7/29/2019 InterestRate_ReviewSheet

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Rate and Volatility Times - 1

Review Sheet: Interest Rate Time, Volatility Time, andExponential/Natural Log Functions

TB bill/note/bond price in nominal terms of a zero coupon claim that matures at

time T, the current (spot) price, also bond/note/billT

PV .

TR simple interest rate in nominal terms for funds invested or lent to time T. For

intervals T less than or equal to a year, the rate is a simple interest rate.For intervals T greater than a year, the rate is on a periodic bond-equivalentzero coupon basis.

TZ Zero coupon continuously compounded interest rate for funds invested in

zero coupon bonds (strips) or borrowed in a zero coupon issue for maturityT.

All other borrowed (short) and invested (long) bonds are aggregates of theT

B s.

T < 1 year, TB = T

1

1 R * T=

*T-Z Te

Example 1%,1

R1

B =

10.0099503*11

1 0.01*0.9901 e ,Z 0

1.99503%

T T T TGiven market R , solve for Z : Z ln 1+ R T / T , TZTZ ln e , ln inverseof e

Continuous discounting is maximal. Given TB , TZ < TR . Quoted rates drop as

compounding grows more frequent:

1 1,221 1,2

1 1,1 1 R 1 R 2

B R R 0.9975% ,

R Z1, 1n

1,

1Lime e

n 1 R n

Analogous to compounding, borrowing or investment can be split into a current/spot

and forward component. For 1% six month and one year simple rates, a forward

six month rate into the remaining six months of the year is implied, : 1/2 1R .

1/2 1/2 1 1z 2 z 2 z

1 1/2 1/2 1

1 1 1 e e e1 R 1 R 2 1 R 2

Solving

11/2 1

1/2

10.9

R2 1

195 ,

R 202%R 1/2 1 1/2 1Z 2 ln 1+ R 2 0.992558%

7/29/2019 InterestRate_ReviewSheet

2/2

Rate and Volatility Times - 2

Evolving, maturity T is broken up into n pieces, and h=T/n is each interval length. Inan interval, a normal random draw moves the price up or down. Volatility doesngrow with interest rate time, in t, but in the square root of time, t .

t hv h.t h t hS S e , v ~ N(0,1)

This equation updates value in each interval.

Since we must take logs to standardize the relation, the type of value evolution is called log-normal: h 0 h hLn S S v h, v ~ N(0,1). We use the square root because it matches the growth of risk/standard deviation

with reality, and because the formulation leads to the elegant Ito process for security prices. There are two logicaconditions needed for a successful process definition: 1) the series is reasonable over time, and 2) the valueoutcomes at any maturity should match the observed distribution of outcomes:

For example, we have 10 normal observations a day for a year, and the first day vhs are :{1.00643,0.38259,0.20658,0.73489,0.67623,0.79946,0.63541,0.75845,0.09569,0.58244}

Multiplying each observation by the 10% volatility, =0.1, and the Sqrt[1/3600]=1/60 of the h interval, we raise e by thpower, for the daily capital gain loss factors:{0.99832,0.99936,0.99966,1.00123,0.99887,0.99867,0.99894,0.99874,0.99984,1.00097}

We compound these daily factors for the value of the $1 in each of the 10 days:{1.,0.99832,0.99769,0.99734,0.99857,0.99744,0.99611,0.99506,0.9938,0.99364,0.99461}

Random Normal, vh

2 4 6 8 10

-1.0

-0.5

0.5

Value Sequence

2 6 8 10

0.995

0.996

0.997

0.998

0.999

1.000

One year of values

500 1000 1500 2000 2500 3000 3500

1.05

1.10

1.15

Five hundred such outcomes:

500 1000 1500 2000 2500 3000 3500

0.8

0.9

1.0

1.1

1.2

1.3

1.4

End of First Quarter (Black 900intervals), and Year End (Blue 3600 Intervals) Outcomes