7 Solutions – Time Value of Money
7.0.1 Solution to Exercise 1 — Nominal vs. Effective Rates
A bank advertises a nominal rate (j = 9%) convertible quarterly ((m = 4)).
- Effective annual interest rate
[ i_{} = (1 + )^m - 1 = (1 + )^4 - 1 = (1.0225)^4 - 1 . ]
So the effective annual rate is about 9.308%.
- Accumulation function
For a nominal rate (j) convertible (m)-thly, in years:
[ a(t) = (1 + )^{mt} = (1.0225)^{4t},t . ]
- Future value at (t = 5)
[ A(5) = 12{,}000 , a(5) = 12{,}000 (1.0225)^{20} 000 18{,}726.11. ]
- Comparison with 9.2% effective
Second bank: (i_2 = 0.092).
[ A_2(5) = 12{,}000 (1.092)^5 000 18{,}633.50. ]
Bank 1 yields CHF 18,726.11 vs. CHF 18,633.50 from Bank 2.
Bank 1 is preferable over 5 years.
7.0.2 Solution to Exercise 2 — Force of Interest, Inflation, Real Rates
[ (t) = 0.03 + 0.002t, t , ] inflation (j = 2.5%) effective.
- Accumulation function
[ a(t) = !(_0^t (s),ds) = !(_0^t (0.03 + 0.002s),ds). ]
[ _0^t (0.03 + 0.002s),ds = [0.03 s + 0.001 s^2]_0^t = 0.03 t + 0.001