Valuation Models

BondsCommon stock

Key Features of a Bond

Par value: face amount; paid at maturity. Assume $1,000.

Coupon interest rate: stated interest rate. Multiply by par value to get dollar interest payment. Generally fixed.

Maturity: years until bond must be repaid. Declines over time.

Issue date: date when bond was issued.

Value = + . . . .

How can we value assets on the basis of expected future cash flows?

CF1

(1 + k)1

CF2

(1 + k)2

CFn

(1 + k)n

The discount rate k is the opportunity cost of capital and depends on: riskiness of cash flows. general level of interest rates.

How is the discount rate determined?

An annuity (the coupon payments).A lump sum (the maturity, or par, value to

be received in the future).

Value = INT(PVIFAi%, n ) + M(PVIFi%, n).

The cash flows of a bond consist of:

0 1

1001,000

Value = = $1,000.

Find the value of a 1-year 10% annual coupon bond when kd = 10%.

$1,1001.10

0 10

1001,000

1 2

100 100

Find the value of a similar 10-year bond.

Way to Solve

Using tables:

Value = INT(PVIFA10%,10)+ M(PVIF10%,10).

= 100*0.9090 + 100*0.9091

= 1000

Rule: When the required rate of return (kd) equals the coupon rate, the bond value (or price) equals the par value.

What would the value of thebonds be if kd = 14%?

1-year bond

Using tables:

Value = INT(PVIFA14%,1)+ M(PVIF14%,1).

= 100*0.9772 + 1000*0.8772

= 964.92

10-year bond

When kd rises above the coupon rate,bond values fall below par.They sell at a discount.

Using tables:

Value = INT(PVIFA14%,10)+ M(PVIF14%,10).

= 100*5.2164 + 1000*0.2697

= 791.34

What would the value of the bonds be if kd = 7%?

1-year bond

Using tables:

Value = INT(PVIFA7%,1)+ M(PVIF7%,1).

= 100*0.9346 + 1000*0.9346

= 1028.06

10-year bond

When kd falls below the coupon rate,bond values rise above par.They sell at a premium.

Using tables:

Value = INT(PVIFA7%,10)+ M(PVIF7%,10).

= 100*7.0236 + 1000*0.5083

= 1210.66

Value of 10% coupon bond over time:

13721211

1000

791 775

M

kd = 10%

kd = 7%

kd = 13%

30 20 10 0Years to Maturity

Summary

If kd remains constant: At maturity, the value of any bond must

equal its par value. Over time, the value of a premium bond

will decrease to its par value. Over time, the value of a discount bond

will increase to its par value. A par value bond will stay at its par valu

e.

Semiannual Bonds

1. Multiply years by 2 to get periods = 2n.2. Divide nominal rate by 2 to get

periodic rate = kd/2.3. Divide annual INT by 2 to get PMT

= INT/2.

INPUTS

OUTPUT

2n kd/2 OK INT/2 OK

N I/YR PV PMT FV

2(10) 14/2 100/220 7 50 1000N I/YR PV PMT FV

788.10

Find the value of 10-year, 10% coupon, semiannual bond if kd = 14%.

INPUTS

OUTPUT

Using tables:

Value = INT(PVIFA7%,20)+ M(PVIF7%,20).

= 50*10.5940 + 1000*0.2584

= 788.10

00 11 22 33 . . .. . . 88

100 100 100 . . . 100100 100 100 . . . 100

What is the cash flow stream of aperpetual bond with an annual

coupon of $100?

A perpetuity is a cash flow stream of equal payments at equal intervals into infinity.

Vperpetuity = .PMT

k

V10% = = $1000.

V13% = = $769.23.

V7% = = $1428.57.

V10% = = $1000.

V13% = = $769.23.

V7% = = $1428.57.

$1000.10$1000.10

$1000.13$1000.13

$1000.07$1000.07

P0 = + + . . . . ^ D1

(1 + k)

D2

(1 + k)2

Dฅ

(1 + k)ฅ

Stock value = PV of dividends

D1 = D0(1 + g)D2 = D1(1 + g)

.

.

.

Future Dividend Stream:

P0 = = .^ D1

ks - g

D0 (1 + g)

ks - g

If growth of dividends g isconstant, then:

Model requires: ks > g (otherwise results in negative

price).g constant forever.

D0 = 2.00 (already paid).

D1 = D0(1.06) = $2.12.

P0 = = =$21.20.

Last dividend = $2.00; g = 6%.

What is the value of Bon Temps’ stock given ks = 16%?

^ D1

ks - g

$2.12

0.16 - 0.06

P1 = D2/(ks - g) = 2.247/0.10 = $22.47.

^

What is Bon Temps’ value one year from now?

Note: Could also find P1 as follows:

P1 = P0 (1 + g) = $21.20(1.06) = $22.47.

^

^

ks = + g

= + 0.06 = 16%.

D1

P0

$2.12

$21.20

Constant growth model can berearranged to solve for return:

^

V =

= = $13.25.

Pmtk

If a stock’s dividends are not expected to grow over time (g = 0), then it is a perpetuity.

$2.12 0.16

Zero growth

Subnormal or Supernormal Growth

Cannot use constant growth model

Value the nonconstant & constant growth periods separately

If we have supernormal growth of 30% for 3 years, then a long-run constant

g = 6%, what is P0?^

0 ks=16% 1 2 3 4

g = 30% g = 30% g = 30% g = 6%

D0 = 2.00 2.60 3.38 4.394 4.658 2.241 2.512 2.815 P3 = = 46.5829.84237.41 = P0

4.658 0.10

00 11 22 33 44

$2.00$2.00 $2.00$2.00 $2.00$2.00 $2.12$2.12

0%0% 0%0% 0%0% 6%6%. . .. . .

Suppose g = 0 for 3 years, then g is constant at 6%.

ฅ

(1) PV 3-year, $2 annuity, 16% PV = PMT(PVIFA 16%,3)

= 2 * 2.2459 = $4.492.

(2)P3 = = $21.20.

PV(P3) = $13.58.

P0 = $4.49 + $13.58 = $18.07.

$2.120.10

What is the price, P0?