2.3 - Quantitative Foundations

Question 1

Simple Interest

Answer 1

• Most Basic, actual return in a period

(End/Start) - 1 =R

Question 2

Compounding Returns

Answer 2

• non annual compounding: quoted rate does not equal simple rate (or actual rate)
• more frequent compounding = higher equivalent annual simple rate

R = (1 + (actual rate/m))^m - 1

Where m is number of compounding periods per year

Question 3

Continuous Compounding

Answer 3

• As ‘m’ gets larger toward infinity

R = e^quoted rate - 1

‘e’ is just a constant —> 2.718

Ex: if a continuously compounding return is 10% what is the simple return?

Ans: R = e^0.10 - 1 = .1052 or 10.52%

Question 4

Log Returns

Answer 4

• Continuously compounding returns also called log returns
• going from a simple (actual) return R to a continuously compounding return:

Rcontcomp. = ln(1 + R)

Ex: if simple return is 10.52% the cont.comp. rate is ln(1.1052) = 10%

Question 5

Multi-period Returns

Answer 5

• Simple interest returns are combined using multiplication (geometric average):

[(1 + R1)(1+ R2)(1+R3)…. ] ^1/n - 1

**note - raising to a fraction is same as taking nth root (ie: (1.2749)^1/3 is same as cubed root (3 √1.2749)

• Log Returns combined using addition (arithmetic average):

[ln(1+R1)+ln(1+R2)+ln(1+R3)…] / n

Question 6

Converting Arithmetic Mean Return to Geo. Mean Return

Answer 6

e^(arithmetic mean log return) - 1

Question 7

Returns on Notional Principal

Answer 7

• returns based on exposure
• forward contracts have zero value at initiation— causes problems in calculating returns
• fully collateralized
  Rcoll = ln(1+R) + Rf
• partially collateralized
  Rpcoll = L * ln(1+R) + Rf

Question 8

IRR Calculation

Answer 8

CF0 + [CF1 / (1+IRR)] + [CF2 / (1+IRR)] + ….. = 0