8/3/2019 Invst. Cheat Sheet

1/1

Returns with dividends: R t+1 =(Pt+1 + Dt+1 - Pt) / Pt

Value in 2 periods: Pt+2 = Pt(1 + rt+1)(1 + rt+2)

2 year net return: rt;t+2 = (1+rt+1)(1+rt+2)-1

Geometric avg for 3 yrs: [(1+r)*(1+r2)*(1+r3)]^(1/3) - 1

Real returns = nominal inflation

Variance: = p1(r1 - E[r ])^2 + _ _ _ + pn (rn - E[r ])^2

The price today is the expected payoff divided by 1 + r : Pt = E[Pt+1 + Dt+1 ] / (1 + r) r is the disc. Rate

Price = Sum of Discounted Expected Future Cash Flows

Present Value = (Year-T Cash Flow) / (1 + r )^TPrice of an annuity : P = C/r (1- (1/(1+r)^t)

Zero coupon bond price : P = B1/100 * C + (B2/100 * C) (B2/100 * (C+FV))

Yield-to-maturity : Instead of discounting the cash flows back at different rates, what one rate (for all dates) would give us

the same answer

FWD RATE = SPOT RATES: (1+r2)^2 = (1+r1)(1+f1;2)

forward rate = E[spot rate] + liquidity premium

Duration: Compute bond's yield-to-maturity, Compute each cash flow's present value using ytm (important), Weight

payment dates by the present value of each cash flow, divided by the bond price

Duration = PV(CF1)/ P *1 + PV(CF2)/ P *2..

If we solve for the price change, _P, we get = change P ~ 1/(1+y) * duration * price * Change in YTM

Immunization: P new = - P old * (Duration old/ duration new)

Utility function: U(E; stdev) = E y*variance higher y means more risk averse. useutility functions to makeinvestment choices

3 risky assets: ~rp = w1r1 + w2r2 + (1 - w1 - w2)r3

1 risky 1 riskless the variance is w^2 variance mkt^2

Optimal weight formula: w* = ((E[rmkt] - rf ))/ 2*y*variance mkt

Covariance: E[(r1 - E(r1))( r2 - E(r2))]

var(~rp) = var(w1~r1 + w2~r2) = w1^2 variance + w2^2 variance2 + 2w1w2covariance

Portfolio variance : var(~rp) = w1^2 variance + w2^2 variance2+ w3^2 variance3 +2w1w2covariance + 2w1w3covariance

+ 2w3w2covariance

Note: square weights for variance but not probabilities

Minimum variance portfolio : w1variance1 + w2 covariance 1,2 = 1 and w2variance2 + w1 covariance 1,2 = 1

Low covariance ! an asset is attractive if it has low or negative covariance with the assets you already own reduces risk

We should increase the slope of the line that connects the riskless asset to a risky portfolio as much as we canIAs the

line steepens, we reach better and better portfolios We cannot go any further when we just barely touch the investment

opportunity set of risky assets When the line is as steep as it can be, the risky portfolio that we touch is known as the

tangency portfolio (it max the sharpe ratio)

This slope is sharpe ratio : (E(r portfolio) - rf)/ std dev of portfolio

We _nd the tangency portfolio by shifting toward assets with high \marginal" Sharpe ratios:

Marginal" Sharpe Ratio(j) = (E[~r j ] rf ) / covariance j,p

The portfolio's Sharpe ratio can be improved until the ratios are the same: (E[~r 1 ] rf ) / covariance 1,p = (E[~r 2 ]

) / covariance 2,p ..

If we have the tangency portfolio, all marginal risk-return ratios have to be equal to some number C: (E[~r 2 ] rf ) /

covariance 2,p= C and w1 + w2 = 1 (scale the weights down)

CAPM: Beta= covariance (r1, rm)/ variance rmwe can write the CAPM as a time-series regression of excess returns on excess market returns

~ri,t - ~rf,t = alpha + beta (~rm,t - ~rf,t ) + error

Predictive results: rt = alpha + beta (dt+1)/(pt-1) + error

The portfolio-weighted average of factor-one betas is 2

The portfolio-weighted average of factor-two betas is 1

Expected return of tracking portfolio: rf + beta1 *lambda1 + beta2 * lambda2