4-1

Chapter 4

Time Value of Money

4-2

Example of a Time Line

Points in time

Time

3210

1 2 3

Periods

4-3

Time Line

1 + R

10

1

1 1.10

Flat Term Structure—One Period

4-4

Time Line

1 + R

20

1

1

(1 + R)(1 + R)

Flat Term Structure—Two Periods

4-5

.21.101.020.01

)10.0()10.0(21)10.1(

RR21)R1(2 timeat value Future

22

22

4-6

Future Value of One Dollar Received at Time 0

N = Number of periods I/YR = Interest ratePMT = 0FV = ?PV = -1

4-7

Time Line

3210

PV1 =

3)R1(1

2)R1(1

1)R1(1

PV2 =

PV3 =

Present Value

4-8

Present Value of One Dollar

N = Number of periods I/YR = Interest ratePMT = 0FV = 1PV = -Price

4-9

Present Value of $1

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264

3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513

4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830

5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209

6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645

7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132

8 0.9235 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665

9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

4-10

Annuities

Suppose that there is an annuity of 1 dollar at each future point in time.

3210

$1 Time

. . . n

. . .$1 $1 $1

4-11

In the case of an annuity for 2 periods, the present value is

.74.1$

)10.1(1

10.11

)R1(1

R11luePresent Va

2

2

4-12

n2 )R1(1

)R1(1

R11PVA

annuityof value

Present

.n time

at $1PV

2 timeat $1

PV

1 timeat $1

PV

.)R1(

11R1PVA

n

4-13

Present Value of a 1 Dollar Annuity

N = Number of periods I/YR = Interest ratePMT = 1FV = 0PV = ?

4-14

Present Value of an Annuity of $1

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.8080 1.7833 1.7591 1.7355

3 2.9410 2.8839 2.8286 2.7751 2.7232 2.6730 2.6243 2.5771 2.5313 2.4869

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3872 3.3121 3.2397 3.1688

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 4.1002 3.9927 3.8897 3.7908

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.7665 4.6229 4.4859 4.3553

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.3893 5.2064 5.0330 4.8684

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.9713 5.7466 5.5648 5.3349

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.5152 6.2469 5.9952 5.7590

10 9.4713 8.9826 8.5302 8.1109 7.7217 7.3601 7.0236 6.7101 6.4177 6.1446

1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

4-15

Future Value of a 1 Dollar Annuity

N = Number of periods I/YR = Interest ratePMT = 1FVA = ?PV = 0

.1)1(

)1(0

R

R

RPVAFVAn

nn

4-16

Time30

20 1

40

.33.60$)10.1(

4010.1

30

)R1(40

R130luePresent Va

2

2

Adding Present Value

4-17

Bond Cash Flows Received by Lender

Points in time

-P

210

+c

. . . n

. . .+c +c + par

4-18

.n time

parcPV

2 timec

PV

1 timec

PV

)R1(parc

)R1(c

R1cPPrice

n2

Price = PV of Future Cash Flows

4-19

.)R1(

par)R1(

11R1c

R)(1par

annuity ofvaluePresent cPPrice

nn

n

Two-period Bond

Time8

20 1

8 + 100P

4-20

53.96$)10.1(

10081.10

8P2

22 )10.1(

100

)10.1(

11

10.0

18

P

4-21

Price of a Bond

N = Number of periods I/YR = Interest ratePMT = CouponFV = Par valuePV = ?

4-22

Cash Flows

n0

parS

.)R1(

parPriceS

nStrip

Strips

4-23

Yield to Maturity

.n time

parcPV

2 timec

PV

1 timec

PV

…

y)(1parc

)y1(c

y1cPPrice n2

…

4-24

Finding the Yield to Maturity by Trial and Error

Suppose that we have a 2-period bond with a par value of $100, a coupon of nine dollars, and a price of $98.26.

20 1

Time98.26 = +

y19 2)y1(

109

4-25

.100)09.1(

1091.09

9PV2

Since 9% gives too high a present value, we try a higher rate.

Suppose we try 9%.

.26.98)10.1(

1091.10

9PV2

4-26

Yield to Maturity for Bond

N = Number of periods I/YR = ?PMT = CouponFV = Par valuePV = - Price

4-27

Yield to Maturity for Treasury Strips

The yield to maturity for treasury strips can be found analytically as follows.

It can also be found by using the calculator.

4-28

%.808.0y

08.173.85$

100$y1

Pricepar

y1

y)(1par

Price

2/1

n/1

Strip

nStrip

4-29

Relationship between Bond Price and Coupons

Price

Coupon

Premium

Par

Discount

Strip = Zerocoupon

C0 Cpar

P = c[PVA] + par[PV]

Slope

Intercept = par[PV]= Price of Strip

4-30

Yield to Maturity (y), Current Yield (c/P) and Stated Yield (c/par)

Discount bond y > c/P > c/parP < par

Par bond y = c/P = c/parP = par

Premium bond y < c/P < c/parP > par

4-31

9

20

98.26

1

9+100

y = 10%c/P = 9/98.26 = 9.1594c/par = 9%y > c/P > c/par

4-32

9

20

98.26

1

9 + 100

y = ?

9

20

98.26

1

9 + 98.26

4-33

Perpetual Bonds

Perpetual bonds are bonds that never mature.

They simply pay a coupon indefinitely.

20 1

TimeP c c . . .

4-34

YieldCoupon

yc P Price

2)y1(

cy1

cPPrice …

.Pcy

.80$10.08$

ycP Price

4-35

P1 + c

0

P0

1

4-36

0

P0

1

= P1 + c

1 + HPR

HPR = one-period holding period return

4-37

.P

cPPHPR

price Beginning

Couponprice

Beginningprice

Ending

returnperiod

Holding

0

01

Holding Period Return

4-38

.yieldCurrent

(loss) gaincapitalPercent HPR

Pc

PPPHPR

priceBeginning

Coupon

priceBeginning

priceBeginning

priceEnding

returnperiod

Holding

00

01

4-39

Bond Price and Maturity for Constant Interest Rates

$

Maturity timeIssue date

Par

Premium

Discount

4-40

Bond Price and Maturity for Non-constant Interest Rates

$

Maturity timeIssue date

Par

Constant rates

Constant rates

.

.

Rising rates

1

Falling rates

4-41

Semi-annual Coupons If half of the annual coupon is paid every

six months, the price of a bond is the present value of the payments discounted at the “semi-annual yield to maturity.”

The semi-annual yield to maturity is technically the annualized, semi-annually compounded yield to maturity.

We will denote the semi-annual yield to maturity by the letter i.

4-42

2.00 1.0

TimeP

Semi-annual Coupons1.50.5

2c

2c

2c par

2c

.

yearsntimeat

receivedparc/2

PV

year1timeat

receivedc/2PV

months 6timeat

receivedc/2PV

i/21

parc/2

i/21c/2

i/21c/2P

n221

4-43

Bond Cash Flows for Semi-annual Coupons

Points in Time (Years)

0 0.50 1.00 1.50 2.00

-P +$4.50 +$4.50 +$4.50 +$104.50

.2270.98$

/210.0150.4$

/210.0150.4$

/210.0150.4$P

321

4/210.01

100$50.4$

4-44

Semi-annual Coupon.Finding Bond Price.

N = Number of periods = 2x number of years

I/YR = Semi-annual yield divided by 2PMT = Coupon = c/2FV = Par valuePV = ?

4-45

Bond with Semi-annual Coupon. Finding the Yield to Maturity.

N = Number of periods = 2x number of years

I/YR = ?PMT = Coupon = c/2FV = Par valuePV = -Price

4-46

Partial Periods until Next Coupon

Suppose that a half year has k days and that j days have elapsed since the last coupon payment.

2.00 1.0

Years

Now

1.50.5

j

k

4-47

The seller of a bond is entitled to receive accrued interest for the days that the seller has held the bond, that is, for j days.

The seller receives j/k% of the next coupon.

.couponannual-Semi

period month6 for Days

couponlast since Days

2c

kj

interestAccrued

4-48

Cash Flows

n0 1.0

P

1.50.5

2c

2c

2c par

2c

Now

j

k

. . .

4-49

2c

kj

P

2c

kj

P

.value0 Time

now to0 time from

forward Brings

interestAccruedPrice

n

n

i221

j/k

j/k-2

j/k-3j/k-2j/k-1

i/21

parc/2

i/21

c/2

i/21

c/2

21

i/21

parc/2

i/21

c/2

i/21

c/2

i/21

c/2

4-50

Suppose j = 90 Days and k = 180 Days.

.65.100$

/210.01100$50.4$

/210.0150.4$

/210.0150.4$

/210.0150.4$50.4$

21P

-1/243-1/2

2-1/2-1/21

4-51

6527.100$25.2$P

2270.98$024695.125.2$P

50.4PMT5YR/I100FV4N05.150.4$

2

1P

Inflows Future of Value0 Time05.150.4$

2

1P

2/1

2/1

Buyer pays $100.65Bond price = $100.65 – $2.25 = $98.40

4-52

.40.98$

50.4$21

/210.01100$50.4$

/210.0150.4$

/210.0150.4$

/210.0150.4$P

1/2-41/2-3

1/2-21/2-1

4-53

Truth in Lending Law

APR = Annual percentage rate

Annual compounding: y = APR

Semi-annual compounding: ii/2 = Half year rateAPR = (1 + i/2)2 – 1 = i + (i2/4)

Quarterly compounding: qq/4 = Rate per quarterAPR = (1 + q/4)4 – 1

Monthly compounding: mMonthly rate = m/12APR = (1 + m/12)12 – 1