1. Deferred annuity
S.Y.Tan

2. Deferred annuity - an annuity wherein the first payment interval does not coincide with the first interest period. The first payment is put off to some later date.
Deferred annuity of n payments
d+n
d+(n-1)
d+(n-2)
d-1
d-2 2 1 R d
d+1
d+2
1st payment starts on the (d+1)th period
NO payment for d periods
S.Y.Tan

3. Amount of a deferred annuity of n payments is equivalent to amount of an ordinary annuity of n payments.
The present value of an annuity deferred for d periods is the value of the annuity or the value of the n payments in lump sum amount at the beginning of the term.
Value of same annuity of n payments at one period before 1st payment date which is the end of d period
Ordinary annuity of n payments translated for d periods
Deferred annuity of n payments
discount P for d periods
d+n
d+(n-1)
d+(n-2)
d-1
d-2 2 1 R d
d+1
d+2
1st payment starts on the (d+1)th period
NO payment for d periods
d = period of deferment
S.Y.Tan

4. Formula for the present value of an annuity of n payments deferred for d periods
If the annuity is deferred by d periods, the first payment is then due at the end of the (d+1)th period. Similarly, if the 1st payment in due at the end of kth period, the period of deferment is always (k-1).
S.Y.Tan

5. Ex 1 Find the present value of a P5,000 annuity payable quarterly for 7 years but deferred for 5 ½ years. Money is worth 9.6% converted quarterly. 50 49 48 21 20 2 1
5000 22 23 24
1st payment starts on the 23rd period
period of deferment d = 22
Note: n = (last- first) + 1
28 = last –
Last = = 50
S.Y.Tan

6. Ex 2 To start an IT business, Elmer and Erwin borrowed P4
Ex 2 To start an IT business, Elmer and Erwin borrowed P4.5M with interest at 10% compounded quarterly. They will repay the debt by making 12 equal quarterly payments. If the first payment is due at the end of 18 months, how much is the quarterly payment? 17 16 6 5 2 1 R 7 8
period of deferment d = 5
1st payment starts on the 6th quarter
S.Y.Tan

7. EXERCISES
Find the present value of a P50,000 annuity payable each three months for ten years but deferred for five years. Money is worth 12% compounded quarterly.
On the birth of a child, the father wants to deposit a sum in a savings account. He intends to provide his daughter P20,000 every month for four years, starting on her 18th birthday. If the savings bank pays 10% compounded monthly, how much should the father deposit?
A man borrowed P300,000 from a finance company that charges interest at 16% compounded semi-annually. He promised to pay off the loan in twelve semi-annual payments. The first payment is to be made at the end of two years. Find his semi-annual payment.

8. Ex Find the present value of a P50,000 annuity payable each three months for ten years but deferred for five years. Money is worth 12% compounded quarterly. 60 59 21 20 2 1
50000 22 23
period of deferment d = 20
1st payment starts on the 21st quarter
S.Y.Tan

9. Ex On the birth of a child, the father wants to deposit a sum in a savings account. He intends to provide his daughter P20,000 every month for four years, starting on her 18th birthday. If the savings bank pays 10% compounded monthly, how much should the father deposit?
263
262
216
215 2 1
20000
217
218
period of deferment d = 215
1st payment starts on the 216th month
S.Y.Tan

10. Ex A man borrowed P300,000 from a finance company that charges interest at 16% compounded semi-annually. He promised to pay off the loan in twelve semi-annual payments. The first payment is to be made at the end of two years. Find his semi-annual payment. 14 13 4 3 2 1 R 5 6 15
period of deferment d = 3
1st payment starts on the 4th period
S.Y.Tan