TVM Calculation - When to start discounting a delayed annuity?

CFA Level I Quant
#1

Specific question is:

“An investor will receive an annuity of $4,000 a year for 10 years. The first payment is to be received five years from today. At a 9% discount rate, this annuity’s worth today is closest to:”

Correct answer is:
“Two steps: (1) Find the PV of the 10-year annuity: N = 10; I/Y = 9; PMT = –4,000; FV = 0; CPT → PV = 25,670.63. This is the present value as of the end of Year 4; (2) Discount PV of the annuity back four years: N = 4; PMT = 0; FV = –25,670.63; I/Y = 9; CPT → PV = 18,185.72. (LOS 1.e)”

If the first payment is to be received 5 years from today, why do we discount the PV back four years? Could someone please help with the logic here?

#2

Draw a time line.

Because:

(Emphasis added.)

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#3

You could also calculate the PV at time 5 using BGN and discount that amount by 5 years. The solution above assumes you used END, and so the PV is at time 4.

Using BGN
PV at time 5
N 5 I 9 PMT 4000 FV 0 CPT PV 27,980.98758

PV at time 0
N 5 I 9 PMT 0 FV 27,980.98758 CPT PV 18,185.722

The CF worksheet saves you a step and some time:

CF0 0 C01 0 F01 4 C02 4000
NPV
I 9 CPT NPV 18,185.722

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