Quantitative Methods Note 1


備考CFA階段，特此整理筆記，用于后期回顧。

Interest Rate

Simple interest:

Simple interest = Principal x i x n

i: Interest rate

n: Duration of the loan ( e.g. year, month, day)

Compounding interest:

Compounding\ interest = Principal *(1+i)^n-Principal

i: Interest rate

n: Duration of the loan ( e.g. year, month, day)

Effective annual rate:

EAR?= (1+\frac{r_{s} }{n} )^n-1

r_{s} : annual interest rate/ state, quoted interest rate

m: Compounding frequency?

\frac{r_{s} }{m} : Periodic interest rate

Continuous compounding:

| EAR= e^{r_{s}} -1

Time Value of Money Problem

Relationship between PV and FV:

FV = PV *(1+r)^n

r: periodic rate

n: number of periods

For Continuous compounding:

Annuity:

1. Ordinary annuity (END)

2. Annuity due (BGN)

用時間價值五要素通過計算器計算（注意：END / BGN Mode 切換）

Perpetuity:

PA=\frac{A}{r} ?

A: the periodic payment to be received forever

r: rate of return

當unequal cash flow 的時候，求FV and PV:

運用計算器的'CF'功能

‘CF'按鈕（輸入每個CF('C0'), 跳過F0） --'NPV'按鈕--CPT NPV/NFV

Evaluation of Cash Flow Streams

Net Present Value (NPV):

NPV = CF_{0}+\frac{CF_{1}}{(1+r)^1 }+\frac{CF_{2}}{(1+r)^2} +...++\frac{CF_{n}}{(1+r)^n }

注：\frac{CF_{1}}{(1+r)^1 } 可以想成 \frac{FV}{(1+r)^1 }=PV_{1}

if NPV > 0, undertake the project, if there are several projects, choose the higher positive NPV.

if NPV<=0, give it up?

Internal rate of return (IRR):

NPV =0= CF_{0}+\frac{CF_{1}}{(1+IRR)^1 }+\frac{CF_{2}}{(1+IRR)^2} +...++\frac{CF_{n}}{(1+IRR)^n }

if IRR > opportunity cost of capital, take it

if IRR <= opportunity cost of capital, give it up

NPV vs. IRR

if decision of? NPV and IRR are conflict, choose the result of NPV provided

Portfolio Return Measurement

Holding period return (HPR):

HPR=\frac{P_{1}-P_{0}+D_{1}}{P_{0}} =\frac{dividend + (end \ value - initial \ value )}{initial\ value}

不考慮時間（持有期為任何時間段）情況下的實際收益率。如果比較兩種金融產品的HPR，則其持有期必須一致（起點和終點都一樣），否則無法比較。實際當中很少用HPR，而是采用實際有效年收益率（Effective annual yield, EAY）

Time-weighted return (TWR) = Geometric Mean Return:

TWR=[(\frac{End\ Value_{1} }{Begin\ Value_{1}} )*(\frac{End\ Value_{2} }{Begin\ Value_{2}} )*...*(\frac{End\ Value_{n} }{Begin\ Value_{n}} )]^\frac{1}{N} -1

Money-weighted return (MWR):

|?CF_{0}+\frac{CF_{1}}{(1+MWR)^1 }+\frac{CF_{2}}{(1+MWR)^2} +...++\frac{CF_{n}}{(1+MWR)^n } = 0

Holding period yield (HPY)?[非年化]：

HPY = (\frac{Ending\ Value}{Beginning\ Value} )-1

Bank discount yield (BDY)?[年化]:

BDY = (\frac{Discount}{Face\ Value} )*(\frac{360}{Days to maturity})

-Discount rate, simple interest, 360-day annualized

銀行貼現(xiàn)收益率是按照單利計算，而且是按照360天每年計算。BDY的缺點在于計算收益率是以面值為計算基準，而實際上投資者付出的價格低于面值。因此引入貨幣市場收益率（Money Market Yield）

Money Market Yield (MMY) [年化]:

MMY = (\frac{Discount}{Price} )*(\frac{360}{Days to maturity})

-Add-on rate, simple interest, 360-day annualized

把BDY的分母變成投資者實際購買票據的價格，更能反映投資者的實際收益率，但一般是用作短期的，所以都計算單利，而且按照360天。

Bond Equivalent Yield (BEY) [年化]:

BEY = (\frac{Discount}{Price} )*(\frac{365}{Days to maturity})

-Add-on rate, simple interest, 365-day annualized

投資者可能支付9852元購買面額1w元為期91天的短期債券，到期時，投資者將收入1w元款項，利息總額為148元，短期國債沒有票面利率，通過BEY計算年利率。

Effective annual yield (EAY) [年化]:

EAY = (1+HPY)^\frac{365}{Days} -1

-Add-on rate, compound interest, 365-day annualized

年有效收益率（EAY）是指考慮到各種復利情況下，債券一年內的收益率。年有效收益率反映的是實際收益率?？梢苑奖闩c其他債券的收益率進行比較。

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