Interest, Discount & Return Rates

Learning Objectives

Present and future value Discount rates Rate compounding

Nominal and effective rates Interest rates Inflation adjustments

Nominal and real rates Return rates

Simple and natural log Mean return rates

Arithmetic and Geometric

2

Present Value: No Intermediate Cash Flow 3

N

N

k)(1PV FV

k)(1FV

PV

0 1 2 N

PV

FV

FV: Future valuePV: Present valuek: effective periodic discount or future value rateN: number of periods

: Discount factor

: Future value factor

Nk)(11

Nk)(1

Present Value w/ No Intermediate Cash Flow

Example k = annual effective discount rate = 5.116% N = 5 years PV =$100.00

FV = PV·(1+.05116)5 = $128.33

i=0 1 2 3 4 5

PV

FV

4

Present Value w/ periodic compounding and no intermediate cash flow

N

Nm

k1PV

mk

1PVFV

Annual effective rate includes effect of

periodic compounding Annual nominal rate does not include

effect of periodic compounding Example

5% annual compounded monthly k = 5%, annual nominal rate m = 12, compounding frequency

Annual effective rate is

N is number of years Effective and nominal monthly rate

%116.5112%5

1k12

%417.1%)116.51(m%5 m

1

5

521

%116.51FV

PV

125%

1

FVPV

5

Using annual nominal rate

Using annual effective rate

N

Nm

k1

FV

mk

1

FVPV

ki is effective annual rate

kj is nominal annual rate

Present Value w/ periodic compounding and intermediate cash flow

6

N

1ii

i

i0 )k1(

CFV

i 0 1 2 m·N

PV

CFi

Nm

1jj

j

j0

mk

1

CFV

Interest Rates

Rate of return on debt securities Bonds

Fixed ‘coupon’ rate Certificates of deposit Notes

Floating rate Mortgages Commercial paper

7

Govt Rates

BLS CPI

BLS CPI Chart

BLS FAQs

CD Rates

Interest Rates

BBA Libor rates for 1 June April 2011 (Simple annual rates)

Jun-121-Jun

EUR USD JPY CHF AUDs/n-o/n 0.25750 s/n-o/n 0.15700 s/n-o/n 0.10871 s/n-o/n 0.01500 s/n-o/n 3.898001w 0.27157 1w 0.19210 1w 0.11800 1w 0.02333 1w 3.942002w 0.27643 2w 0.21475 2w 0.12514 2w 0.02917 2w 3.969001m 0.33179 1m 0.23975 1m 0.14429 1m 0.04250 1m 3.989002m 0.41714 2m 0.34675 2m 0.15786 2m 0.06667 2m 4.022003m 0.59050 3m 0.46785 3m 0.19571 3m 0.09583 3m 4.110004m 0.69814 4m 0.56810 4m 0.23857 4m 0.12333 4m 4.192005m 0.78843 5m 0.65065 5m 0.29371 5m 0.15250 5m 4.264006m 0.90079 6m 0.73790 6m 0.33586 6m 0.17833 6m 4.316007m 0.95979 7m 0.80000 7m 0.38586 7m 0.19833 7m 4.364008m 1.01786 8m 0.85470 8m 0.43229 8m 0.22750 8m 4.404009m 1.06571 9m 0.90955 9m 0.47443 9m 0.26283 9m 4.4460010m 1.11643 10m 0.96265 10m 0.50157 10m 0.30033 10m 4.4920011m 1.16607 11m 1.01295 11m 0.52729 11m 0.34450 11m 4.5430012m 1.22393 12m 1.07070 12m 0.55229 12m 0.38617 12m 4.59500

s/n is spot/next and o/n is overnight

Inflation Adjustments

n = nominal rate r = real rate i = inflation rate

Example n=3% i=2% r =0.98% 1%

Cash flows and discount rates must be congruent Nominal is typical

inr

1i)(1n)(1

r

i)(1r)(1n)(1

9

5.000%

5.020%

5.040%

5.060%

5.080%

5.100%

5.120%

5.140%

0 5 10 15 20

Effec

tive

Annu

al R

ate

Annual Compounding Periods (m)

Continuous Compounding 10

?mk

1iml

m gcompoundin continous For

mk

1PVFV

m

w

m

k is annual nominal rate, m is number of compounding periods per year

5% annual nominal rate is e.05 – 1 continuously compounded annual effective rate: 5.1271%

kkw

w

m

m

1w

w

ew1

1imlmk

1iml

ew1

1iml

)m : w as 1k (For

kwm and mk

w1

Therefore

km

w Define

Continuous Compounding 11

1ii

1i

ii

v

1i

i

v1ii

v

SlnSln

SS

lnv

eSS

eSS

ePVFV

i

i

FV = PV·ek

k = 5%

ek is the future value factore.05 = 1.051271

e-k is the discount factor e-.05 = 0.951229

ek-1 is the continuously compounded ratee.05-1 = 0.051271

Now utilize with the stock price model

Continuously compounded future value factor

Natural log rate of return

Mean Rate: Simple Return Rates 12

S

SS

SSr

1i1i

1iii

What’s the average or mean quarterly simple rate of return?

%6691.4

3.4483%5.4545%3.7736%6.0000%

41

a

t i Si ri

0.00 0 100.00$ 0.25 1 106.00$ 6.0000%0.50 2 110.00$ 3.7736%0.75 3 116.00$ 5.4545%1.00 4 120.00$ 3.4483%

Example: Quarterly historical price record for 1 year

Compute the sequence of simple rates of return from security price, S

rn1

am

1ii

n = number of periods in a historical return record, associated with n+1 prices

m = number of periods in a year

Mean Rate: Simple Return Rates 13

03.120$)046691.01(100$ )a1(SS 4404

No, it over estimates the price

What’s the mean rate of return that results in the actual price, S4 ?

Does this mean rate over 4 quarters reproduce the stock price at the end of 1 year ?

That’s the geometric mean rate of return, g

1SS

1)r1(gm1

0

mm1

m

1ii

4.6635%1100120

g

4.6635%11.0344831.0545451.0377361.060000g

41

41

00.120$)046635.01(100$)g1(SS 4404

Periodic Rate

MeanPeriodic

Mean Rate

Arithmetic aGeometric g

v Arithmetic u

r

Mean Rate: Simple Return Rates

a is the periodic (e.g., quarterly) arithmetic mean rate of return

g is the periodic (e.g., quarterly) geometric mean rate of return

‘Periodic’ herein means daily, weekly, monthly, quarterly, but not annual

So how do we time scale these periodic mean return rates? For example: Scale the quarterly mean rates to an annual mean return

Via multiplication ?

Via compounding

NO

026% 20.

1- 4.6691%1 1-a1

18.6541% 4.6635% · 4 g · m 18.6764% 4.6691% · 4 a · m

4m

%000.021- 4.6635%1 1-g1 4m

But compounding the geometric mean rate does produce the annual rate – by definition - but ignores the intermediate rate fluctuations

Mean Rate: Log Return Rates 15

1ii

1i

ii

SlnSln

SS

lnv

n

1iiv

n1

u

The periodic mean natural log return rate is

Now the natural log rate of return

%5580.4

3.3902%5.3110%3.7041%5.8269%41

u

18.2322%4.5580%4u4μ

Multiply the quarterly natural log mean return rate by 4 to get the annual log mean return rate?

t i Si ri vi

0.00 0 100.00$ 0.25 1 106.00$ 6.0000% 5.8269%0.50 2 110.00$ 3.7736% 3.7041%0.75 3 116.00$ 5.4545% 5.3110%1.00 4 120.00$ 3.4483% 3.3902%

Average 4.6691% 4.5580%

Mean Rate of Return 16

$120.00 e$100.00eS S

$120.00 e$100.00eSS

.182322μ04

.045580*4u404

Now check whether the natural log mean return rate reproduces the year end stock price

Annual and other accumulated rates of return can be determined by multiplying the log mean periodic rate of return

factor discount annual e

factor value future annual e

returnof rate annualμ

μ

μ

Another Example17

%0000.06.7659%-2.7652%-14.6603%5.1293%-41

u

%3800.06.5421%-2.7273%-15.7895%5.0000%-41

a

%0000.01100$100$

%0000.010.03460.97271579.10.9500g

41

41

00.100$eSeSS 000.0*40

u404

00.100$)0000.01(100$)g1(SS 4404

53.101$)3800.01(100$)a1(SS 4404

t i Si ri vi

0.00 0 100.00$ 0.25 1 95.00$ -5.0000% -5.1293%0.50 2 110.00$ 15.7895% 14.6603%0.75 3 107.00$ -2.7273% -2.7652%1.00 4 100.00$ -6.5421% -6.7659%

Average 0.3800% 0.0000%

1eg

g1lnueg1

eg1 :Note

u

u

umm

0100200300400500600700800900

1000110012001300140015001600

Jan-50 Jun-55 Dec-60 Jun-66 Nov-71 May-77 Nov-82 May-88 Oct-93 Apr-99 Oct-04 Mar-10

SPX (^GSPX) Monthly Prices: 1950 - 201218

753 Monthly PricesJanuary 1950 to September 2012

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

Jan-50 Jun-55 Dec-60 Jun-66 Nov-71 May-77 Nov-82 May-88 Oct-93 Apr-99 Oct-04 Mar-10

SPX Monthly Return Rates: 1950 - 201219

752 monthly return ratesJanuary 1950 to September 2012

SPX Monthly Return Rates: 1950 - 201220

-40% -35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20%Monthly Natural Log Return Rates

Simple mean rate of return

Future value factor

Natural log mean rate of return

SPX Monthly Ln Return Rates: 1950 - 201221

%979.3vr1ln

%059.104er1

%059.4r

ii

vi

i

i

End Date Adj Close S r 1+r ln(1+r) v ev

9/4/2012 1437.92 2.228% 102.228% 2.204% 2.204% 102.228%8/1/2012 1406.58 1.976% 101.976% 1.957% 1.957% 101.976%7/2/2012 1379.32 1.260% 101.260% 1.252% 1.252% 101.260%6/1/2012 1362.16 3.955% 103.955% 3.879% 3.879% 103.955%5/1/2012 1310.33 -6.265% 93.735% -6.470% -6.470% 93.735%4/2/2012 1397.91 -0.750% 99.250% -0.753% -0.753% 99.250%3/1/2012 1408.47 3.133% 103.133% 3.085% 3.085% 103.133%2/1/2012 1365.68 4.059% 104.059% 3.979% 3.979% 104.059%1/3/2012 1312.41 4.358% 104.358% 4.266% 4.266% 104.358%

12/1/2011 1257.60 0.853% 100.853% 0.850% 0.850% 100.853%11/1/2011 1246.96 -0.506% 99.494% -0.507% -0.507% 99.494%10/3/2011 1253.30 10.772% 110.772% 10.231% 10.231% 110.772%

9/1/2011 1131.42 -7.176% 92.824% -7.447% -7.447% 92.824%

SPX Monthly Mean Rates: 1950 - 201122

.59148%

1)]r(11)]r(1g752

1752

1ii

n1

n

1ii

%58973.

v752

1 )rln(1

n1

u 752

1ii

n

1ii

r 1+r ln(1+r) v ev

Mean E[r]=a E[1+r] E[ln(1+r)] E[v]=u E[ev]0.68045% 100.68045% 0.58973% 0.58973% 100.68045%

%68045.

r752

1r

n1

a752

1ii

n

1ii

g0.59148%

0100200300400500600700800900

1000110012001300140015001600

Jan-50 Jun-55 Dec-60 Jun-66 Nov-71May-77Nov-82May-88 Oct-93 Apr-99 Oct-04 Mar-10

actual

arith mean

geom mean

nat log mean

SPX Monthly Prices: 1950 - 201123

u1ii

1ii

1ii

es s

g1s sa1s s

SPX Monthly Variance Rates: 1950 - 20112424

0018077.

%58973.v257

1

uvn1

s

svarvr1lnarv

752

1i

2i

n

1i

2i

2

2

0017835.

%68045.r752

1

arn1

d]e[arv]r1[arv]r[arv

752

1i

2i

n

1i

2i

2v

r 1+r ln(1+r) v ev

SD[r]=d SD[1+r]=d SD[ln(1+r)]=s SD[v]=s SD[ev]=d

0.17835% 0.17835% 0.18077% 0.18077% 0.17835%Var[r]=d2 Var[1+r]=d2 Var[ln(1+r)]=s2 Var[v]=s2 Var[ev]=d2

0.0017835 0.0017835 0.0018077 0.0018077 0.0017835

Standard Deviation

Variance

Next Topics Building These Concepts

Equities Stochastic Processes

Martingales PDF (probability density function) for future value factors

Central Limit Theorems PDF for return rates Convert statistics between return rate PDFs

Bonds Yield to maturity Mortgages etc

25