Annuities

 

Alice and Bob take out a reducing balance loan of $20000 at a rate of 8% compounded half-yearly for 3 years and agree to repay 3815.24 every six months.
(a) What will be the balance after 2 years.
(b) How much interest have they paid so far?

Calculate the rate per period

\[ \begin{align} i &= \frac{8.0}{200} \\ &= 0.04 \end{align} \ \]

Calculate the balance outstanding after 2 years

The number of repayments after 2 years is 4, so \[ \begin{align} V_1 &= 20000 (1.04) - 3815.24 \\ &= 16984.76 \\ V_2 &= 16984.76(1.04) - 3815.24 \\ &= 13848.91 \\ V_3 &= 13848.91 (1.04) - 3815.24 \\ &= 10587.63 \\ V_4 &= 10587.63 (1.04) - 3815.24 \\ &= 7195.89 \\ \end{align} \ \] So after 4 repayments the balance outstanding is 7195.89.

Interest Paid is Repayments + Balance Outstanding - Loan Amount

\[ \begin{align} \text{Balance Outstanding} &= 7195.89 \\ \text{Repayments } &= 3815.24 \times 4 = 15260.96 \\ \text{Interest Paid } &= 15260.96 + 7195.89 - 20000 \\ &= 2456.85 \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Alice and Bob take out a reducing balance loan of $25000 at a rate of 7% compounded half-yearly for 3 years and agree to repay $4691.71 every six months.
(a) What will be the balance after 18 months.
(b) How much interest have they paid so far?