張智星 (Roge Jang) jang@mirlab.org http://mirlab.org/jang 台大資工系 多媒體檢索實驗室 Compounding 複利 張智星 (Roge Jang) jang@mirlab.org http://mirlab.org/jang 台大資工系 多媒體檢索實驗室

The Euler’s Number Euler’s number: Proof by binomial theorem (二項式定理): Quiz! Quiz!

Compounding (複利) 故事 國王和棋盤 一莫耳的新台幣有多少？為什麼領18%是老賊？ Credit cards (Annual interest rate: 2.74%~20%) George & Mary現金卡廣告（救急免利） 2006年，台灣有70萬人淪為卡奴，平均欠款數100萬新台幣… (wiki) 20K payment per month, 4 years to clear the debt  Wrong! It takes 8 years to clear it! 10K payment per month, 8 years to clear the debt Wrong! You cannot even pay off the interests!

Basic Formula for Compounding Example Initial investment: 1000 Annual interest: 3% No. of periods: 20 years Quiz! This is the formula used by banks!

Periodic Compounding Periodic compounding Formula: Example Monthly compounding Daily compounding 年利率=r  月利率=r/12  日利率=r/365

Continuous Compounding Formula: Example:

Comparison Effect of compounding at various frequencies, with an initial investment of 1000 and 20% annual interest. (wiki)

Rule of 70 Goal Also known as Reference Formula Example: A quick method to estimate the doubling time of an investment Also known as Rule of 72 Rule of 69 Reference Wiki Formula T*r=70 T=doubling time in year r=annual interest rate (%) Example: r=1%  T=70 r=3%  T=23.3 r=18%  T=3.9 r=20%  T=3.5 Quiz!

How to Derive the Rule of 70 3 Ways of compounding to derive the rule Formulas: Examples: Quiz!

Time Value of Money Time value of money 現金流量表 average principal amortization Time Value of Money Time value of money The value of money usually increases with time (if invested properly). 現金流量表 在時間軸上秀出收入及支出，並以期末（或期初）總值來判斷投資是否划算 期末總收入 當期收入 當期支出 期末總支出

Mortgage Computation Problem 銀行傳統 銀行貸款100萬，20年還清，利率固定為3％，請問每個月還款金額？ 利率  年利率 複利計算方式  以「月」為單位來計算複利

Types of Mortgage Payment Typical types of mortgage computation (source) 本息平均攤還 本金平均攤還 Examples Payment derivation Comparison 線上試算 本息定額攤還、本息等額攤還

本息平均攤還：月付額 (1/2) 現金流量圖 Formula Here r is monthly interest rate

本息平均攤還：月付額 (2/2) 現金流量圖 Formula Quiz

本息平均攤還： 本金及利息每期分攤 現金流量圖 本金及利息每期分攤 Quiz:

本金平均攤還法 (1/2) 現金流量圖

本金平均攤還法 (2/2) 現金流量圖

Useful Functions Two functions (utility toolbox) available loan.m Monthly payment  x=loan(p, 3/100, 20) saving.m Present value  p=saving(x, 3/100, 20, ‘initial’); Final value  f=saving(x, 3/100, 20, ‘final’); 年利率3% 為期20年 Moreover, self demo of the functions…

Rule of 200 p/x vs. r Plot of p/x vs. r Rule of 200 (by Roger): 房貸總額/200=每月應付款 （假設20年貸款，利率2%） Quiz! 社會新鮮人的困惑： Monthly income=22k  Mortgage = 7k (= 1/3 of income) 台北市哪裡去找140萬的房子？

References References Wiki: Time value of money Wiki: Compound interest