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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
Bond Valuation - Terminology
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
Bond Valuation - Terminology
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
Besley: Chapter 7 11
Tabular Solutions
M = 1,000N = 10 yearsKd = 10%
INT = 100
Input:
Output:N I/Y PV PMT FV10 10
1,000
100 1,000
Besley: Chapter 7 12
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
Besley: Chapter 7 13
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)
Besley: Chapter 7 14
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%
Besley: Chapter 7 15
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
Besley: Chapter 7 16
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%
Besley: Chapter 7 17
Yields
Current Yield: 10.00%Capital Gains Yield: 35.54%Total Yield: 45.54%
Besley: Chapter 7 18
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.
Besley: Chapter 7 19
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.
Besley: Chapter 7 20
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.
Besley: Chapter 7 21
Bond Values with Semiannual Compounding
2Nd
2N
1tt
dd
2k1
M
2k1
2INT
V
Besley: Chapter 7 22
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.
Besley: Chapter 7 23
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..
Besley: Chapter 7 24
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
Besley: Chapter 7 25
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
Besley: Chapter 7 26
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
$200.00
1 2 3 4 5
Interest Rate Risk
Effect of Reinvestment Rate Risk
Effect of Interest Rate Risk
Besley: Chapter 7 27
Interest Rate RiskReinvestment Risk
Current Market Interest Rate, kd
1-Year Bond 14-Year Bond5% 1,095.24$ 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
Besley: Chapter 7 28
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