Compound interest, DCA, APR, mortgage and buy vs. rent comparison
FV = PV × (1 + r/n)^(n·t)
Excel:
=B1*(1+B2/B4)^(B4*B3)
B1=PV, B2=r (decimal), B3=t, B4=n
Holding-period test § 4 Czech ITA: securities
held ≥ 3 years are exempt from tax.
Monthly rate: i = r/12
For each month m = 1…N:
if (m mod 12 == 1 and m > 1) and inflation on:
deposit = deposit × (1 + π)
balance = balance × (1 + i) + deposit
Excel (no inflation, FV formula):
=FV(B3/12/100; B4*12; -B2; -B1; 0)
B1=PV, B2=monthly deposit, B3=r, B4=years
Annuity: PMT = P × i / (1 − (1+i)^−n)
where i = r/12, n = number of months
APR: solve the cash-flow IRR:
t=0: +(P − one-time fee)
t=1…n: −(PMT + monthly fee)
Then APR = (1 + i_month)^12 − 1
Excel:
monthly payment: =-PMT(B2/12/100; B3; B1)
APR (via IRR): =((1+IRR(CF_range))^12)-1
or use the IRR/RATE function.
Annuity payment:
PMT = P × i × (1+i)^n / ((1+i)^n − 1)
i = r/12, n = term in months
For each month:
interest = balance × i
principal = PMT − interest
balance = balance − principal
Excel: =-PMT(B3/12/100; B4*12; B2)
| Year | Payment | Interest | Principal | Balance |
|---|
BUY SCENARIO:
- Down payment = price × (1 − LTV)
- Monthly cost = annuity + maintenance/12
- Property value grows by π_prop per year
- Net worth = value − loan balance
+ invested difference, if rent was more expensive
RENT SCENARIO:
- Down payment invested in an index fund
- Rent grows by π_rent per year
- If the buy monthly cost > rent,
the difference is invested
- Net worth = value of investments
