Question

Find the real interest rate, when the compound interest is done quarterly, and for this interest rate it is required a capital of 40,000 euros in order to get a final investement of 52,000 euros in 11 years.

Answer 1

When compounded quarterly, effective annual real interest rate = (1+ Quarterly interest rate)*-1 # of quarters left to reach the goal, NPER Quarterly deposits, PMT Initial deposit today, PV Target maturity value in 11 years, FV Interest compounding type, TYPE Quarterly rate of interest Number of quarters per year 44 [11 years x 4 --> As there will be 4 quarters per year] € 40,000 € 52,000 0 = 0 if compounded at the end of each quarter; 1 if compounded at the beginning of each quarter 0.59806% [Ref: NOTE] 4 2.41% [((1+0.59806%)^4) - 1] Effective annual real interest rate NOTE: Quarterly rate of interest is calculated using EXCEL FUNCTION RATE(nper,pmt,pv,fv,type) where nper=44; pmt=0; pv=40000; fv=-52000; Here, pv is positive as it denotes cash outflow; fv is negative as it denotes cash inflow; type=0 as interest is compounded at the end of each quarter only. Н K # of quarters left to reach the goal, NPER 2 Quarterly deposits, PMT 3 Initial deposit today, PV =11*4 [11 years x 4 --> As there will be 4 quarters per year] € 40,000 € 52,000| 0 = 0 if compounded at the end of each quarter; 1 if compounded at the beginning of each quarter 4 Target maturity value in 11 years, FV Interest compounding type, TYPE 5 6 Quarterly rate of interest 7 Number of quarters per year =RATE(B1,B2,B3,-B4,B5) [Ref: NOTE] 4 =((1+B6)^B7)-1 [((1+0.59806%)^4) - 1] Effective annual real interest rate 8

Alternatively, When compounded quarterly, effective annual real interest rate = (1+ Quarterly interest rate)*-1 Present value of a future lumpsum receipt = Future lumpsum receipt / (1+ Discount rate) Number of periods Discount rate) Number of periods (1 = Future lumpsum receipt / Present value of a future lumpsum receipt (1/Number of periods) 1 + Discount rate = (Future lumpsum receipt / Present value of a future lumpsum receipt) (1/Number of periods) - 1 Discount rate = (Future lumpsum receipt / Present value of a future lumpsum receipt) € 52,000 [Target maturity amount] € 40,000 [Amount invested today] 44 [11 years x 4 Future lumpsum receipt Present value of future lumpsum receipt Number of periods --> As there will be 4 quarters per year] Quarterly discount rate Number of quarters per year 0.59806% {[(52000 / 40000)^(1/44)] - 1 = 0.0059806 = 0.59806%} 4 2.41% [((1+0.59806%)^4) - 1] Effective annual real interest rate