CAS Exam 2Financial Mathematics (FM)

Calculate the present value of an annuity-immediate paying 1,000 at the end of each year for 10 years at 6% annual effective.

A nominal rate of 8% convertible quarterly is equivalent to what annual effective rate?

A 10-year 1,000 face bond pays 5% annual coupons and is redeemable at par. Calculate its price to yield 6% annual effective.

A 100,000 loan at 5% annual effective is repaid with 20 level annual payments. Calculate the outstanding balance immediately after the 5th payment.

A liability of 1,000 is due in 5 years. At 4% annual effective, calculate the modified duration of this liability.

An insurer must pay 10,000 in 3 years. It can buy 1-year and 5-year zero-coupon bonds at a flat 5% yield. Determine the amounts to invest in each bond to Redington-immunize the liability and verify the convexity condition.

A fund begins the year with 100. A deposit of 50 is made at time 0.5, immediately after which the fund is valued at 120 (i.e., the fund grew to 120 before the deposit). The fund ends the year at 170. Calculate the time-weighted annual return.

An insurer enters a 1,000,000 notional interest-rate swap, paying a fixed rate of 4% and receiving the floating rate, settled annually net. At the first reset the floating rate is 5%. Calculate the net cash flow to the insurer at that settlement.

The nominal annual interest rate is 8% and expected inflation is 3%. Using the exact Fisher relationship $1+i_{nom}=(1+i_{real})(1+\pi)$, calculate the real rate of interest.

Calculate the present value of an increasing perpetuity-immediate that pays 1 at the end of year 1, 2 at the end of year 2, 3 at the end of year 3, and so on, at an annual effective rate of 5%.

A borrower takes a 10,000 loan for 10 years under a sinking-fund arrangement: the lender is paid interest only each year at 8%, and the borrower separately deposits into a sinking fund earning 6% annually to accumulate 10,000 at the end of 10 years. (a) Calculate the annual sinking-fund deposit. (b) Calculate the total annual outlay. (c) Calculate the equivalent annual effective interest rate the borrower is actually paying, and explain why it exceeds 8%.