8.3 Coupon Bonds, Current yield, and Yield to Maturity

Financial Economics, Spring 2011

8.3 Coupon Bonds, Current yield, and Yield to Maturity

? Relationships between zero rates, bond price and yield to maturity

? Yield to maturity, YTM, is an internal rate of return, IRR for a bond. ? Internal rate of return is interest rate such that NPV becomes zero. YTM may not be equal to zero rate. ? Zero rate is interest rate which makes price of pure discount bond Equal to PV of its face value. Spot rate is another name for zero rate.

From zero rates to YTM 1. Coupon bond has periodic coupon payments. 2. Each coupon can be considered as a zero coupon, i.e. pure discount bond. 3. Evaluate these coupons as pure discount bond. 4. Sum their PV's. 5. Set it equal to price. 6. For a given price, find internal rate of return for that bond. This IRR is "yield to maturity."

? Example: 3-year Bond with 10% Coupon

Suppose that, in the bond market, we observe the following set of spot rate, i.e., zero rates; numerical example in 8.2. We apply annual compounding.

Pure Discount Bond

Maturity price per $1 spot rate

of face value as APR

1 year 2 years 3 years

0.95 0.88 0.80

5.26 6.60 7.72

Table 1.

Suppose also that there is a 3-year bond with 10% coupon. We like to find its theoretical price and its yield to maturity. Face value is $1,000. Coupon is paid once a year. Each coupon is $100. We apply annual compounding.

? Price is equal to sum of PV' s of coupons and face value

There are three cash inflows from the coupon bond.

time of receipt 1 year 2 year 3 year

dollar amount

100 100 100 1000

We evaluate these cash flows as pure discount bonds. If we apply zero rates,

then we calculate the following equation;

100

100

100 1000

PV

1 0.0526 1 0.0662 1 0.07723

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Financial Economics, Spring 2011

In[950]:=

ClearPV; PV

100

100

100 1000

;

1 0.0526 1 0.0662 1 0.07723

Print "PV", PV

PV1063.05

Alternatively, we can take a short cut. Table 1 implies the following relationships.

price of pure discount bond and zero rate

0.95

1 10.0526

0.88

1

0.80

1

10.0662

10.07723

We calculate the following; PV = 100?0.95 + 100?0.88 + (100 + 1000)?0.8.

In[952]:= ClearPV; Print"PV ", 100 0.95 100 0.88 100 1000 0.8

PV 1063.

We calculate the following; PV = 100?0.95 + 100?0.88 + (100 + 1000)?0.8. Then, PV= 1063. Due to the rounding errors, figures do not coincide. Interest rates should be more precise. Let's use $1063 in this example.

Set price equal to PV. Bond price is $1, 063.00 dollars.

In[953]:=

Let's check precise interest rates Clearx1, x2, x3

NSolve0.95 1 , x1 1 x1

NSolve0.88

1

, x2

1 x22

NSolve0.80

1

, x3

1 x33

Out[954]= x1 0.0526316

Out[955]= x2 2.066, x2 0.0660036

Out[956]= x3 1.53861 0.932898 , x3 1.53861 0.932898 , x3 0.0772173

? Yield to Maturity

YTM is the value of single discount rate y which makes NPV equal to zero. In other words, YTM is a solution for the following equation. Let y be APR for annual compounding.

price

100 100

1y 1y2

100 1000

1y3

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Financial Economics, Spring 2011

In[957]:=

Clearprice, y, ans; price 1063;

ans NSolveprice 100

100

100

1000 ,

y;

1 y 1 y2

1 y3

y y . ans3; Print"Yield to maturity is equal to ", 100 y , "."

Yield to maturity is equal to 7.57415.

? Current Yield

? Example 10% coupon bond with one year remaining

An example given on page 228 of the text book: Face value is $1, 000. Coupon rate 10 %. Coupon is paid once a year. We apply annual compounding. One year zero rate is 5 %.

Since zero rate is 5%, bond price is given by the following;

price

1001000 10.05

1047.62

Price is higher than its face value. Such a bond is premium bond.

There is another kind of "yield", called current yield. Its definition is given by

current yield

coupon

current bond price

Current yield of the above example is 100 0.0955 . This yield ignores capital loss. At maturity, you won't receive

1047.62

principal amount of your investment, which is $1047.62. You receive coupon plus face value which is $1100. It is smaller

than you paid. Your yield to maturity of one year investment is given by

YTM coupon Face value price 100 1000 1047.62 0.05

price

1047.62

It is 5%. Here, 1000-1047.62 = - 47.62 is capital loss.

? Bond Pricing Principle

Depending on the relationships between price and face value, bonds are categorized into three groups; par, premium and discount bonds. 1. par bond : P= F; 2. P>F: premium bond; 3. PF fl c > y . Principle 3 says P coupon rate If a coupon bond has a price higher than its face value, its yield to maturity is less than its coupon rate. And vice versa.

? Reason of these relationship ?

Principle 1 P=F ?c=y This can be shown using the formula of sum of geometric sequence. P=F? c=y This can be shown using graph of value of cash flow as a function of YTM.

Principle 2 and 3 follow from the above proof.

? Graphical Presentation of Bond Pricing Principles

Let g(y) be PV of bond as a function of YTM y. part 1 Shape of function for y>-1 If y > -1 then, the first derivative is always negative and that the second derivative is positive. For y > -1. g'(y)0. In addition, as y? -1, g(y) goes to infinity. And as y ? +? , g(y) ? 0.

Such a shape of function g(y) means that, for a given bond price p , an equation g(y) = p has always a solution in the range of y > -1. Also there exits only one solution because of the shape of g(y). part 2 c=y fl P=F If YTM equals to coupon rate, then PV equals to bond price. This can be proved using a formula of sum of geometric sequences.

This means that g(c)=F where c is coupon rate. This equality always holds for any value of c. Result of part1 tells that solution is unique. So the coupon rate c is the only solution for equation g(y)=F. It implies that if bond price equals to F, then YTM equals to coupon rate.

Graphically, the above results tell that 1. g(y) intersects with horizontal line of F when y=c. Par bond 2. For y > c, then g(y) < F Discount bond 3. For y < c, then g(y)> F Premium bond

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Financial Economics, Spring 2011

? Example: coupon rate 6%, 5 years to maturity

In[961]:=

Clearg, x, c, F, T, y; c 0.06; F 100; T 5;

T

gy_ :

cF

F

t1 1 yt 1 yT

Plotgy, y, 0.05, 1, ImageSize 250,

PlotLabel "Shape of gy where g'y0 and g''y0",

AxesLabel y, present value

Shape of gy where g'y0

present value

150

Out[963]= 100 50

y

0.2

0.4

0.6

0.8

1.0

Principle 1 holds for any remaining years. Let's see examples for T= 3, 5, 7,

In[964]:=

ClearT, graph; T 3; graph3 Plotgy F, y, 0.055, 0.065,

ImageSize 200, PlotLabel "T3", PlotStyle Dotted; ClearT; T 5; graph5 Plotgy F, y, 0.055, 0.065, ImageSize 200, PlotLabel "T5"; ClearT; T 7; graph7 Plotgy F, y, 0.055, 0.065,

ImageSize 200, PlotLabel "T7", PlotStyle Thick; Showgraph3, graph5, graph7, PlotLabel T 3, 5, 7 , AxesLabel y, P F

P - 100

3, 5, 7

1.0

Out[970]=

0.5

-0.5 -1.0

y 0.058 0.060 0.062 0.064

8.5 Why Yields for the Same Maturity May Differ

Often bonds with the same maturity have different YTM. There are following reasons. 1. effect of coupon rate 2. effect of default risk 3. callability and convertibility

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