北京大学 光华管理学院 赵龙凯 证券投资学 第二讲

Байду номын сангаас Yield-to-Maturity

Definition:

The Yield to Maturity of a bond (YTM) is the single interest rate that makes the discounted value of the bond’s future payments equal to its price.
Investment
Lecture 2
Fixed-income Securities

Bond Characteristics Bond Pricing and YTM Relationships among:

Yield to Maturity (YTM) Spot Rates Discount Factors Forward Rates
Holding-Period Return: Single Period
I (P0 P1 ) HPR P0
where I = interest payment P1 = price in one period P0 = purchase price
Holding-Period Example

Puttable Bonds


Convertible Bonds

Default Risk

Investment Grade Bonds

Bonds rated BBB or above (S&P) or Baa and above (Moody’s) Lower rated bonds

Speculative Grade Bonds or Junk Bonds
T
Ct M Price t T (1 r ) t 1 (1 r )

e.g. a two year pure-discount bond sells for 857.34. Show the YTM is 8%.
YTM Example
950

35 (1r)t t 1
20
CR = 8% ; YTM = 8%; Semiannual Compounding In 6M the rate falls to 7%; N=10 years P0 = $1000 P1 =$1068.55
40 (1068 .55 1000) HPR 1000
HPR = 10.85% (semiannual)
Bond Values Decrease if Discount Rates Increase
2500 2000
Value of Bond
1500
1000
500
0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Discount Rate
Bond Prices and Interest Rates

Zero Coupon Bonds

Bonds that pay no interest payments

Floating Rate Bonds

Coupon rates are reset periodically to specific market rates The payments of the bonds are tied to specific price indices
Example

Two-year bond selling at par, 10% coupon paid once a year. First coupon is reinvested at 8%. Then:
FV 1,100 100 1.08 1,208
P (1 y)2 1,208 y (realized (1.208)0.5 1 )
Callable Bonds

The issuer can repurchase the bonds at specific call prices during the call period The investor of the bonds can sell the bonds at par value during the call period The investor of the bonds can exchange the bonds for a specific number of common stocks

Bond’s payments:

Face Value (M) received at maturity, Coupon (C) received periodically (semiannually for government bonds)
YTM

The YTM is the discount rate, r, that implicitly solves the following equation:
Realized Compound Yield vs. YTM


Requires actual calculation of reinvestment income Solve for the Internal Rate of Return using the following:


Future Value: sale price + future value of coupons Investment: purchase price

FVt = FVt-1* (CPIt/CPIt-1)

The coupon payment Ct depends on the real interest rate rt and the face value FVt:

Ct = rt * FVt
Characteristics of Bonds

Bid 142:10
Asked 142:11
Chg 3
Ask Yld 4.35
How much interest do you receive on this bond? How much do you have to pay for this bond?
Characteristics of Bonds

Spot rates are convenient ways to quote prices:
Mt Pt t (1 st )
Spot Rates

Spot rates can sometimes be backed out of coupon bond prices.

This uses a no-arbitrage style relationship to solve a system of linear equations:

Indexed Bonds Floaters and Reverse Floaters
Bond Characteristics

Face or par value Coupon rate

Zero coupon bond Accrued Interest, “dirty price”

Compounding and payments
Bond Pricing: Example


Let’s look at the 9.25% Treasury Bond with maturity on February 1st, 2019 What is the value of this bond?

The bond has a face value of $1,000 Treasury bonds pay semi-annual coupons Assume that the discount rates for all maturities equal 4%
Different Issuers of Bonds


Treasury bonds Provincial government bonds Corporations Municipalities International Governments and Corporations Innovative Bonds
Price Paths of Coupon Bonds
Price Premium bond
1,000
Discount bond 0
Maturity date
Time
Spot Rates

Definition:

The current yield-to-maturity on a purediscount security.



Prices and market interest rates have an inverse relationship When interest rates get very high the value of the bond will be very low When rates approach zero, the value of the bond approaches the sum of the cash flows
Bond Pricing
PB

Ct t t 1 ( r) 1
T

Par ValueT (1 r)T
PB = price of the bond Ct = interest or coupon payments T = number of periods to maturity r = the appropriate semi-annual discount rate

Indexed Bonds

Treasury Inflation Protected Bonds (TIPS)

Treasury Inflation Protected Bonds (TIPS)

The face value of the bond FVt increases with the consumer price index CPIt
M1 P 1 1 s1 C2 P2 1 s1
M2 (1 s2 ) 2
Using Spot Rates to price Coupon Bonds




A coupon bond can be viewed as a series of zero coupon bonds To find the value, each payment is discounted at the zero coupon rate Once the bond value is found, one can solve for the yield It’s the reason for which similar maturity and default risk bonds sell at different yields to maturity

Provisions of Bonds


Secured or unsecured Registered or bearer bonds Call provision Convertible provision Retractable and extendible bonds Floating rate bond
Realized Yield versus YTM

Reinvestment Assumptions Holding Period Return


Changes in rates affects returns Reinvestment of coupon payments Change in price of the bond

Quoted Bond Prices

The WSJ of February 25, 2006 gives the following information on a Treasury bond with a face value of $1,000:
Rate 9.25

Maturity Feb 16

1000 T (1r)
10 yr Maturity Coupon Rate = 7% Price = $950 Solve for r = semiannual rate
r = 3.8635%
Yield Measures
Bond Equivalent Yield 3.86% x 2 = 7.72% Effective Annual Yield (1.0386)2 - 1 = 7.88% Current Yield (Annual Interest/Market Price) $70 / $950 = 7.37 %