# What is the present value of an immediate annuity over 12 years with 4 yearly payments and an interest of i = 2%?

See the question above, the result should be 10.689. I tried using the temporary annuity-due formula (see below):

$$\ddot{\mathbf{a}}_{n}^{[m]}=\frac{1-v^{n}}{d^{[m]}}$$

where:

$$d^{[m]}=m \cdot\left[1-(1+i)^{-\frac{1}{m}}\right]$$

This is already answered but you should be more clear as I can see this is a problem for the FM exam. You probably meant $$i^{(4)}=2\%$$ and the annuity is due (first payment is due immediately), you also need to clarify the amount of each payment which is probably 1 annually payable quarterly (0.25 dollars every 3 months).You can use the formula $$\ddot a_{n,j}=\frac{1-v_j^n}{d_j}$$ where $$j=\frac{i^{(4)}}{4}=0.5\%$$ is the effective quarterly rate, $$v_j=(1+j)^{-1}$$ is the quarterly discount factor and $$d_j=1-(1+j)^{-1}$$ is the effective quarterly discount rate. The payment is 0.25 so the PV is $$0.25\ddot a_{48,0.5\%}$$. Note that n represents the number of payments, not the number of years. This is what noob2 did.

There is no need for the $$\ddot a_{n,i}^{(m)}$$ function since the compounding and payment frequencies are equal. You typically use that function when payments are made more frequently than interest is compounded.

Assume the following:

An annuity due of 1 dollar per year, paid in 4 quarterly installments (i.e. 0.25 per quarter). For 12 years. Interest of 2% a year.

Then we calculate:

periodic interest rate = iper = 0.02/4 = 0.005

number of periods = n = 4*12 = 48

annuity due value = pmt x (1+iper) x (1/iper) x (1-1/(1+iper)^n)

= 0.25*(1.005)*(1/0.005)(1-1/(1.005)^48) = 10.6983

(The given answer 10.689 might be a typographical error).