SOA FMFinancial Mathematics

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 $t=0.5$, immediately after which the fund is valued at 170. The fund value just before the deposit (at $t=0.5$) was 120, and the year-end value is 180. Calculate the time-weighted annual return.

Using the same fund as above (begin 100, deposit 50 at $t=0.5$, end value 180), calculate the (exact) money-weighted annual return $i$ solving $100(1+i)+50(1+i)^{0.5}=180$.

One-, two- and three-year spot rates are 4%, 5% and 6% annual effective. Calculate the fixed swap rate $R$ for a 3-year interest-rate swap with annual settlements, using $R=\dfrac{1-v_3}{v_1+v_2+v_3}$.

Under the Fisher equation and standard term-structure theory, an increase in expected future inflation, holding the real risk-free rate and risk premia constant, should:

Calculate the present value of a perpetuity-due paying 1,000 at the start of each year, at 5% annual effective.