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Investments (EF5052) Investments (EF5052) |^ Dr.^ Ferenc^ Horvath

Week^1 Dr.^ Ferenc^ Horvath September^ 8,^2018

Contents • Risk and Return  Bodie, Kane, and Marcus, Chapter 5 • Highly suggested exercises:  Bodie, Kane, and Marcus, Chapter 5, Problems5. a‐c, 6. a‐b, 7, 10, 11, 12, 13, 15. (Solutions areavailable on Canvas.)  Bodie, Kane, and Marcus, Chapter 5, CFA Problems1‐7. (Solutions are available on Canvas.)Investments (EF5052) | Dr. Ferenc Horvath

Nominal^ rates and^ •^ $1 in T‐bills from 1926–2015 grew to $20.25 but witha real value of only $1.55.Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

Inflation,^1926 ‐^2015

-^ Tax liabilities are levied on nominal income. •^ After‐tax real interest rate:^ ௥೙೚೘ Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

After‐Tax Real^ Interest^ Rate ൈ ଵି௧ି ௜^ , ଵା௜^ where^ is the tax rate.

-^ Example:^ A one‐year bank deposit pays 8% interest. The inflation will be 5%over the next one‐year period. The tax on interest is 15%.What is the before‐tax real interest rate? What is the after‐tax realinterest rate?^ Answer:^ The before‐tax real interest rate is Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

Real^ and Nominal^ Interest^ Rates ଴.଴ଷ^ ൌ 0.0286.ଵ.଴ହ^ (The before‐tax real interest rate is approximately^ 0.08 െ 0.05 ൌ 0.03 ). ଴.଴଼ൈ^ ଵି଴.ଵହି ଴.଴ହ The after‐tax real interest rate is^ ൌ 0.0171. ଵା଴.଴ହ^

-^ Effective^ annual^ rate Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

(EAR)^ of^ a^ T‐yearinvestment: rate of return with annual (i.e.,once per year) compounding.

-^ Annualized percentage rate (APR)

of a T‐year

Two^ useful^ rate^ of^ return investment:^ rate of return with (1/T)‐times‐per‐year compounding.

concepts

State^ Prob. of State Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

r^ in State

a)^ 0.^

b)^ 0.^

c)^ 0.^

d)^ 0.^

Expected Return: •

Example

Variance^ and^ Standard • Variance (VAR): • Standard Deviation (STD):Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

Deviationof Return^2      ^2 S T D 

2 p^ s^ r^ s^ s

E^ r

^ ^ ^

Time^ Series^ Analysis of • True means and variances are unobservable. • Thus, they must be estimated.Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

Past^ Returns

-^ Estimated expected return: Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

-^ Note: the geometric mean^ ଵ^ ଶ^

೙ leads to a^ biased^ estimated expected return! • Thus, do^ not^ use geometric mean for estimation! • The arithmetic^ mean^ (see^ above)

is^ an^ unbiased

Estimating^ the^ Expected estimator.

Return

Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

௙ ௙ where^ •^ is^ the^ excess^ return௙^

(i.e.,^ the^ difference

Sharpe^ Ratio between the investment’s return and the risk‐freereturn), • and is the standard deviation of the excess௙ return. • The Sharpe ratio tells us how much excess returnwe can expect per volatility of excess return.

The^ Normal Investments (EF5052) |^ Dr.^ Ferenc^ Horvath

Distribution Mean = 10%, SD = 20%

-^ Third moment:^ 𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 ൌ 𝐸 Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

What^ if^ excess^ returns^ are^ not^ normally^ distributed? ଷ𝑟 െ 𝐸 𝑟^. 𝜎 • Fourth moment:ସ𝑟 െ 𝐸 𝑟^ 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 ൌ 𝐸^. 𝜎 • Note: many times, excess kurtosis is used:^ 𝑒𝑥𝑐𝑒𝑠𝑠 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 ൌ 𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 െ 3.

Normal^ and^ Skewed Investments^ (EF5052) |^ Dr.^ Ferenc^ Horvath

Distributions

Mean = 6%STD = 17%