and FV positive Yield measures Nominal yield = the annualised coupon rate Current yield = annual coupon payment/Bond payment

VALUATION OF FIXED INTEREST SECURITIES Yield measures Yield to maturity (YTM)  Rate equating coupon rates plus capital gain to the market price of the bond  IRR  Two assumptions  Bond held to maturity date  All coupon payment reinvested in equivalent rate Example Remember that current yield is always annualised, this one isn’t

VALUATION OF FIXED INTEREST SECURITIES Yield measures Yield to call (YTC)  Callable bonds  Have to calculate YTM and YTC  ‘Callable amount becomes the FV’ Price-Yield Relationship Figure 12.1 Inverse between price and YTM Relationship between coupon rate, YTM and price

VALUATION OF FIXED INTEREST SECURITIES Yield-maturity relationship Assume working with bonds with similar characteristics bar maturity Normal curve – Figure 12.2 The longer the maturity period the steeper the graph Yield curves though change BEASSA yield curve used as industry standard

VALUATION OF FIXED INTEREST SECURITIES Effect of reinvestment on realised return Assumption You only realise the stated yield if you reinvest all the interest payments received on the bond Yield illusion Popularity of zero-coupon bonds Try to calculate the Realised Compound Yield (RCY)

VALUATION OF FIXED INTEREST SECURITIES RCY Calculation  Calculate FV of the reinvested interest payments  Add the face value of the bond to that FV to get a new FV for the bond  Calculate the yield in a normal bond calculation with the new FV Example

VALUATION OF FIXED INTEREST SECURITIES Determinants of interest rates Fisher equation  Risk free + inflation + bond premium Yield spread (*)  Between two similar bonds Credit spread  Credit quality

VALUATION OF FIXED INTEREST SECURITIES Computing a bond’s value using spot rates Ripping bonds cash flow streams apart Building zero-coupon bonds Must make zero coupon bonds PV equal to what the previous bond’s was Need to work out the rates Example is taking the rate from the previous bond and applying it for that period Seems to behave like a variable bond

VALUATION OF FIXED INTEREST SECURITIES Calculating forward rates from the spot rate curve Forward is market consensus of future interest rates Example of two different period bonds – one is longer Rates are annualised – therefore need to divide Find the values of the bonds The interest rate to get from one value to the other is the forward rate (annualise it) More examples

VALUATION OF FIXED INTEREST SECURITIES Calculating forward rates from the spot rate curve Alternative formula to calculate the forward rate quickly (1 plus rate in next period) / (1 plus rate in previous period) Forward rate agreements Parties contract on basis of an unknown future rate (only known at end)

VALUATION OF FIXED INTEREST SECURITIES Forward rate agreements Almost like betting Need to work out value Formula on page 206 Working out payoff amount payable by borrower Example Yield curve theory Pure expectations hypothesis Liquidity preference theory Segemented market hypothesis

VALUATION OF FIXED INTEREST SECURITIES Bond price volatility Principles related to bond volatility Bond move inversely to yield and maturity Price volatility related to bond maturity  Larger discounting effect from change in rates Volatility inversely related to coupon  Lower coupon bonds more volatile  Related to how the YTM moves relative to the coupon

VALUATION OF FIXED INTEREST SECURITIES Bond price volatility There lowest coupon and longest maturity bonds have the greatest volatility (almost like beta’s) Duration Trying to combine interest rate and maturity variation into one Macaulay duration

VALUATION OF FIXED INTEREST SECURITIES Duration Duration equals 1.Get cash flows for each period 2.Discount back with YTM 3.Add up 4.Divide each component by the total above 5.Multiple each value by the time period 6.Add all those numbers up Example Shows the ‘weighted average duration’ to realise cash flows Always less than actual time

VALUATION OF FIXED INTEREST SECURITIES Modified Duration Normal duration divided by (1 + YTM/number of payments per year) Price effect Change in yield percentage points multiplied by negative modified duration equals price effect Shows how much price varies if YTM increases by a certain value Relativistic Remember that 1% = 100 points Examples

VALUATION OF FIXED INTEREST SECURITIES Modified Duration Normal duration divided by (1 + YTM/number of payments per year) Price effect Change in yield percentage points multiplied by negative modified duration equals price effect Shows how much price varies if YTM increases by a certain value Relativistic Remember that 1% = 100 points Examples

VALUATION OF FIXED INTEREST SECURITIES Convexity Figure 12.5  Modified duration is a linear function  Curve shows real price as a function of yield  Need to work out the ‘convexity’ Methodology to calculate  For each period square the period itself and then add it to itself  Then multiply that number by the cash flow’s present value  Add up those numbers  Discount it back on the basis of years  Divided that value by the PVIF and then again by the compounding per year squared

VALUATION OF FIXED INTEREST SECURITIES Convexity Approximation formula on page 212 Callable bonds Above calculations can’t deal with embedded options Figure 12.6 Ceiling on bond if callable at a certain value

VALUATION OF FIXED INTEREST SECURITIES Callable bonds Effective duration Effective convexity Formula – page 212 Example