【補足】6 つの係数の導出:まとめ!

この記事は約3分で読めます。

この記事は「【補足】資金計画に関する 6 つの係数の導出」の連載です。直リンクでここに来た方向けに、以下の記事に概要が記されています:)


 以上連続する 3 つの記事で「資金計画に関する 6 つの係数」:終価係数&原価係数年金終価係数&減債基金係数資本回収係数&年金原価係数一般式の導出とななみんの本に与えられている表の数値が正しいことの確認をしました。

 そこで最後に、 1 記事分を割いてこれらに関するまとめを書いておき、さらに各係数間に成り立つ関係を見ておくことにします。また、グラフは運用期間を \( n = 5 \) 年に固定したものしか示してこなかったので、運用期間 \( n \) を変えてみたときの各係数の挙動の変化も示しておくことにします。

6 つの係数の一般式まとめ

  1. 終価係数

\[ k_1 = (1+r)^n \,\, . \]

  1. 原価係数

\[ k_2 = \frac{1}{(1+r)^n} \,\, . \]

  1. 年金終価係数

\[ k_3 = \frac{ (1+r)^n – 1 }{r} \,\, . \]

  1. 減債基金係数

\[ k_4 = \frac{r}{ (1+r)^n – 1 } \,\, . \]

  1. 資本回収係数

\[ k_5 = \frac{ (1+r)^n r }{(1+r)^n – 1} \,\, . \]

  1. 年金原価係数

\[ k_6 = \frac{(1+r)^n – 1}{ (1+r)^n r } \,\, . \]

6 つの係数間の関係

 まず各係数の定義から、\( k_1 k_2 = 1 \)、\( k_3 k_4 = 1 \)、\( k_5 k_6 = 1 \) が成り立つことは明らかです(というか、\( k_1, k_3, k_5 \) に対して、これらの式を以て \( k_2, k_4, k_6 \) を定義していると言ってもよいです)。さらに、上の 6 つの式をよく見ると、たとえば \( k_1 k_4 = k_5 \) などが成り立つことが分かります。以上を一つの表にまとめると下の表の通りです:

表:6 つの係数が満たす掛け算の関係式。

グラフ集(オマケ)

 これまでは 6 つの係数を別々のグラフに示してきましたが、全部を一つのグラフに示すと下の図の通りです:

図:6 つの係数を一つのグラフに示した図。運用期間はこれまでと同様に \( n = 5 \) 年。

 また、これまでは運用年数を \( n = 5 \) 年に固定してグラフを示していましたが、たとえば終価係数 \( k_1 \) について、運用年数を \( n = 5, 15, 25, 35, 45 \) 年と変えて示してみると、下の図のようになります:

図:終価係数 \( k_1 \) の運用年数 \( n = 5, 15, 25, 35, 45 \) 年に対する挙動。

まとめ!

 これまでに導出した「資金計画に関する 6 つの係数」:終価係数&原価係数・年金終価係数&減債基金係数・資本回収係数&年金原価係数の一般式を列挙した。各ペアは掛けると 1 になる(互いに逆数の関係にある)。また、たとえば \( k_1 k_4 = k_5 \) のように、二つを掛けると他の係数を生み出すような組がある。これまでは運用年数 \( n \) を固定してきたが、例として終価係数 \( k_1 \) について、 種々の \( n \) に対する挙動を示してみた。

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