1. Bond Valuation by Binam Ghimire

2. Learning Objectives Bond valuation
Understand the relationship between bond and interest rate
Compute Yield to Maturity
Work out bond valuation in Excel
The alternative bond yields that are important to investors
Spot rates and forward rates and how do you calculate these rates from a yield to maturity curve
What is the spot rate yield curve and forward rate curve
How and why use the spot rate curve to determine the value of a bond

3. The Fundamentals of Bond Valuation
The present-value model
Where:
Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond

4. The Present Value Model
For example. 8% coupon, 30-year maturity bond with par value of $1,000 paying 60 semi-annual coupon payments of $40 each. Suppose interest rate is 8% p.a or 4% per-6month. What is the price of the bond?

5. The Present Value Model
Then bond price is = 𝑡= $ 𝑡 $1, = $810.71

6. The Price Yield Curve
When we increase the required rate of return, the market price falls down

7. The Price Yield Curve
If yield < coupon rate, bond will be priced at a premium to its par value
If yield > coupon rate, bond will be priced at a discount to its par value
Price-yield relationship is convex (not a straight line)

8. The Price Yield Curve
X Company has just issued 8 percent, 10-years, £ 1,000 par bond. Current market interest rate is 8 percent. What is the price of the bond?
What will be the bond price if interest rate were to a) rise to 10% b) fell to 6%

9. The Price yield curve
The results:

10. The Yield Model
The expected yield on the bond may be computed from the market price
Where:
i = the discount rate that will discount the cash flows to equal the current market price of the bond

11. YTM Discussed before but see again
The Yield to Maturity (YTM) of a bond represents the rate of return investors earn if they buy the bond at a specific price and hold it until maturity
YTM is the interest rate that makes the present value of a bond’s payments equal to its price
So when price of bond = face value of bond then YTM = Coupon Interest Rate

12. YTM
In October 2007 Tesco raised $2bn (£990m) of debt in its first dollar-denominated bond issue.
The bond issue includes 10-year notes paying 5.5 per cent interest (US$ 850m) and 30-year notes paying 6.15 per cent interest (US 1150m).
The proceeds of the debt raising, which was jointly arranged by Citigroup and JP Morgan Cazenove, would be used for "general corporate purposes“.
What does this bond offer?
the first one pays 5.5/2 = 2.75% every six months until November 2017 then it pays the coupon and the par value of $100.
The observed price of the first bond in Datastream was $ Find the YTM for the first bond

13. YTM

14. Computing Bond Yields Yield Measure Purpose Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held to maturity
Promised yield to call
Measures expected rate of return for bond held to first call date
Realized (horizon) yield
Measures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.

15. Nominal Yield
Measures the coupon rate that a bond investor receives as a percent of the bond’s par value

16. CY = Ci/Pm Current Yield Similar to dividend yield for stocks
Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond

17. Promised Yield to Maturity
Widely used bond yield figure
Assumes
Investor holds bond to maturity
All the bond’s cash flow is reinvested at the computed yield to maturity
Solve for i that will equate the current price to all cash flows from the bond to maturity, similar to IRR

18. Computing the Promised Yield to Maturity
Two methods
Approximate promised yield
Easy, less accurate
Present-value model
More involved, more accurate

19. Approximate Promised Yield
Coupon + Annual Straight-Line Amortization of Capital Gain or Loss
Average Investment =

20. Computing the Promised Yield to Maturity
Example
8%, 20 Year bond, is priced at $900, what is the promised yield to maturity?
Answer: 4.45%

21. Promised Yield to Call Approximation
May be less than yield to maturity
Reflects return to investor if bond is called and cannot be held to maturity
Where:
AYC = approximate yield to call (YTC)
Pc = call price of the bond
Pm = market price of the bond
Ct = annual coupon payment
nc = the number of years to first call date

22. Promised Yield to Call Present-Value Method
Where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond

23. Realized Yield Approximation
Where:
ARY = approximate realized yield to call (YTC)
Pf = estimated future selling price of the bond
Ci = annual coupon payment
hp = the number of years in holding period of the bond

24. Realized Yield Present-Value Method

25. Calculating Future Bond prices
We need to compute a future price (Pf) when estimating the expected realised (horizon) yield performance of bonds

26. Calculating Future Bond Prices
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
i = expected semiannual rate at the end of the holding period

27. Calculating Future Bond prices
Assume you bought 10% 25 year bond at $842 giving it a promised YTM of 12%. Based on the analysis of the economy and the capital market, you expect this bond’s market YTM to decline to 8% in 5 years. Therefore you want to compute its future price at the end of year 5 to estimate the expected rate of return, assuming you are correct in your assessment of the decline in overall market interest rate.
Above you estimated the holding period of 5 years which means the remaining life is 20 years and estimated future market YTM is 8%
Find out the future selling price of the bond

28. Calculating Future Bond Prices

29. Calculating Future Bond Prices
50 ( ) + 1,000 (0.2083)
$1,197.94

30. Bond Valuation using spot rates
We said we discount all CFs by one common yield but having one YTM is not realistic
Investors at any point in time require a different rate of return for flows at different times
For example in a zero coupon bond, investors will expect different rates for bond maturing at 2, 5 or 10 years
The rates that is used to discount a CF at a certain point are called spot rates.

31. Bond Valuation using spot rates
See the excel file
It shows the desire for different rates.
Analysts recognise that it is inappropriate to discount all the flows for a bond at one single rate
For example see page 615, Brown and Reilly (2012) Analysis Investments And Management Of Portfolios, 10th Ed., Cengage

32. Zero Coupon Bonds and Treasury Strips
Brown and Reilly (2012) Analysis Investments And Management Of Portfolios, 10th Ed., Cengage, Page: 443
Determinants of Bond Safety

33. Further Exercise
You can find more exercise on different bond yields in Bodie, Kane and Marcus (2008) Investments, International Edition. Pages: ,

34. Thank you