PRICING SECURITIES

Chapter 6

Bank ManagementBank Management, 5th edition.5th edition.Timothy W. Koch and S. Scott MacDonaldTimothy W. Koch and S. Scott MacDonaldCopyright © 2003 by South-Western, a division of Thomson Learning

Future value and present value

PV(1+i) = FV1 Example: 1,000 PV and 1,080 FV

means: i = $80 / $1,000

= 0.08 = 8% If we invest $1,000 at 8% for 2 years:

$1,000 (1+0.08) (1+0.08) = $1,080 (1.08)= $1,166.40

In general, the future value is…

FVn = PV(1+i)n

Alternatively, the yield can be found as: i = [FVn / PV](1/n) - 1

Example: $1,000 invested for 6 years at 8%: FV6 = $1,000(1.08)6 = $1,586.87

Example: Invest $1,000 for 6 years, Receive $1,700 at the end of 6 years. What is the rate of return?

i = [$1,700 / $1,000](1/6) - 1 = 0.0925

Future value and present value…multiple payments

The cumulative future value of a series of cash flows (CFVn) after n periods is: CFVn =CF1(1+i)n +CF2(1+i)n-1 +...+CFn(1+i)

The present value of a series of n cash flows: PV = [CF1 / (1+i)] + [CF2 / (1+i)2] + [CF3 / (1+i)3]

. . . +[CFn / (1+i)n] Example:

rate of interest = 10%, what is the PV of a security that pays $90 at the end of the next three years plus $1,000 at the end of three years?

PV = 90/(1.1) + 90/(1.1)2 + 1090/(1.1)3 = $975.13

Simple interest versus compound interest

Simple interest is interest that is paid only on the initial principal invested: simple interest = PV x (i) x n

Example, simple interest: if i = 12% per annum, n = 1 and principal of $1,000: simple interest = $1,000 (0.12) 1 = $120

Example, interest is paid monthly: monthly simple interest = $1,000 (0.12 / 12) 1= $10

Compounded interest is interest that is paid on the interest: PV (1 + i/m)nm = FVn

and PV = FVn / (1 + i/m)nm

Compounding frequency

Example: $1,000 invested for 1 year at 8% with interest compounded monthly: FV1 = 1,000 (1 + 0.08/12)12 = 1083.00

The effective annual rate of interest, i* can be calculated from: i* = (1 + i/m)m - 1

In this example: i* = (1 + 0.08/12)12 - 1 = 8.30%

A. What is the future value after 1 year of $1,000 invested at an 8% annual nominal rate?

Compounding Interval

Number of Compounding Intervals in I Year (m)

Future Value (FVI)*

Effective Interest Rate*

Year 1 $1080.00 8.00% Semiannual 2 1081.60 8.16 Quarter 4 1082.43 8.24 Month 12 1083.00 8.30 Day 365 1083.28 8.33 Continuous infinite 1083.29 8.33

B. What is the present value of $1,000 received at the end of 1 year with compounding at 8%?

Compounding Interval

Number of Compounding Intervals in 1 Year (m)

Present Value (PV)*

Effective Interest Rate*

Year 1 $925.93 8.00% Semiannual 2 924.56 8.16 Quarter 4 923.85 8.24 Month 12 923.36 8.30 Day 365 923.12 8.33 Continuous ∞ 923.12 8.33

The effect of compounding on future value and present value

000,10$047.1

000,10

047.1

470Price

6

6

1

t

t

$9,847.731.05

10,000

1.05

470Price

6

6

1tt

Bond prices and interest rates vary inversely

Consider a bond which pays semi-annual interest payments of $470 with a maturity of 3 years.

If the market rate of interest is 9.4%, the price of the bond is:

If the market rates of interest increases to 10%, the price of the bond falls to $9,847.72:

Relationship Impact Market interest rates and bond prices vary inversely.

Bond prices fall as interest rates rise and rise as interest rates fall.

For a specific absolute change in interest rates, the proportionate increase in bond prices when rates fall exceeds the proportionate decrease in bond prices when rates rise. The proportionate difference increases with maturity and is larger the lower a bond's periodic interest payment.

For the identical absolute change in interest rates, a bondholder will realize a greater capital gain when rates decline than capital loss when rates increase.

Long-term bonds change proportionately more in price than short-term bonds for a given change in interest rates from the same base level.

Investors can realize greater capital gains and capital losses on long-term securities than on short-term securities when interest rates change by the same amount.

Low-coupon bonds change proportionately more in price than high-coupon bonds for a given change in interest rates from the same base level.

Low-coupon bonds exhibit greater relative price volatility than do high-coupon bonds.

Price and yield relationships for option-free bonds that are equivalent except for the feature analyzed

In general

Par Bond Yield to maturity = coupon rate

Discount Bond Yield to maturity > coupon rate

Premium Bond Yield to maturity < coupon rate

Relationship between price and interest rate on a 3-year, $10,000 option-free par value bond that pays $270 in semiannual interest

10,155.24

10,000.00

9,847.73

8.8 9.4 10.0 Interest Rate %

$’s

= +$155.25

= -$152.77

Bond Prices Change Asymmetrically to

Rising and Falling Rates

For a given absolute change in interest rates, the percentage increase in a bond’s price will exceed the percentage decrease.

This asymmetric price relationship is due to the convex shape of the curve-- plotting the price interest rate relationship.

The effect of maturity on the relationship between price and interest rate on fixed-income, option free bonds

$’s

For a given coupon rate, long-term bonds price changes proportionately more in price than do short-term bonds for a given rate change.

10,275.13

10,155.24

10,000.00

9,847.739,734.10

8.8 9.4 10.0 Interest Rate %

9.4%, 3-year bond

9.4%, 6-year bond

The effect of coupon on the relationship between price and interest rate on fixed-income, option free bonds

% change in priceFor a given change in market rate, the bond with the lower coupon will change more in price than will the bond with the higher coupon.

+ 1.74

+ 1.55

0

- 1.52- 1.70

8.8 9.4 10.0Interest Rate %

9.4%, 3-year bond

Zero Coupon, 3-year bond

Duration and price volatility

Maturity simply identifies how much time elapses until final payment.

It ignores all information about the timing and magnitude of interim payments.

Duration is a measure of effective maturity that incorporates the timing and size of a security's cash flows.

Duration captures the combined impact of market rate, the size of interim payments and maturity on a security’s price volatility.

Duration versus maturity

1.) 1000 loan, principal + interest paid in 20 years.2.) 1000 loan, 900 principal in 1 year,

100 principal in 20 years. 1000 + int

|-------------------|-----------------| 0 10 20

900+int 100 + int|----|--------------|-----------------|

0 1 10 20

What is the maturity of each? 20 yearsWhat is the "effective" maturity?

2.) = [(900/100) x 1]+[(100/1000) x 20] = 2.9 yrsDuration, however, uses a weighted average of the present values.

1

2

Duration…approximate measure of the price elasticity of demand

Price elasticity of demand = % in quantity demanded / %in price

Price (value) changes Longer duration larger changes in price

for a given change in i-rates. Larger coupon smaller change in price

for a given change in i-rates.

Duration…approximate measure of the price elasticity of demand Solve for Price:

P -Duration x [i / (1 + i)] x P

Price (value) changes Longer maturity/duration larger changes in

price for a given change in i-rates. Larger coupon smaller change in price for a

given change in i-rates.

Δi

%ΔΔ

i+1ΔiP

ΔP

DUR

In general notation, Macaulay’s duration (D):

Example: 1000 face value, 10% coupon, 3 year, 12% YTM

Sec. the ofPV r)+(1(t)CF

r)+(1CF

r)+(1(t)CF

=D

n

1=tt

t

k

1=tt

t

k

1=tt

t

rs yea2.73 = 951.96

2597.6

(1.12)1000

+ (1.12)

100(1.12)

31000 +

(1.12)3100

+ (1.12)

2100+

(1.12)1100

D 3

1=t3t

332

1

Measuring duration

1136.16(1.05)

3*1000 +

(1.05)3*100

+ (1.05)

2*100+

(1.05)1*100

D332

1

years2.75 = 1136.16

3127.31D

Measuring duration

If YTM = 5%1000 face value, 10% coupon, 3 year, 5% YTM

years2.68 = 789.35

2131.95D

Measuring duration

If YTM = 20%1000 face value, 10% coupon, 3 year, 20% YTM

definition (by 3

(1.12)1000

(1.12)31000

D

3

3

Measuring duration

If YTM = 12% and Coupon = 01000 face value, 0% coupon, 3 year, 12% YTM

1000|-------|-------|-------|0 1 2 3

i i+1

Duration

P

ΔP

Compare price sensitivity

Duration allows market participants to estimate the relative price volatility of different securities:

Using modified duration:modified duration

= Macaulay’s duration / (1+i) We have an estimate of price volatility:

%change in price = modified duration x change in i

Type of Bond 3-Yr. Zero 6-Yr. Zero 3-Yr. Coupon 6-Yr. Coupon

Initial market rate (annual) 9.40% 9.40% 9.40% 9.40% Initial market rate (semiannual) 4.70% 4.70% 4.70% 4.70% Maturity value $10,000 $10,000 $10,000 $10,000 Initial price $7,591.37 $5,762.88 $10,000 $10,000 Duration: semiannual periods 6.00 12.00 5.37 9.44 Modified duration 5.73 11.46 5.12 9.02 Rate Increases to 10% (5% Semiannually) Estimated P -$130.51 -$198.15 -$153.74 -$270.45 Estimated P / P -1.72% -3.44% -1.54% -2.70% Initial elasticity 0.2693 0.5387 0.2406 0.4242

Comparative price sensitivityindicated by duration

P = - Duration [i / (1 + i)] P P / P = - [Duration / (1 + i)] i

where Duration equals Macaulay's duration.

Valuation of fixed income securities

Traditional fixed-income valuation methods are too simplistic for three reasons:

1. Investors do not hold securities until maturity

2. Present value calculations assumes all coupon payments are reinvested at the calculated Yield to Maturity

3. Many securities carry embedded options, such as a call or put, which complicates valuation since it is unknown if the option will be exercised.

Total return analysis

Market participants attempt to estimate the actual realized yield on a bond by calculating an estimated total return

= [Total future value / Purchase price](1/n) - 1

$101.03 (1.035)

$100

(1.035)

$3.658

8

1tt

Total return for a 9-year 7.3% coupon bond purchased at $99.62 per $100 par value and held for 5-years.

Assume: semiannual reinvestment rate = 3% after five years; a comparable 4-year maturity bond will be priced to yield 7% (3.5% semiannually) to maturity

Coupon interest: 10 x $3.65 = $36.50Interest-on-interest:

$3.65 [(1.03)10 -1] / 0.03 - $36.50 = $5.34

Sale price after five years:

Total future value: $36.50 + $5.34 + $101.03 = $142.87

Total return: [$142.87 / $99.62]1/10 - 1 = 0.0367

or 7.34% annually

Money market yields

Interest rates for most money market yields are quoted on a different basis.

In particular, some money market instruments are quoted on a discount basis, while others bear interest.

Some yields are quoted on a 360-day year rather than a 365 or 366 day year.

Interest-bearing loans with maturities of one year or less

The effective rate of interest depends on the term of the loan and the compounding frequency.

A one year loan which requires monthly interest payments at 12% annually, carries an effective yield to the bank of 12.68%: i* = (1 + 0.12/12)12 - 1 = 12.68%

If the same loan was made for 90 days: i* = [1 + 0.12 / (365/90)](365/90) - 1 = 12.55%

In general: i* = [1 + i / (365 / h)](365/h) - 1

360-day versus 365-day yields

Some securities are reported using a 360 year rather than a full 365 day year.

This will mean that the rate quoted will be 5 days too small on a standard annualized basis of 365 days.

To convert from a 360-day year to a 365-day year: i365 = i360 (365/360)

Example: one year instrument at 8% nominal rate on a 360-day year is actually an 8.11% rate on a 365-day year: i365 = 0.08 (365/360) = 0.0811

Discount yields

Some money market instruments, such as Treasury Bills, are quoted on a discount basis.

This means that the purchase price is always below the par value at maturity.

The difference between the purchase price and par value at maturity represents interest.

The pricing equation for a discount instrument is:idr = [(Pf - Po) / Pf] (360 / h)

where idr = discount rate Po = initial price of the instrument Pf = final price at maturity or sale, h = number of days in holding period.

The bond equivalent rate on discount securities The problems of a 360-day year for a rate quoted on a

discount basis can be handled by converting the discount rate to a bond equivalent rate: (ibe) ibe = [(Pf - Po) / Po] (365 / h)

Example: consider a $1 million T-bill with 182 days to maturity, price = $964,500.The discount rate is 7.02%,idr = [(1,000,000 - 964,500) / 1,000,000] (360 / 182)

= 0.072The bond equivalent rate is 7.38%:idr = [(1,000,000 - 964,500) / 964,500] (365 / 182)

= 0.0738The effective annual rate is 7.52%:i* = [1 + 0.0738 / (365 / 182)](365/ 182) - 1 = 0.0752

Yields on single-payment interest-bearing securities Some money market instruments, such as large

negotiable CD’s, Eurodollars, and federal funds, pay interest calculated against the par value of the security and make a single payment of interest and principal at maturity.

Example: consider a 182-day CD with a par value of $1,000,000 and a quoted rate of 7.02%. Actual interest paid at maturity is: (0.0702)(182 / 360) $1,000,000 = $35,490 The 365 day yield is:

i365 = 0.0702(365 / 360) = 0.0712 The effective annual rate is 7.31%:

i* = {1 + [0.0712 / (365 / 182)]}(365/182) - 1 = 0.0724

Summary of money market yield quotations and calculations Simple Interest is:

Discount Rate idr:

Money Mkt 360-day rate, i360

Bond equivalent 365 day rate, i365 or ibe:

Effective ann. interest rate,

DefinitionsPf = final valuePo = initial valueh=# of days in holding

periodDiscount Yield quotes:

Treasury billsRepurchase agreementsCommercial paperBankers acceptances

Interest-bearing, Single Payment:

Negotiable CDsFederal funds

o

ofs p

ppi

h

360

p

ppi

f

ofdr

h

360

p

ppi

o

of360

h

365

p

ppi

o

ofbe

1365/h

i1i

365/h*

PRICING SECURITIES

Chapter 6

Bank ManagementBank Management, 5th edition.5th edition.Timothy W. Koch and S. Scott MacDonaldTimothy W. Koch and S. Scott MacDonaldCopyright © 2003 by South-Western, a division of Thomson Learning