Immunization + Duration

Duration Of a Portfolio 𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑖𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑎 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑜𝑓 3 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 : 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑖𝑠 𝑝𝑜𝑟𝑡𝑖𝑓𝑜𝑙𝑖𝑜. 𝑀𝑎𝐶𝐷=2∙ 1,000 6,000 +3∙ 2,000 6,000 +5∙ 3,000 6,000 =3.83333 Investment Price Duration A 1,000 2 B 2,000 3 C 3,000 5

Comparing Durations 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝑪 𝟏 𝑪 𝟐 𝑪 𝟑 𝑪 𝟒 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 1 2 3 4 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝐹𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑜𝑛𝑡𝑎𝑛𝑡 𝑟 𝒓𝑪 𝟏 𝒓𝑪 𝟐 𝒓𝑪 𝟑 𝒓𝑪 𝟒 1 2 3 4 𝑴𝒂𝑪 𝑫 𝑨 =𝑴𝒂𝑪 𝑫 𝑩

Comparing Durations 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝑪 𝟏 𝑪 𝟐 𝑪 𝟑 𝑪 𝟒 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 1 2 3 4 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝐹𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑜𝑛𝑡𝑎𝑛𝑡 𝑟 𝒓𝑪 𝟏 𝒓𝑪 𝟐 𝒓𝑪 𝟑 𝒓𝑪 𝟒 1 2 3 4 𝟏∙𝑪 𝟏 𝒗+ 𝟐∙𝑪 𝟐 𝒗 𝟐 + 𝟑∙𝑪 𝟑 𝒗 𝟑 + 𝟒∙𝑪 𝟒 𝒗 𝟒 𝑪 𝟏 𝒗+ 𝑪 𝟐 𝒗 𝟐 + 𝑪 𝟑 𝒗 𝟑 + 𝑪 𝟒 𝒗 𝟒 = 𝟏∙𝒓𝑪 𝟏 𝒗+ 𝟐∙𝒓𝑪 𝟐 𝒗 𝟐 + 𝟑∙𝒓𝑪 𝟑 𝒗 𝟑 + 𝟒∙𝒓𝑪 𝟒 𝒗 𝟒 𝒓 𝑪 𝟏 𝒗+ 𝒓𝑪 𝟐 𝒗 𝟐 + 𝒓𝑪 𝟑 𝒗 𝟑 + 𝒓𝑪 𝟒 𝒗 𝟒 𝟏∙𝑪 𝟏 𝒗+ 𝟐∙𝑪 𝟐 𝒗 𝟐 + 𝟑∙𝑪 𝟑 𝒗 𝟑 + 𝟒∙𝑪 𝟒 𝒗 𝟒 𝑪 𝟏 𝒗+ 𝑪 𝟐 𝒗 𝟐 + 𝑪 𝟑 𝒗 𝟑 + 𝑪 𝟒 𝒗 𝟒 = 𝒓[𝟏∙𝑪 𝟏 𝒗+ 𝟐∙𝑪 𝟐 𝒗 𝟐 + 𝟑∙𝑪 𝟑 𝒗 𝟑 + 𝟒∙𝑪 𝟒 𝒗 𝟒 ] 𝒓 𝑪 𝟏 𝒗+ 𝑪 𝟐 𝒗 𝟐 + 𝑪 𝟑 𝒗 𝟑 + 𝑪 𝟒 𝒗 𝟒

Example 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝟏𝟓𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟐𝟎𝟎 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟑𝟎𝟎𝟎 𝟐𝟎𝟎𝟎 𝟐𝟎𝟎𝟎 1 2 3 4 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟑𝟎𝟎𝟎 𝟐𝟎𝟎𝟎 𝟐𝟎𝟎𝟎 𝟐𝟒𝟎𝟎 1 2 3 4

Example 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝟏𝟓𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟐𝟎𝟎 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟐∙𝟏𝟓𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 1 2 3 4 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟐∙𝟏𝟓𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 𝟐∙𝟏𝟐𝟎𝟎 1 2 3 4

Example 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝟏𝟓𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟎𝟎𝟎 𝟏𝟐𝟎𝟎 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟐∙𝟏𝟓𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 1 2 3 4 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝟐∙𝟏𝟓𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 𝟐∙𝟏𝟎𝟎𝟎 𝟐∙𝟏𝟐𝟎𝟎 1 2 3 4

Example 𝑇ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $1000 𝑖𝑠 5.6 𝑦𝑒𝑎𝑟𝑠. 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $50000.

Example 𝑇ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $1000 𝑖𝑠 5.6 𝑦𝑒𝑎𝑟𝑠. 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $50000. 𝑴𝒂𝑪𝑫=𝟓.𝟔

Comparing Durations 𝑑 𝐴 =𝑡+ 𝑑 𝐴 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 1∙𝐶 1 𝑣+ 2∙𝐶 2 𝑣 2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 = (𝑡+1)∙𝐶 1 𝑣 𝑡+1 + (𝑡+2)∙𝐶 2 𝑣 2 𝐶 1 𝑣 𝑡+1 + 𝐶 2 𝑣 𝑡+2 1∙𝐶 1 𝑣+ 2∙𝐶 2 𝑣 2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 = 1∙ 𝐶 1 𝑣 𝑡+1 +𝑡∙ 𝐶 1 𝑣 𝑡+1 + 2∙𝐶 2 𝑣 𝑡+2 + 𝑡∙𝐶 2 𝑣 𝑡+2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 1∙𝐶 1 𝑣+ 2∙𝐶 2 𝑣 2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 = 𝑡 𝐶 1 𝑣 𝑡+1 + 𝐶 2 𝑣 𝑡+2 +1∙ 𝐶 1 𝑣 𝑡+1 +𝑡∙+ 2∙𝐶 2 𝑣 𝑡+2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 1∙𝐶 1 𝑣+ 2∙𝐶 2 𝑣 2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 =𝑡∙ 𝐶 1 𝑣 𝑡+1 + 𝐶 2 𝑣 𝑡+2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 + 1∙ 𝐶 1 𝑣 𝑡+1 +𝑡∙+ 2∙𝐶 2 𝑣 𝑡+2 𝐶 1 𝑣+ 𝐶 2 𝑣 2 𝑑 𝐴 =𝑡+ 𝑑 𝐴 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐴 𝑪 𝟏 𝑪 𝟐 𝟏 𝟐 𝐼𝑛𝑣𝑒𝑠𝑚𝑒𝑛𝑡 𝐵 𝐹𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑜𝑛𝑡𝑎𝑛𝑡 𝑡 𝑪 𝟏 𝑪 𝟐 𝒕+𝟏 𝒕+𝟐

Example 𝑇ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $1000 𝑖𝑠 5.6 𝑦𝑒𝑎𝑟𝑠. 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑑𝑢𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $5000.

Example 𝑇ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑖𝑚𝑚𝑒𝑑𝑖𝑎𝑡𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $1000 𝑖𝑠 5.6 𝑦𝑒𝑎𝑟𝑠. 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 10–𝑦𝑒𝑎𝑟 𝑎𝑛𝑛𝑢𝑖𝑡𝑦–𝑑𝑢𝑒 𝑤𝑖𝑡ℎ 𝑎𝑛𝑛𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 $5000. 𝒅=𝟓.𝟔−𝟏=𝟒.𝟔

Immunization

First : Redington Immunization 𝑇𝑒𝑟𝑚𝑖𝑛𝑜𝑙𝑜𝑔𝑖𝑒𝑠: 𝑳𝒊𝒂𝒃𝒊𝒍𝒊𝒕𝒊𝒆𝒔 :𝑃𝑎𝑦𝑚𝑒𝑛𝑡 𝑡ℎ𝑎𝑡 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑖𝑠 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝑚𝑎𝑘𝑒 . 𝑨𝒔𝒔𝒆𝒕𝒔 𝑪𝒂𝒔𝒉𝒇𝒍𝒐𝒘𝒔 :𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑖𝑠 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝑚𝑎𝑘𝑒. 𝑺𝒖𝒓𝒑𝒍𝒖𝒔 :𝑆= 𝑃 𝐴 − 𝑃 𝐿 𝑰𝒎𝒎𝒖𝒏𝒊𝒛𝒂𝒕𝒊𝒐𝒏:𝑇ℎ𝑒 𝐴𝑐𝑡 𝑜𝑓 𝑝𝑟𝑜𝑡𝑒𝑐𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑢𝑟𝑝𝑙𝑢𝑠 𝑓𝑟𝑜𝑚 𝑎 𝑠𝑢𝑟𝑝𝑙𝑢𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒.

Redington Immunization : Conditions 𝑖. 𝑆 𝑖 =0 𝑖𝑖. 𝑆 ` 𝑖 =0 𝑖𝑖𝑖. 𝑆 `` 𝑖 ≥0 𝑅𝑒𝑚𝑎𝑟𝑘 :Redington Immunization protects against only small changes in i . Surplus Interest Rate

Redington Immunization :Alternative Conditions Redington Immunization Conditions : 𝑖. 𝑃 𝐴 = 𝑃 𝐿 𝑖𝑖. 𝑑 𝐴 = 𝑑 𝐿 𝑖𝑖𝑖. 𝑐 𝐴 > 𝑐 𝐿 𝑅𝑒𝑚𝑎𝑟𝑘 :Redington Immunization protects against only small changes in i . Surplus Interest Rate

Full Immunization : Conditions Redington Immunization Conditions : 𝑖. 𝑃 𝐴 = 𝑃 𝐿 𝑖𝑖. 𝑑 𝐴 = 𝑑 𝐿 𝑖𝑖𝑖.𝑇ℎ𝑒𝑟𝑒 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑎𝑛 𝐴𝑠𝑠𝑒𝑡 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 𝑏𝑒𝑓𝑜𝑟𝑒 𝑎𝑛𝑑 𝑎𝑓𝑡𝑒𝑟 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠. 𝑅𝑒𝑚𝑎𝑟𝑘 :Protects against changes of any size.

Dedication (Exact Matching) 𝐴 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑚𝑢𝑠𝑡 𝑝𝑎𝑦 𝐿 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤 𝑎𝑛𝑑 2𝐿 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤 . 𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑏𝑢𝑦𝑠 𝑡𝑤𝑜 𝑏𝑜𝑛𝑑𝑠 𝑡𝑜 𝑚𝑎𝑡𝑐ℎ 𝑡ℎ𝑒 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 . 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑏𝑜𝑛𝑑 𝑖𝑠 𝑎 1000 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑏𝑜𝑛𝑑 . 𝑇ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑏𝑜𝑛𝑑 𝑖𝑠 2500 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟 𝑏𝑜𝑛𝑑 𝑏𝑜𝑡ℎ ℎ𝑎𝑣𝑒 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟 . 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝐿 . "𝑬𝒂𝒄𝒉 𝒂𝒔𝒔𝒆𝒕 𝒄𝒂𝒔𝒉𝒇𝒍𝒐𝒘 𝒊𝒔 𝒅𝒆𝒅𝒊𝒄𝒂𝒕𝒆𝒅 𝒕𝒐 𝒂 𝒔𝒊𝒏𝒈𝒍𝒆 𝒍𝒊𝒂𝒃𝒊𝒍𝒊𝒕𝒚" 𝑳 𝟐𝑳 1000 1000𝑟 2500 2500𝑟 2500𝑟

Dedication (Exact Matching) 𝐴 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑚𝑢𝑠𝑡 𝑝𝑎𝑦 𝐿 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤 𝑎𝑛𝑑 2𝐿 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟𝑠 𝑓𝑟𝑜𝑚 𝑛𝑜𝑤 . 𝑇ℎ𝑒 𝑐𝑜𝑚𝑝𝑎𝑛𝑦 𝑏𝑢𝑦𝑠 𝑡𝑤𝑜 𝑏𝑜𝑛𝑑𝑠 𝑡𝑜 𝑚𝑎𝑡𝑐ℎ 𝑡ℎ𝑒 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑐𝑎𝑠ℎ𝑓𝑙𝑜𝑤𝑠 . 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑏𝑜𝑛𝑑 𝑖𝑠 𝑎 1000 𝑜𝑛𝑒 𝑦𝑒𝑎𝑟 𝑏𝑜𝑛𝑑 . 𝑇ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑏𝑜𝑛𝑑 𝑖𝑠 2500 𝑡𝑤𝑜 𝑦𝑒𝑎𝑟 𝑏𝑜𝑛𝑑 𝑏𝑜𝑡ℎ ℎ𝑎𝑣𝑒 𝑐𝑜𝑢𝑝𝑜𝑛 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑟 . 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝐿 . "𝑬𝒂𝒄𝒉 𝒂𝒔𝒔𝒆𝒕 𝒄𝒂𝒔𝒉𝒇𝒍𝒐𝒘 𝒊𝒔 𝒅𝒆𝒅𝒊𝒄𝒂𝒕𝒆𝒅 𝒕𝒐 𝒂 𝒔𝒊𝒏𝒈𝒍𝒆 𝒍𝒊𝒂𝒃𝒊𝒍𝒊𝒕𝒚" 𝑳=𝟏𝟎𝟎𝟎+𝟏𝟎𝟎𝟎𝒓+𝟐𝟓𝟎𝟎𝒓 𝟐𝑳=𝟐𝟓𝟎𝟎+𝟐𝟓𝟎𝟎𝒓 𝟐∙ 𝟏𝟎𝟎𝟎+𝟏𝟎𝟎𝟎𝒓+𝟐𝟓𝟎𝟎𝒓 =𝟐𝟓𝟎𝟎+𝟐𝟓𝟎𝟎𝒓 𝒓= 𝟏 𝟗 𝑳=𝟏𝟎𝟎𝟎+𝟏𝟎𝟎𝟎 𝟏 𝟗 +𝟐𝟓𝟎𝟎 𝟏 𝟗 =𝟏𝟑𝟖𝟖.𝟖𝟗