Exercise 1. A loan of S = 100 000 euros needs to be reimbursed, together with the interest I, after T = 3 years. I is calculated applying an annual interest rate of i1 = 3% for the first year, i2 = 4% for the second, and i3 = 5% for the third. Determine the periodic interest rate j(0, 3) of the loan, the interest I, the annual compound rate iC and the annual simple interest rate iS. The debtor can decide to redeem (payback) the debt at time T′ < T , reimbursing S together with the interests I′ accrued until T′ considering an annual compound interest rate rule to accrue interest within the year. Determine T′ such that I′ = 10 000.
I = S ∗(1+3%)(1+4%)(1+5%)−S = 12 476 euros, j(0, 3) = (1+3%)(1+4%)(1+5%)−1 = 12.476%, iC = (1+j(0, 3))(1/3)−1 = 3.9967948%, iS = j(0, 3)/3 = 4.1586667%. Since the interests matured after two years are S(1 + i1)(1 + i2) − S = 7 120 < I′, it follows that T′ > 2. Then, T′ can be found by solving S(1 + i1)(1 + i2)(1 + i3)T
′−2 = S + I′ with respect to T′, obtaining T′ = 2.54377 years.
Exercise 2. Fill the non standard amortization plan , with S = 100 000 euros, 4 annual instalments at interest rate i = 4%
and where (a) the first two amortization quotas are equal, (b) the last two instalments are equal and (c) the sum of the last two amortization quotas is equal to 50% of the initial debt.
R1 = 29 000.00, R2 = 28 000.00, R3 = 26 509.80, R4 = 26 509.80, C1 = 25 000.00, C2 = 25 000.00, C3 = 24 509.80, C4 = 25 490.20, I1 = 4 000.00, I2 = 3 000.00, I3 = 2 000.00, I4 = 1 019.61, D1 = 75 000.00, D2 = 50 000.00, D3 = 25 490.20, D4 = 0.00.
Exercise 3. At time t = 0 the following term structure of interest rates applies: i(0, s) = 1.1% + 0.1% s (s in years). Compute the following quantities (annual basis, rates in %): p(0, 1), p(0, 2), p(0, 3), h(0, 2), i(0, 1, 3), p(0, 1, 3). Determine then the forward price at T = 1 of the contract {10, 20}/{2, 3} (euro/years).
p(0, 1) = (1+i(0, 1))−1 = 0.9881, p(0, 2) = (1+i(0, 2))−2 = 0.9745, p(0, 3) = (1+i(0, 3))−3 = 0.9591, h(0, 2) = 1.2916%, i(0, 1, 3) = 1.5001%, p(0, 1, 3) = 0.9707, P (0, T, {10, 20}) = 10 ∗ p(0, 1, 2) + 20 ∗ p(0, 1, 3) = 29.22751 euros.
Exercise 4. A CCT (Italian government floating rate notes with coupons every 6 months) with maturity 7 years ju- st issued has 60 basis point spread over each coupon. Under a flat interest rate term structure with annual interest rate i = 4.04%, compute the market price and the duration (in years). Determine then how to split an investment of 100 000 euros between the CCT and a ZCB with maturity 10 years, so that the duration of the portfolio is 3 years.
P = 100 + 0.6 ∗ 1−vi 14
sem = 107.26375, where v = 1+i1
sem = 1
1+2%, D = 0.709204 years which is equal to the duration of the CCT without the spread (0.5 years) plus the duration of the annuity made by the spreads over the coupons. Calling the quota to be invested in the CCT α, 1 − α will be invested in the ZCB. To compute the quotas to invest such that the portfolio duration is 3, it is necessary to solve the following equation: αDCCT + (1 − α)DZCB = 3, that is α0.709204 + (1 − α)10 = 3.
Hence, α = 10−0.709210−3 = 0.753438. Hence, 75 343.38 euros are invested in the CCT and the rest in the ZCB.
