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現值與終值
現值（Present value PV）與終值（Future Value FV）是一組對應的概念。各類金融資產會給投資者在未來不同時間點帶來現金流（既可以是現金流入，也可以是現金流出）。現值PV是指將金融資產各個階段的現金流按複利形式折算到投資期初的價值之和。終值FV：是指將金融資產各階段的現金流按照複利形式折算到投資期末的價值之和。不難看出，現值和終值實際上是同一組現金流在不同時間點的價值，兩者之間的相互轉換公式如下：
Present value and final value are a set of corresponding concepts. Various types of financial assets will bring cash flow to investors at different time points in the future (either cash inflows or cash outflows). The present value PV refers to the sum of the value of the cash flow of each stage of financial assets converted to the beginning of the investment period in the form of compound interest. Final value FV: refers to the sum of the value of the cash flow of each stage of financial assets converted to the end of the investment period in the form of compound interest. It is not difficult to see that the present value and the final value are actually the values of the same group of cash flows at different time points. The conversion formulas between the two are as follows:
其中，n表示投資期，r表示計息期利率。Among them, n denotes the investment period and R denotes the interest rate of the interest period.
從上面的兩個公式可以看出現值、終值與利率水平之間的關係：（1）給定利率水平r，FV隨著計息週期數n上升而上升（2）給定計息週期數n，FV隨著利率水平r上升而上升（3）在給定時間段內，利率水平r越大，現值PV越小。
From the above two formulas, we can see the relationship between present value, final value and interest rate level.(1) Given the interest rate level r, FV rises with the increase of the number of interest-bearing cycles n.(2) Given the number of interest-bearing cycles n, FV rises with the increase of interest rate level R.(3) In a given period of time, the higher the interest rate level r, the smaller the present value PV.
例題1假定利率是10%，每年計息一次，初期投資2000元，求兩年後的終值。解析從現在開始必須養成畫現金流圖的習慣，對後續理解複雜的估值與定價模型有很大的幫助Example1Assuming that the interest rate is 10%, the interest rate is calculated once a year, and the initial investment is 2,000yuan, the final value after two years is calculated.analysisFrom now on, we must develop the habit of drawing cash flow graphs, which is very helpful to understand complicated valuation and pricing models in the future.
現金流圖中，橫軸標識時間軸，數字代表各時期序列號。例如，數字0表示第0期，即投資期初：數字1所在位置表示第一期末或第二期初。直接利用FV=PV×(1+r)n公式可計算兩年後終值FV=2000×(1+10%)2=2000×1.21=2420 In the cash flow chart, the horizontal axis identifies the time axis and the number represents the serial number of eachperiod. For example, the number 0 denotes the beginning of the investmentperiod: the location of the number 1 denotes the end of the first period or thebeginning of the second period.The final value after two years can becalculated directly by FV=PV×(1+r)nformula.FV = 2000×(1 +10%) 2 = 2000×1.21 = 2420
例題2:假定利率是10%，每年計息一次，若兩年後要收回2000元，現在應該投資多少元解析:先畫現金流圖Example 2Assuming that the interest rate is 10%, theinterest rate is charged once a year. If 2,000 yuan is to be recovered aftertwo years, how much should we invest now?analysisDraw a cash flow chart first
解析：同樣將現金流示意圖畫出，利用公式PV=可以求出Analysis:Similarly, the cash flow sketch is drawn, andthe formula PV=FV/(1+r)n can be used to calculate it.PV=2000÷(1+10%)2=1652.9
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