Quantitative Methods Note 1

作者: _JKing_ | 来源:发表于2019-02-23 11:37 被阅读0次

备考CFA阶段，特此整理笔记，用于后期回顾。

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)

用时间价值五要素通过计算器计算（注意：END / BGN Mode 切换）

Perpetuity:

$PA=\frac{A}{r}$

A: the periodic payment to be received forever

r: rate of return

当unequal cash flow 的时候，求FV and PV:

运用计算器的'CF'功能

‘CF'按钮（输入每个CF('C0'), 跳过F0） --'NPV'按钮--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):

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

不考虑时间（持有期为任何时间段）情况下的实际收益率。如果比较两种金融产品的HPR，则其持有期必须一致（起点和终点都一样），否则无法比较。实际当中很少用HPR，而是采用实际有效年收益率（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

银行贴现收益率是按照单利计算，而且是按照360天每年计算。BDY的缺点在于计算收益率是以面值为计算基准，而实际上投资者付出的价格低于面值。因此引入货币市场收益率（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的分母变成投资者实际购买票据的价格，更能反映投资者的实际收益率，但一般是用作短期的，所以都计算单利，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

投资者可能支付9852元购买面额1w元为期91天的短期债券，到期时，投资者将收入1w元款项，利息总额为148元，短期国债没有票面利率，通过BEY计算年利率。

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）是指考虑到各种复利情况下，债券一年内的收益率。年有效收益率反映的是实际收益率。可以方便与其他债券的收益率进行比较。