chapter three understanding interest rates slide 3–3 present value four types of credit...

Chapter Three

Understanding Interest Rates

3

Present Value

Four Types of Credit Instruments1. Simple Loan – all principal & interest due at maturity2. Blended Payment Loan3. Coupon Bond4. Discount Bond

Concept of Present Value A dollar today is worth more than a dollar to be received in the

future. Yield to maturity

The interest rate that equates today's market value with the present value of all future payments

Equivalent to the IRR (Internal Rate of Return)

4

Yield to Maturity: Loans

1. Simple Loan Example: You borrow $1,000 today. Repayment of $1,469.33 is due in 5 years. What is the yield to maturity on the loan?Solution: Solve for i in the formula

6

1

$1,469.33 1,000 1

8%

tFV PV i

i

i

5

Yield to Maturity: Loans

2. Blended Payment LoanExample: You borrow $10,000. You will repay the loan in 60 monthly payments of $212.47. What is the nominal annual yield to maturity for the loan?Solution: You can either use trial & error or use a financial calculator. We will opt for the calculator.

10,000 PV0 FV212.47 +/- PMT60 N

i 0.833326% per month

or 10% per year

6

Yield to Maturity: Bonds

2. Coupon BondsExample: A 10 year, $1,000 face value bond with a 6% annual pay coupon is currently selling for $929.76. What is the bond’s yield to maturity?Solution: You can either use trial & error or use a financial calculator. We will opt for the calculator.

1,000 FV60 PMT929.76 +/- PV10 N

i 7%

Bond Page of the Newspaper

Copyright © 2004 Pearson Education Canada Inc. Slide 3–7

8


9


10

Relationship Between Price and Yield to Maturity

Three interesting facts in Table 11. When the bond sells at par, yield equals coupon rate

2. Price and yield are negatively related

3. Yield is greater than the coupon rate when the bond’s price is below par value

4. Yield is less than the coupon rate when the bond’s price is above par value

11

Decomposing a Bond’s Yield to Maturity

The yield to maturity on a bond can always be decomposed into: Current yield Capital gain or loss

For example, assume that we are holding the 10 year bond reviewed earlier. It has a $1,000 face value, a 6% annual pay coupon and it is currently selling for $929.76, providing a 7% YTM.

We want to decompose the YTM into its constituent parts

12

Decomposing Yield to Maturity Step #1: Calculate the Current Yield

60

929.766.453%

CouponCurrent Yield =

Price

=

Step #2: Calculate the Capital Gain or Loss First, calculate the market price of the bond, assuming one year has passed but all else remains equal.1,000 FV

60 PMT9 N

7 i

PV $934.85

13

Decomposing Yield to Maturity

Then calculate the capital gain

1 0

0

934.85 929.76

929.760.5472%

P PCapital Gain =

P

=

Now add up the current yield plus the capital gain to obtain the YTM

/

6.453 0.5472

7.00%

YTM = Current Yield Capital Gain Loss

Where:

P1 = Price at end of period

P0 = Price at start of period

14

Yield to Maturity: Bonds

4. Discount Bonds

Example: A 91 day, $100,000 face value Treasury bill is currently selling for $98,600. What is the bond’s annualized yield to maturity?

Solution:

100,000 -98,600 365

98,600 91

5.695%

T Bill

FacePrice =

n1+i

B

Face - Price Bi =

Price n

Where:Price = current market priceFace = face value of T billB = Annual basis (365 in Canada)n = days to maturity

15

Clean & Dirty Bond Prices

The usual formula for calculating the market price of a bond is:

This formula assumes that the sale transaction is taking place on a coupon payment date, which is unrealistic for most market transactions

Therefore, have to calculate two prices: Clean price – the quoted price Dirty price – the clean price plus accrued interest since the last

coupon payment date

1 1$

1

t

Bond t

YTM FaceValuePV Coupon

YTM YTM

16

Graphically, it appears like this:


+

17

Example: Assume that it is March 1, 2006 and we are holding a $1,000 face value bond with a 6% YTM and a 7% coupon paid semi-annually. The bond matures December 31, 2010. Coupon payment dates are December 31 and June 30. Calculate the clean & the dirty price of the bond.


Dec 31, 2006

Dec 31, 2007

Dec 31, 2008

Dec 31, 2009

Dec 31, 2010

June 30 June 30 June 30 June 30

Today – March 1,

2006

The clean price is the PV of all future cash flows not yet received, as of the transaction date.

$35 $35 $35 $35 $35 $35 $35 $1035

18


7.6667

7.6667

1 1$

1

1 1.03 1,000$35

.03 1.03

1,033.80

t

Bond t

YTM FaceValuePV Coupon

YTM YTM

Calculating the Clean Price

Calculating the Dirty Price

Dirty Price = Clean Price + Accrued Interest

21,033.80 $35

6

$1,045.47

19

Day Count Conventions

In the capital markets, there are a number of ways that days between dates are computed for interest rate calculations. Many of these conventions were developed before the wide spread introduction of computers. The historical rationale for many of these calculations was to simplify the math involved in performing normally complex financial calculations. And as in most industries with a long history, many of these conventions have stayed with us despite considerable advances in computers and computational methods.

The day count basis indicates the manner by which the days in a month and the days in a year are to be counted. The notation utilized to indicate the day count basis is (days in month)/(days in year).

The five basic day count basis are the following: Actual/360 Actual/365 Actual/Actual 30/360 30/360 European

20

Day Count Conventions: Actual/360

This calculates the actual number of days between two dates and assumes the year has 360 days. Many money market calculations with less than a year to maturity use this day count basis.

For example, a $1 Million six month CD issued on 4/15/2006 and maturing on 10/15/2006, with an 8% coupon would pay an interest payment of: Actual days between 4/15/2006 to 10/15/2006 = 183 days Interest = 0.08 x 1,000,000 x (183/360) = $40,666.67

21

Day Count Conventions: Actual/365

This calculates the actual number of days between two dates and assumes the year has 365 days.

Using an Actual/365 day count basis, a $1 Million six month CD issued on 4/15/2006 and maturing on 10/15/2006, with an 8% coupon would pay an interest payment of: Actual days between 4/15/2006 to 10/15/2006 = 183 days Interest = 0.08 x 1,000,000 x (183/365) = $40,109.59

22

Day Count Conventions: Actual/Actual

This day count basis calculates the actual number of days between two dates and assumes the year has either 365 or 366 days depending on whether the year is a leap year. More accurately, if the range of the date calculation includes February 29 (the leap day), the divisor is 366, otherwise the divisor is 365.

Using our CD example, the interest payment would be: Actual days between 4/15/2006 to 10/15/2006 = 183 days Interest = 0.08 x 1,000,000 x (183/365) = $40,109.59

Notice that even if 2006 were a leap year, the denominator used for this calculation would be 365 because February 29 does not fall into the date range of the calculation. If the issue date was before February 29 and the year were a leap year, the divisor would have been 366 instead.

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This day count convention assumes that each month has 30 days and the total number of days in the year is 360 (12 months x 30 days per month). There are adjustments for February and months with 31 days.

The formula for the 30/360 day calculation is as follows: Assume Date 1 is of the form M1/D1/Y1 and Date 2 is of the form M2/D2/Y2. Let Date 2 be later than Date 1.

Then: If D1 = 31, change D1 to 30 If D2 = 31 and D1 = 30, change D2 to 30 Days between dates = (Y2-Y1) x 360 + (M2-M1) x 30 + (D2-D1)

Day Count Conventions: 30/360

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The 30/360 day count basis is different outside of the United States. The Europeans further simplified this calculation as follows. Assume Date 1 is of the form M1/D1/Y1 and Date 2 is of the form M2/D2/Y2. Let Date 2 be later than Date 1.

Then: If D1 = 31, change D1 to 30 If D2 = 31, change D2 to 30 Days between dates = (Y2-Y1) x 360 + (M2-M1) x 30 + (D2-D1)

Day Count Conventions: 30/360 European

25

Distinction Between Real and Nominal Interest Rates

Real interest rate1. Interest rate that is adjusted for expected changes in the price level

ereal Nominali i

2. Real interest rate more accurately reflects true cost of borrowing

3. When real rate is low, greater incentives to borrow and less to lend

e = the expected rate of inflation

26

Distinction Between Real and Nominal Interest Rates (cont.)

Re

Re Re

1 1

1

eNom al

e eNom al al

i i i

i i i i

Fisher Equation: developed by Irving Fisher (1867 – 1947)

Cross product term

When inflation is low, use the approximation.

When inflation is high, use the exact method

27

Distinction Between Real and Nominal Interest Rates (cont.)

Real Rate Inflation Rate

Nominal Rate

After-tax Nominal

After-tax Real Rate

5% 0% 5% 2.5% 2.5%

5% 5% 10% 5% 0%

5% 10% 15% 7.5% -2.5%

Nominal rate = Real rate + Inflation rate

Assumptions: 50% marginal tax rate

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Inflation as a Form of Taxation

Any tax is a transfer of purchasing power from A to B

The inflation tax is a transfer of purchasing power from creditors to debtors

The largest debtor is the federal government The government issues a long term bond with a fixed

coupon During the life of the bond, inflation rises The bundle of goods & services able to be purchased at

the maturity of the bond is less than that purchased at the bonds issue date

29

U.S. Real and Nominal Interest Rates

Figure 3: Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2001

Sample of current rates and indexeshttp://www.martincapital.com/charts.htm

30

Real Return Bonds Issued by the Government of Canada Principal is grossed up every 6 months, based on the change

in the CPI. The coupon is paid based on the grossed-up face value Protects the holder against unexpected inflation

31

Key Facts about the Relationship Between Rates and Returns

32

Burton Malkiel’s Five Bond Theorems

1. Bond prices move inversely tobond yields

2. Long bonds have greater pricevolatility than short bonds

3. For a given change in yield, price volatility increases but at a decreasing rate as term to maturity increases

4. The capital gain due to a fall in market yields is always larger than the capital loss due to a rise in market yields

5. High coupon bonds have less price volatility than low coupon bonds

33

Reinvestment Rate Risk

Question: You buy a $1,000, three year bond with a 6% coupon at a price of $1,027.23 to yield 5% to maturity. If you hold the bond to maturity, are you assured of realizing a yield of 5% on your bond investment?

34


Answer: Contrary to popular thought, you are not assured of earning a realized yield of 5%, unless you can reinvest each of the coupons at 5%. Assume that shortly after you purchase the bond, market rates drop to 4%. What is your realized yield if you hold to maturity?

0 1 2 3

1,027.23 60 60 1,060.00

62.40

64.90

1187.30Now solve for the yield required for $1,027.23 to grow to $1,187.30 in three years (see next page for solution).

35


1,187.30 FV0 PMT1027.23 +/- PV3 Ni 4.9456%

3

1

1,187.30 1027.23 1

4.9456%

tFV PV r

r

r

Or, using Algebra

36

Reinvestment Risk

1. All coupon bonds are subject to reinvestment rate risk

2. The higher the coupon, the greater the reinvestment rate risk

3. The only bonds not subject to reinvestment rate risk are zero coupon or strip bonds

37

Duration

Three factors affect the price volatility of a bond. These are: Term to maturity Size of coupon General level of interest rates

Duration captures all three factors in one number. Duration was introduced by Frederick Macaulay in

1938

38

Two Definitions of Duration

1. Duration is the approximate percentage change in the price of a bond, given a 1% change in market yields

2. Duration is that point in time when the capital gain or loss due to a change in the YTM is exactly offset by the change in the reinvestment rate on the coupons.

39

Calculating Durationi =10%, 10-Year 10% Coupon Bond

Calculating Durationi = 20%, 10-Year 10% Coupon Bond

Copyright © 2004 Pearson Education Canada Inc. Slide 3–40

41

1

1

1

1

nt

tt

nt

tt

Ct

iDUR

C

i

Formula for Duration

Where: Dur = Macaulay’s Duration t = the number of time periods C = the cash flow at time period t i = the bond’s initial yield to maturity

42

Formula for Duration

• Numerator1. First calculates the PV of each cash flow

2. Then multiplies the PV by the number of time periods until the cash flow occurs

• Denominator1. Is equal to the current market price of the bond

43

What are we Measuring with Duration?YTM

PriceP0

YTM0

True price-yield relationship

Price-Yield relationship as measured by duration

44

Duration

Duration is a linear relationship that is attempting to measure something that is not linear

This leads to errors in measurement As the size of the interest rate shock grows, so does

the size of the error Duration will always overstate the capital loss and

understate the capital gain Can correct the duration error using convexity

45

Modified Duration

1 Initial

DurationModified Duration

YTM

%

1 Initial

YTMP Duration

YTM

Modified duration reduces some of the measurement error

To calculate the dollar & the percentage change in the price of a bond using modified duration

$

1 Initial

YTMP Duration Price

YTM

46

Using Modified Duration

Assume that you are holding a three year, $1,000 bond with a 7% coupon and a YTM of 8%. Calculate the bond’s duration Calculate the dollar change in the price of the bond using

modified duration, assuming the YTM rises by 20 basis points

Calculate the percentage in the price of the bond using modified duration, assuming the YTM rises by 20 basis points

47

Duration

Duration of a coupon bond is always less than maturity

Duration of a zero coupon bond is equal to duration Duration will rise as maturity increases Duration rises as the coupon decreases Duration rises as the YTM decreases

48

Duration of a Portfolio

Assume that you have a portfolio consisting of 4 assets, as shown on the right side of the page:

Asset Duration

(Di)

Proportion of Portfolio

(Xi)

#1 1.5 10%

#2 3.0 25%

#3 5.0 35%

#4 8.0 30%

1 1 2 2 ...

0.10 1.5 0.25 3.0 0.35 5.0 0.30 8.0

5.05

Portfolio N ND X D X D X D

49

Duration & Immunization - Banks

A Bank can immunize its Balance Sheet against changes in interest rates by matching the duration (rather than the maturity) of its assets & liabilities. Why might it want to do this?

To calculate the duration of the assets and liabilities, first calculate the duration of each instrument. Then multiply the duration of each instrument by its market value weight (See formula on next page).

50

Duration of the Assets & Liabilities

Duration of the assets & liabilities

Where: DA = Duration of the assets

DL = Duration of the liabilities

XA1 = The proportion of the 1st asset

D1A = The duration of the 1st asset

51

Calculating the Change in Market Value of the Assets & Liabilities When Rates Change

We use the duration formula to calculate the change in the market value of the assets & liabilities due to a change in market interest rates (remember modified duration)

But what we really want to know is, what is the change in equity due to a change in interest rates?

52

Calculating the Change in Equity

Where: k = a measure of leverage

53

Understanding the Formula

Can now decompose the effect of a change in market interest rates on the bank’s equity into three separate effects: the leverage adjusted duration gap (DA - DLk)

Is measured in years Reflects the exposure of the B/S to interest rate shocks

Size of the bank (reflected by the size of A) The size of the interest rate shock

The change in equity captures the exposure of the bank to an interest rate shock

54

Convexity

Duration is an accurate measure of price sensitivity for small changes in interest rates. It is not accurate for large changes.

R

Price

True Price/Yield relationship

Duration Error

R1

R2

P1P2 P3

When rates rise from R1 to R2, duration suggests that price will fall from P1 to P2,but

in fact, the true fall in price is only to P3. The distance from P2 to P3 is the error due

to duration.

55

Convexity

Duration Overstates the decline in price due to a rise in interest

rates Understates the increase in price due to a fall in

interest rates

Convexity is used to correct for this pricing inaccuracy

Convexity is: desirable for an asset undesirable for a liability

56

Calculating Price Changes, Including Convexity

To calculate the percentage change in price, use the following formula:

Where: CX = convexity

And where CX is calculated as follows:

57

Using Convexity

Example: To calculate the convexity of a $1,000, 6 year, 8% coupon, 8% YTM Eurobond, first calculate the capital loss and the capital gain from a 1 basis point change in interest rates. Then multiply by a scaling factor of 10 to the 8th power:

58

Calculating the Change in Price

The percentage change in price is now calculated as follows (assuming a 2% increase in interest rates, from 8% to 10%):

59

Convexity: Things to Remember

Convexity increases with maturity

Convexity increases as coupon size decreases

When duration is the same, zero coupon bonds are less convex than coupon bonds

chapter three understanding interest rates slide 3–3 present value four types of credit...

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