1. Computational Finance 1/37 Panos Parpas Bonds and Their Valuation 381 Computational Finance Imperial College London

2. Computational Finance 2/37 Topics Covered   Bonds and their valuation   bond prices, yield to maturity   duration and convexity   term structure of interest rates   spot rates,   forward rates

3. Computational Finance 3/37 Introduction to Bonds   fixed income security specifies series of future payments   contract specifies timing & amounts of cash flows over time   Timing - how often the payments are made: coupon payment time - pre-specified period of time that the loan has to be paid: maturity date   Payments - coupon payments represent the interest on bonds - par (face) value is amount paid to bond holder at maturity

4. Computational Finance 4/37 Introduction to Bonds   securities which establish creditor relationship between purchaser & issuer – – issuer   such as corporations, governments and government agencies  receives a certain amount of money in return for the bond   is obligated to repay the principal at maturity to purchaser   corporate bond – issued by a corporation   municipal bond – issued by municipality   treasury bond – issued by government, maturity more than 10 years, 2xin a year coupon payments   treasury notes – similar to treasury bond, issued for 1-10 years   treasury bills (zero coupon bonds) – held for a shorter time period, matures in 3,6,9 months

5. Computational Finance 5/37 Elements of Bond   Bonds require coupon or interest payments determined as part of the contract   Coupon payments represent interest on the bond   Final interest payment and principal are paid at specific date of maturity   face (par) value: amount paid to bondholder at maturity   coupon payments: interest paid   maturity (or term): the end of life time of a bond

6. Computational Finance 6/37 Bond Table   Issuer: company, state or country   Coupon: fixed interest rate that issuer pays to lender (investor)   Maturity date: date when borrower will pay the lenders (investor) principal back   Bid price: price that someone is willing to pay the lenders   Yield: indicates annual return until the bond matures

7. Computational Finance 7/37 Bond Price bond specifies a series of future payments (cash flows) – price of the security is the value today of future cash flows market price of the bond – P principal payment (face value) – F the number of coupon payments per year – m annual coupon rate of the bond (yearly coupon payment) – M per year periodic coupon rate – c = M/m the number of years to maturity – t total number of periods – n = m.t annual percentage rate (APR) – R effective periodic interest rate – i = R/m Price of a zero coupon bond

8. Computational Finance 8/37 Example: Coupon Bonds Consider a 20% coupon bond with par value of £100. It matures in 3 years from now and pays the coupon semi-annually. The annual percentage rate is 13%. What is the price of the bond? 20% coupon bond will have a coupon of £20 in a year and £10 after 6 months

9. Computational Finance 9/37 Bond Price Dynamics  effects of changes in the interest rate on the present value of the cash flows  direction of change is determined by the first derivative of price function.  Consider price function of a zero coupon bond.  gives the change of price of bond in response to change in the level of interest rate  negative sign - the price of the zero coupon bond decreases with an increase in the interest rate.

10. Computational Finance 10/37 Bond Price Dynamics  First derivative of the price function is a first order (linear) approximation to the slope of the function  The price response to a change in the interest rates by is approximately  The percentage response of the bond price is obtained by dividing derivative by the value of the bond  The percentage price change of the zero coupon bond is proportional to maturity of the bond:

11. Computational Finance 11/37 Example: Bond Price Dynamics Consider a zero coupon bond with a term to maturity of 5 years, a face value of £1000 and the interest rate is 8%.  The change of price of the bond in response to change in the level of interest rate is computed as  When interest rate changes by one percentage point, bond price changes by £31.51 Interest rateBond ValueChange(£)Change(%) 8%680.5832 9%649.9314-30.6518-4.5038 =(-30.6518*100/680.5832) 7%712.9862+32.4030+4.7611 =(32.4030*100/680.5832) Average 31.5274 4.6324

12. Computational Finance 12/37 Bonds Volatility is absolute value of percentage price change  Factors affecting bond volatility are  Level of yield  Time to maturity  Coupon rate Bonds Volatility

13. Computational Finance 13/37 Yield to Maturity (YTM) the percentage rate of return measuring total performance of a bond (coupon payments as well as capital gain or loss) from purchase time to maturity.  Bond Valuation,  future payments are discounted at same interest rate R.  reverse the present value procedure – given the bond price, solve for the interest rate that equates PV(cash flow of coupon payments) = P  solution is interest rate at which the present value of stream of payments is exactly equal to current price  bond’s yield is interest rate implied by the payment structure

14. Computational Finance 14/37 Yield to Maturity (YTM)  Suppose that a bond with a face value F makes m coupon payments of M/m each period and there are n periods. The current price is P.  YTM satisfies  For a zero coupon bond:  With continuous compounding:

15. Computational Finance 15/37 Example: (YTM) Consider a zero-coupon bond with par value £100. It matures in 6 years and trading £55. What is the yield under annual semi-annual and continuous compounding?  The more periods per year, the lower the yield – Continuous compounding is the lowest!

16. Computational Finance 16/37 Price-Yield Curve Price-yield chart Bonds with 5%,10%,15% coupon rates, 10 years to maturity and face value of £100  Price-yield curve describes how yield and price are related  the price-yield curve rises as the coupon rate increases  price and yield have inverse relation (negative slope): if yield goes up, price goes down  when YTM=0, the bond is priced as if it offered no interest  price of bond must tend to zero as the yield increases-large yields imply heavy discounting  when the value of bond =par value, it means that yield =coupon rate – A PAR BOND

17. Computational Finance 17/37 Example: problems using YTM Consider two bonds A and B. They both cost £1000, have 3 years maturity, and compounded annually. The cash flows and yields are Bond A Bond B Price 1000 1000 Year 1145430 Year 2145430 Year 311451430 Yield (%)14.513 Bond A is better than B since it has a higher yield? PV(cash flows for bonds A and B) at interest rates of 10%, 20%, 15% over 1,2,3 periods YTM rule does not always work as a guide to higher returns, as we vary pattern of interests Assumption of equal annual rates of return for computing YTM.

18. Computational Finance 18/37 Time to Maturity  As maturity is increased, price yield curve becomes steeper – this indicates that longer maturity implies greater sensitivity of price to yields  When yield is equal to coupon rate (bond value equal to par value), bond is at par Price of bond with long maturity is more sensitive to interest rate changes than the one with short maturity Price-yield curves 10% coupon rate 3, 10, 15 years maturities Influence of time to maturity on price of the bond

19. Computational Finance 19/37 Example: Price-Time to Maturity Show the price-yield relation in tabular form for bonds paying 8% coupon rate with 1,3,5 and 10 years to maturity and 3%, 5%, 8%, 10% yield values. Face value is £100. Higher maturity implies higher volatility. Time to Yield Maturity3%5%8%10% 1 year104.85102.8510098.18 3 years114.14108.1610095.02 5 years122.89112.9810092.41 10 years142.65123.1610087.71 10 years to maturity more sensitive to yield changes than 1 year to maturity

20. Computational Finance 20/37 Duration Maturity does not give a complete quantitative measure of interest rate sensitivity Duration is  another measure of time length which gives direct measure of interest rate sensitivity  calculated as weighted average of times that cash flow payments are made  weighting coefficients are the present values of individual cash flows: unit of time – intermediate between the first and last cash payments for a zero coupon bond, D = T. for a nonnegative cash flows Duration can be viewed as a generalised maturity measure If maturity is long, then duration is high Higher maturity implies higher volatility

21. Computational Finance 21/37 Macaulay Duration   How to compute PV or what interest rate to use?   When the yield is used, the general duration formula becomes   This may look complicated ! The denominator looks familiar? The denominator is simply the price of a bond, P.

22. Computational Finance 22/37 Duration   m payments per year with payment a k in period k, n periods remaining  factor k/m (in the numerator) is time measured in years

23. Computational Finance 23/37 Example: Duration Suppose a bond has maturity of 5 years pays coupon payments of £50 each and has a principal repayment of £500 upon redemption. It’s current price is £539.93. Find its yield to maturity and duration. Yield to maturity is the rate of interest that equates the PV of future cash flows to the price. Solving equation, YTM is found as 8%. Solving equation, YTM is found as 8%.

24. Computational Finance 24/37 Example Continued  D mac = 2269.73/539.93 = 4.20 years. This is the average time to receipt of the cash flows.

25. Computational Finance 25/37 Modified Duration   Duration measures directly the sensitivity of price to changes in yield   Payments are made m times per year and the yield is based on the same periods

26. Computational Finance 26/37 Modified Duration   It measures the relative change in a bond’s price as yield changes   Using approximation of   the change in price due to a small change in yield can be estimated as

27. Computational Finance 27/37 Example: Modified Duration What does this mean ? –Suppose the yield changes from 8% to 8.1%. The change on price is

28. Computational Finance 28/37 Convexity   D M measures the relative slope of the price-yield curve at a given point; linear approximation to price-yield curve   Convexity is a better approximation which is relative curvature at a given point on the price-yield curve.   Given a cash flow a t at time t for t=1,2,…,n

29. Computational Finance 29/37 Convexity If is small change in the yield to maturity and is the corresponding change on price, then second order approximation to the price-yield curve is

30. Computational Finance 30/37 Example: Convexity

31. Computational Finance 31/37 Term Structure of Interest Rates   Theory is based on - “ interest rate charged or paid for money depends on length of time that money is held”   Term structure of interest rates – – name given to the pattern of interest rates available on instruments of similar credit risk but with different terms to maturity – – measures shape of the interest rate relationship with maturity – – describes relationship between fixed interest securities that differ only in their time to maturity – – differences between interest rates for payments at different maturity reflect expectations about future interest rates and preferences of investors   Theories developed for explaining term structure  Expectation theory  Liquidity preference theory  Preferred Habitat theory

32. Computational Finance 32/37 Spot Rates Interest rate fixed today on a loan that is made today   Basic interest rates defining term structure – yearly rate of interest depends on length of time funds are held.   A spot rate s t is expressed in yearly terms, charged for money held from present time (t=0) until time t: s 2 represents the rate that is paid for money held for 2 years   If an investor puts amount of A to a bank with spot rates of   money grows by a factor of

33. Computational Finance 33/37 Determining Spot Rate   Assume that the 1-year spot rate s 1 is known.   In order to determine the 2-year spot rate, solve the following equation for s 2   This is basically says; price is equal to discounted value of the cash flow stream   Carrying out forward to determine 3-year, 4-year,…spot rates

34. Computational Finance 34/37 Example: Spot Rate Consider two bonds A and B which mature in a year and two years with coupon payments of 10% and 8%. The face value is £100. If the prices of bonds are £98 and £96. What is 1-year and 2-year spot rates?

35. Computational Finance 35/37 Defined as interest rate fixed today on a loan to be made at some future date  In other words, interest rate for money to be borrowed between two dates in the future, but under terms agreed upon today.  Assume that you invest £1 in with spot rates s 1 for 1 year and s 2 for 2 years.  leave £1 in a 2-year account; then money grows  leave £1 in a 1-year account; then money grows and then lend your money for a year with interest rate f.  This loan will get interest at a pre-arranged rate f Forward Rates

36. Computational Finance 36/37 Forward Rates Forward rate between time i and j (i < j) is the interest rate charged for borrowing money at i, which is to be repaid with interest at j   Yearly compounding   Compounding m periods per year   Continuous compounding

37. Computational Finance 37/37 Example: Forward Rates   If the spot rates for 1 year and 2 years are s 1 = 6.3% and s 2 = 6.9%, then what is the forward rate f 1,2 ?