Fin351: lecture 3

Bond valuation

The application of the present value concept

Today’s plan

Interest rates and compounding Some terminology about bonds Value bonds The yield curve Default risk

Interest

Simple interest - Interest earned only on the original investment.

Compounding interest - Interest earned on interest.

In Fin 351, we consider compounding interest rates

Simple interest

Example

Simple interest is earned at a rate of 6% for five years on a principal balance of $100.

Simple interest

Today Future Years

1 2 3 4 5

Interest Earned 6 6 6 6 6

Value 100 106 112 118 124 130

Value at the end of Year 5 = $130

Compound interest

Example

Compound interest is earned at a rate of 6% for five years on $100.

Today Future Years

1 2 3 4 5

Interest Earned 6.00 6.36 6.74 7.15 7.57

Value 100 106.00 112.36 119.10 126.25133.82

Value at the end of Year 5 = $133.82

Interest compounding

The interest rate is often quoted as APR, the annual percentage rate.

If the interest rate is compounded m times in each year and the APR is r, the effective annual interest rate is

11

m

mr

Compound Interest i ii iii iv vPeriods Interest Value Annuallyper per APR after compoundedyear period (i x ii) one year interest rate

1 6% 6% 1.06 6.000%

2 3 6 1.032 = 1.0609 6.090

4 1.5 6 1.0154 = 1.06136 6.136

12 .5 6 1.00512 = 1.06168 6.168

52 .1154 6 1.00115452 = 1.06180 6.180

365 .0164 6 1.000164365 = 1.06183 6.183

Compound Interest

Interest Rates

Example

Given a monthly rate of 1% (interest is compounded monthly), what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

Solution

12.00%or .12=12 x .01=APR

12.68%or .1268=1 - .01)+(1=EAR 12

Interest Rates

Example

If the interest rate 12% annually and interest is compounded semi-annually, what is the Effective Annual Rate (EAR)? What is the Annual Percentage Rate (APR)?

Solution

APR=12% EAR=(1+0.06)2-1=12.36%

Nominal and real interest rates

Nominal interest rate• What is it?

Real interest rate• What is it?

Inflation• What is it?

Their relationship• 1+real rate =(1+nominal rate)/(1+inflation)

Bonds

Bond – a security or a financial instrument that obligates the issuer (borrower) to make specified payments to the bondholder during some time horizon.

Coupon - The interest payments made to the bondholder.

Face Value (Par Value, Face Value, Principal or Maturity Value) - Payment at the maturity of the bond.

Coupon Rate - Annual interest payment, as a percentage of face value.

Bonds

A bond also has (legal) rights attached to it:• if the borrower doesn’t make the required

payments, bondholders can force bankruptcy proceedings

• in the event of bankruptcy, bond holders get paid before equity holders

An example of a bond

A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years.• The coupon payment is $100 annually

• The discount rate is different from the coupon rate.

• In the third year, the bondholder is supposed to get $100 coupon payment plus the face value of $1000.

• Can you visualize the cash flows pattern?

Bonds

WARNINGWARNINGThe coupon rate IS NOT the discount rate used in the Present Value calculations. The coupon rate merely tells us what cash flow the bond will produce.

Since the coupon rate is listed as a %, this misconception is quite common.

Bond Valuation

The price of a bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

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Zero coupon bonds

Zero coupon bonds are the simplest type of bond (also called stripped bonds, discount bonds)

You buy a zero coupon bond today (cash outflow) and you get paid back the bond’s face value at some point in the future (called the bond’s maturity )

How much is a 10-yr zero coupon bond worth today if the face value is $1,000 and the effective annual rate is 8% ?

PV

Facevalue

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Zero coupon bonds (continue)

P0=1000/1.0810=$463.2 So for the zero-coupon bond, the price is

just the present value of the face value paid at the maturity of the bond

Do you know why it is also called a discount bond?

Coupon bond

The price of a coupon bond is the Present Value of all cash flows generated by the bond (i.e. coupons and face value) discounted at the required rate of return.

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Bond Pricing

Example

What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.

Bond Pricing

Example

What is the price of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? Assume a required return of 5.6%.

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Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 6 %?

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Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 15 %?

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Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?

Bond Pricing

Example (continued)

What is the price of the bond if the required rate of return is 5.6% AND the coupons are paid semi-annually?

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Bond Pricing

Example (continued)

Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?

Bond Pricing

Example (continued)

Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?

Time Periods

Paying coupons twice a year, instead of once

doubles the total number of cash flows to be discounted

in the PV formula.

Bond Pricing

Example (continued)

Q: How did the calculation change, given semi-annual coupons versus annual coupon payments?

Time Periods

Paying coupons twice a year, instead of once

doubles the total number of cash flows to be discounted

in the PV formula.

Discount Rate

Since the time periods are now half years, the discount rate is also

changed from the annual rate to the half year rate.

Bond Yields

Current Yield - Annual coupon payments divided by bond price.

Yield To Maturity (YTM)- Interest rate for which the present value of the bond’s payments equal the market price of the bond.

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An example of a bond

A coupon bond that pays coupon of 10% annually, with a face value of $1000, has a discount rate of 8% and matures in three years. It is assumed that the market price of the bond is the same as the present value of the bond. • What is the current yield?

• What is the yield to maturity.

My solution

First, calculate the bond price P=100/1.08+100/1.082+1100/1.083

=$1,051.54 Current yield=100/1051.54=9.5% YTM=8%

Bond Yields

Calculating Yield to Maturity (YTM=r)

If you are given the market price of a bond (P) and the coupon rate, the yield to maturity can be found by solving for r.

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Bond Yields

Example

What is the YTM of a 6 % annual coupon bond, with a $1,000 face value, which matures in 3 years? The market price of the bond is $1,010.77

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Bond Yields

In general, there is no simple formula that can be used to calculate YTM unless for zero coupon bonds

Calculating YTM by hand can be very tedious. We don’t have this kind of problems in the quiz or exam

You may use the trial by errors approach get it.

Bond Yields (3)

Can you guess which one is the solution in the previous example?

(a) 6.6%

(b) 7.1%

(c) 6.0%

(d) 5.6%

The bond price, coupon rates and discount rates

If the coupon rate is larger than the discount rate, the bond price is larger than the face value.

If the coupon rate is smaller than the discount rate, the bond price is smaller than the face value.

The rate of return on a bond

price bondor investmentchange price+incomeCoupon

=return of Rate

Example: An 8 percent coupon bond has a price of $110 dollars with maturity of 5 years

and a face value of $100. Next year, the expected bond price will be $105. If you hold this bond this year, what is the rate of return?

investment ofcost

profit=return of Rate

My solution

The expected rate of return for holing the bond this year is (8-5)/110=2.73%• Price change =105-110=-$5

• Coupon payment=100*8%=$8

• The investment or the initial price=$110

The Yield Curve

Term Structure of Interest Rates - A listing of bond maturity dates and the interest rates that correspond with each date.

Yield Curve - Graph of the term structure.

The term structure of interest rates (Yield curve)

YTM for corporate and government bonds

The YTM of corporate bonds is larger than the YTM of government bonds

Why does this occur?

Default Risk

Default risk• The risk associated with the failure of the

borrower to make the promised payments

Default premium• The amount of the increase of your discount

rate

Investment grade bonds Junk bonds

Ranking bondsStandard

Moody' s & Poor's Safety

Aaa AAA The strongest rating; ability to repay interest and principalis very strong.

Aa AA Very strong likelihood that interest and principal will berepaid

A A Strong ability to repay, but some vulnerability to changes incircumstances

Baa BBB Adequate capacity to repay; more vulnerability to changesin economic circumstances

Ba BB Considerable uncertainty about ability to repay.B B Likelihood of interest and principal payments over

sustained periods is questionable.Caa CCC Bonds in the Caa/CCC and Ca/CC classes may already beCa CC in default or in danger of imminent defaultC C C-rated bonds offer little prospect for interest or principal

on the debt ever to be repaid.