Present Value of a Quarterly-Paid Annuity Due
Summary
The document works through the present value of an annuity due over twelve years when payments are made in four installments per year and the annual interest rate is 2%. The replies clarify the assumptions needed to interpret the problem: each annual payment is split into quarterly installments, and the first installment is due immediately. Under those assumptions, each quarterly payment is one quarter of the annual amount, the periodic rate is the annual rate divided by four, and the number of periods is the number of quarterly payments across the term.
The calculation discounts those installments using the quarterly rate and values the resulting annuity due. One answer states a present value of 10.6983 and suggests that the proposed result of 10.689 may be a typo. The discussion also distinguishes payment count from year count and notes that a temporary annuity-due formula with a different frequency setup is unnecessary here. The result depends on the assumed payment size, timing, and compounding convention.
Key ideas
- An annuity due begins with a payment at the start of the payment schedule.
- Four equal quarterly installments make up each annual payment under the stated interpretation.
- Convert the annual interest rate to a quarterly rate and count all quarterly periods over the term.
- The annuity value depends on payment amount, timing, and interest-conversion convention.
- The worked answer reports a value and flags the question’s suggested result as a possible typo.
Tags
Full text
# What is the present value of an immediate annuity over 12 years with 4 yearly payments and an interest of i = 2%?
# 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:
$$ d^{[m]}=m \cdot\left[1-(1+i)^{-\frac{1}{m}}\right] $$
Thanks for any advice.
## Answer by actuarialboi9 (score 1, accepted)
https://quant.stackexchange.com/a/51754
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.
## Answer by nbbo2 (score 1)
https://quant.stackexchange.com/a/51692
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.