# stalla: replacement chain approach

has anyone ever seen this calculation for the common lives approach: 1. machine 4 years: calculate NPV = 77533 2. machine 6 years: calculate NPV = 107643 discount rate 12 % Common life NPVs: common life for project with 4 and 6 years lives is 12 years machine 1: 77533 + 77533/(1.12)^4 + 77533 / (1.12)^8 machine 2: 107643 + 107643/ (1.12)^ 6 does this seem right to you ?

it does… Think of it as a stream of cash flows. 0…4…8…12 77533…77533…77533… 0…6…12 107643…107643… and you have it.

yup now that makes sense. thx CP!

Aurora just borrowed 10 mil from its bank etc etc current market value of the loan is 10 mil. effective duration of the loan is 8 with a 5 year key rate duration of 2 a 10 year key rate duration of 6 and an effective convexity of 0.4. blahblah charles job is to use the proceeds from the loan to make fixed inc investments. blah blah charles constantly calculates the market value of the firms portfolio and compares it to the market value of the liability (loan) Portfolio has an effective duration of 8 and effective convexity of 0.12 assume there is an immediate 100 basis point upward parallel shift in the yield curve which is then followed by a gradual parallel shift downward totaling 200 basis points over the next year. the most likely immediate and one year effects of Aurora’s net worth is IMMEDIATE - DECREASE ONE YEAR- DECREASE Answer: because the durations are equal, parallel shifts in the yield will have the same duration effect. however the portfolio has a convexity of 0.12 which is lower than the convexity of the loan 0.4. the convexity of the firms net worth is 0.12-0.4= -0.28. as a result when rates immediately rise firm net worth will decline because of the convexity effect. similarly when rates fall over the next year, the net worth will not change because of a duration effect but will decline (compared to today’s value) because of the convexity effect. ok i got the duration part, but still do not understand the convexity part…

U can use the other method, gives you the same answer