Instruction: Review textbook chapter 2 first. Multiple reading might help. Then try to solve the homework problems quickly.
1. Given d(4) = 0.08, compute the equivalent rates i, d, d(m) and i(m) for each m = 2,6,12.
2. Suppose that interest is paid once every 2 years at a nominal interest rate i(1/2) . That is, the borrower pays interest at an effective rate of 2i(1/2) every 2 years. Find an expression for i(1/2) in terms of i.
3. Suppose that the accumulation function a(·) is given by
a(t) = (1.02)t(1 + .03t)(1 − .05t)−1 ; t ≥ 0.
Compute the force of interest δ3 at time t = 3.
4. Esteban borrows $ 20,000 and the loan is governed by compound interest at an annual effective interest rate of 6%. Esteban agrees to repay the loan by making a payment of $ 10,000 at the end of T years and a payment of $ 12,000 at the end of 2T years. Find T .
5. Suppose that there are contributions Ct1,...,Ctn at time t1,...,tn , respectively. We define
.
Here v = 1/(1 + i) and log is the natural logarithm. Verify that T ≥ T for every n.
6. Define the function f(t) = 525(1.1)−2t + 525(1.1)−t − 1000, t ≥ 0 and find the root T∗ , such that f(T∗) = 0 by Newton’s method.
7. Sandra invests $ 10,832 in the Wise Investment Fund. Three months later her balance has grown to $ 11,902 and she deposits $ 2,000. Two months later her holdings are $ 14,308 and she withdraws $ 7,000. Two years after the initial investment, she learns that her holdings are worth $ 12,566. Compute the approximate dollar-weighted annual yield with different methods.
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