1 Valuation Concepts Part 1: Bond Valuation. Besley: Chapter 7 2 Basic Valuation The value of any...

1

Valuation Concepts

Part 1: Bond Valuation

Besley: Chapter 7 2

Basic Valuation

The value of any asset is based on the present value of the future cash flows the asset is expected to produce.

0 n-11 2 n

CF1

›

CF2›

CFn-1

›

CFn

› PV of CFn

›

PV of CF1

›

PV of CF2

›

PV of CFn-1

›

Value

Besley: Chapter 7 3

Basic Valuation

Asset Value = V = CF1/(1+k)1 + CF2/(1+k)2 + . . . + CFn/(1+k)n

Where: CFt = Anticipated cash flow (CF) in period tk = required rate of return for an asset in this

class

A larger asset value (V) will result from: A larger expected CF A lower required return rate

Besley: Chapter 7 4

Bond Valuation - Terminology

Bond: Long term debt instrumentPrincipal Amount: the amount that the debtor

borrows and promises to repay at some future date. Also know as Maturity Value; Par Value and Face ValueBonds typically have a face value of $1,000 or some

multiple thereof.

Besley: Chapter 7 5


Coupon Payment: the fixed dollar payment per period (usually each six months)

Coupon Interest Rate: the annual interest rate paid on the bond (=Coupon Payment/Face Value)

Maturity Date: the date on which the maturity value must be repaid to the bondholder

Original Maturity: the number of years to maturity as of the bonds issue date

Besley: Chapter 7 6


Call Provision: a provision within the bond terms that allows the bond issuer to “call back” the bonds and repay them at a date prior to the maturity date (a call provision usually calls for the issuer to pay the par vale plus one year’s interest when the bonds are called

A Call Provision allows the debtor to take advantage of lowering interest rates by issuing new debt at the lower rate and using the proceeds of the sale to retire the older (more expensive) debt.

Besley: Chapter 7 7

Basic Bond Valuation Model0 n-11 2 n

INT INT INT INTPVA of INT

PV of MBond Value = Vd

M

kd = rate of return on a debt instrument

Bond Value = Vd = INT/(1+kd)1 + INT/(1+kd)2 + . . . + INT/(1+kd)n + M/(1+kd)n

= t=1

n INT(1 + kd)t

M(1 + kd)n

= INT(PVIFAkd,n) + M(PVIFkd

,n)

Besley: Chapter 7 8

Bond Valuation

M = 1,000N = 10 yearsKd = 10%

INT= 100

0 1 2 8543 6 7 9 10

100 100 100 100 100 100 100 100 100 1001,000-90.91

-82.64-75.13-68.30-62.09-56.45-51.32-46.65-42.41-38.55

-385.54-1,000.00

Besley: Chapter 7 9

Numerical Solutions

M = 1,000N = 10 yearsKd = 10%INT = 100

Vd = INT{ [1-(1/(1+kd)N]/kd} + M[1/(1+kd)N]= 100{ [1-(1/(1.10)10]/0.10} + 1,000[1/(1.10)10]= 100{6.1446} + 1,000[0.3855]= 614.4567 + 385.5433= 1,000.00

Besley: Chapter 7 10

Tabular Solutions

M = 1,000N = 10 yearsKd = 10%INT = 100

Vd = INT(PVIFAkd,N) + M(PVIFkd,N)

= INT(PVIFA10%,10) + M(PVIF10%,10) = 100(6.1446) + 1,000(0.3855= 614.46 + 385.5= 999.96 = 1,000.00


Tabular Solutions

M = 1,000N = 10 yearsKd = 10%

INT = 100

Input:

Output:N I/Y PV PMT FV10 10

1,000

100 1,000


Bond Valuation

The price of a bond being purchased between interest payments must be adjusted for the earned interest being purchased.

1,000+(180/360)(100) = 1,050Basic Price

Coupon Payment

Total Time

Time Interest Earned

Bond Price


Bond Valuation

Suppose that 1-year after the bond is issued, interest rates have fallen to 5%kd drops below the Coupon Rate (5% vs. 10%)Coupon Payments and Maturity Value remain

constantVd increases to $1,355.38

Vd = INT(PVIFA5%,9)+M(PVIF5%,9)Vd = 100(7.1078)+1,000(0.6446)


Bond Valuation

Suppose you decide to sell the bond after 1-year after rates have fallen (the new price is the $1,355.38).

Sell Price: $1,355.38Coupons: $100Total Inflows: $1,455.38Price Paid: $1,000.00Net Profit: $455.38Yield: $455.38/$1,000.00 = 45.538%


Yields

The yield (total rate of return) is comprised of two components:

Current Yield (Interest Yield): Annual Coupon Payment divided by its current market value

Capital Gains Yield: Capital Gain on the bond divided by the current market value


Yields

Current YieldCurrent Yield = INT/Vd

= 100/1,000= 10%

Capital Gains YieldCapital Gains Yield = (Vd,End – Vd,Begin)/Vd,Begin

= (1,355.38 – 1,000)/1,000= 35.54%


Yields

Current Yield: 10.00%Capital Gains Yield: 35.54%Total Yield: 45.54%


YieldsImpact of Interest Rate Change on Bond Valuations

500.00

1,000.00

1,500.00

1 2 3 4 5 6 7 8 9 10

Year

Bon

d V

alue

kd=5% kd=10% kd=15%

Year kd=5% kd=10% kd=15%1 1,355.39 1,000.00 761.42 2 1,323.16 1,000.00 775.63 3 1,289.32 1,000.00 791.98 4 1,253.78 1,000.00 810.78 5 1,216.47 1,000.00 832.39 6 1,177.30 1,000.00 857.25 7 1,136.16 1,000.00 885.84 8 1,092.97 1,000.00 918.71 9 1,047.62 1,000.00 956.52 10 1,000.00 1,000.00 1,000.00

This assumes that rates do not change after year one, if interest rates do fluctuate, so will the price.


Bond Movement

Bond prices are inversely related to interest rates: When interest rates increase bond prices will fall and the

bond will trade at a discount (bond price is less than maturity value)

When interest rates decrease bond prices will rise and the bond will trade at a premium (bond price is greater than maturity value)

Market Value will always approach par, the closer it gets to maturity.


Yield to Maturity (YTM)

Yield to Maturity (YTM): The average rate of return earned on a bond if held to maturity.

Approximate YTM = (Annual Interest+Accrued Capital Gains)/Avg. Value of Bond

= INT+[(M-Vd)/N]

[(2(Vd)+M)/3]

This is an approximation and does not include the TVM.


Bond Values with Semiannual Compounding

2Nd

2N

1tt

dd

2k1

M

2k1

2INT

V


Bond Values with Semiannual CompoundingVd = (INT/2)(PVIFAkd/2,2N) + M(PVIFkd/2,2N)

NOTE: The maturity value must be discounted at the same percentage and time period as the annuity.


Interest Rate Risk

Two types of interest rate risks:Interest Rate Price Risk: The risk that changing

interest rates will adversely affect bond prices.Increasing interest rates will force the bond prices down exposing the bond holder to a loss on the bond price.

Interest Rate Reinvestment Risk: The risk that changing interest rates will cause the cash flows generated by the bond to be invested at a rate other than the original bond rate..


Interest Rate RiskBond Price Reaction to Change in Interest Rates

$800.00

$900.00

$1,000.00

$1,100.00

$1,200.00

1 2 3 4 5

Year

k5% k10% k15%

Decrease in Bond Price Due to 5% Increase in Rate


Interest Rate Risk

Effect of Interest Rate RiskYear k5% k10% k15%

1 $177.30 $0.00 ($142.75)2 $136.16 $0.00 ($114.16)3 $92.97 $0.00 ($81.29)4 $47.62 $0.00 ($43.48)5 $0.00 $0.00 $0.00

Net Effect of Interest Rate ChangeYear k5% k10% k15%

1 $172.30 $0.00 ($137.75)2 $120.41 $0.00 ($97.91)3 $59.88 $0.00 ($46.05)4 ($10.33) $0.00 $20.255 ($91.37) $0.00 $103.81

Earnings on Coupon PaymentsYear k5% k10% k15%

1 $5.00 $10.00 $15.002 $15.25 $31.00 $47.253 $31.01 $64.10 $99.344 $52.56 $110.51 $174.245 $80.19 $171.56 $275.37

Effect of Reinvestment Rate RiskYear k5% k10% k15%

1 ($5.00) $0.00 $5.002 ($15.75) $0.00 $16.253 ($33.09) $0.00 $35.244 ($57.95) $0.00 $63.735 ($91.37) $0.00 $103.81


Effect of 5% Drop in Interest Rate to Bond Price and Reinvestment

($150.00)

($100.00)

($50.00)

$0.00

$50.00

$100.00

$150.00

$200.00

1 2 3 4 5

Interest Rate Risk

Effect of Reinvestment Rate Risk

Effect of Interest Rate Risk


Interest Rate RiskReinvestment Risk

Current Market Interest Rate, kd

1-Year Bond 14-Year Bond5% 1,095.24$ 1,989.86$

10% 1,045.45$ 1,368.33$ 15% 1,000.00$ 1,000.00$ 20% 958.33$ 769.47$ 25% 920.00$ 617.59$

Value of


Interest Rate RiskReinvestment Risk

Interest Rate, k d (%)

BondValue

($) 2,000

1,500

1,000

500

0 5 10 15 20 25

14-Year Bond

1-Year Bond

1 Valuation Concepts Part 1: Bond Valuation. Besley: Chapter 7 2 Basic Valuation The value of any...

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