Question

An investor has three possible scenarios for a project as follows: Pessimistic – NOI will be $220,000 for the first year, and then decrease by 3 percent per year over a five-year holding period. The property will sell for $1.7 million after five years. Most likely – NOI will be level at $220,000 per year for the next five years and the property will sell for $2.5 million Optimistic – NOI will be $220,000 the first year and increase 3 percent per year over a five-year holding period. The property will sell for $2.8 million. The asking price of the property is $2.2 million. The investor believes that there is a 30% probability of the pessimistic scenario, a 50% probability for the most likely scenario, and a 20% probability for the optimistic scenario. (a) Compute the IRR for each scenario and the expected IRR for the project. (b) Compute the variance and standard deviation of the IRRs. (c) Would this project be better than one with a 12 percent expected return and a standard deviation of 3 percent? (d) If a loan of $1.2 million is used to purchase the property at an 8 percent interest rate with a 15 year term, calculate the expected IRR and the standard deviation of the return on equity (ignore taxes). Contrast your findings with those obtained in (a) and (b) above.

Answer 1

(a) IRR for each scenario and the expected IRR for the project
Pessimistic(30% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	209000	198550	188622.5	179191.4
Sale value						1700000
Expected cash flows	-2200000	220000	209000	198550	188622.5	1879191
IRR	4.98%
Most likely(50% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	220000	220000	220000	220000
Sale value						2500000
Expected cash flows	-2200000	220000	220000	220000	220000	2720000
IRR	12.14%
Optimistic(20% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	226600	233398	240399.9	247611.9
Sale value						2800000
Expected cash flows	-2200000	220000	226600	233398	240399.9	3047612
IRR	14.61%
Expected IRR for the project=
Sum of (IRR for each scenario*its respective probability)
ie.(30%*4.98%)+(50%*12.14%)+(20%*14.61%)=
10.49%
(b) Variance and standard deviation of the IRRs.
Variance=Sum ((individual IRR-Project IRR)^2*Prob.)
ie. ((4.98%-10.49%)^2*30%)+((12.14%-10.49%)^2*50%)+((14.61%-10.49%)^2*20%)=
0.1386%
Std. devn= Sq.rt. Of variance
(0.1386%)^(1/2)=
3.72%
(c) Would this project be better than one with a 12 percent expected return and a standard deviation of 3 percent?
No.
As this project's expected /probable IRR 10.49% < 12% & also
the std. devn. From the most expected return/mean 3.72% > 3%

d. Loan to purchase property

Calculating the annual-annuity on the loan

Using the PV of ordinary year-end annuity formula,

& plugging in the known values,

1200000=Annuity amt.*(1-1.08^-15)/0.08

we get the annual payment as,

140195

The Loan amortisation table for the 1st 5 yrs.
Year	Annuity	Tow. Int.	Tow. Loan	Loan bal.
0				1200000
1	140195	96000	44195	1155805
2	140195	92464	47731	1108074
3	140195	88646	51549	1056525
4	140195	84522	55673	1000852
5	140195	80068	60127	940726

Now calculating IRR for diff. scenarios as above, with interest cash outflows as in the above Table
Pessimistic(30% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	209000	198550	188622.5	179191.4
Interest on loan		-96000	-92464	-88646	-84522	-80068
Sale value						1700000
Expected cash flows	-2200000	124000	116536	109904	104100.5	1799123
IRR	0.54%
Most likely(50% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	220000	220000	220000	220000
Interest on loan		-96000	-92464	-88646	-84522	-80068
Sale value						2500000
Expected cash flows	-2200000	124000	127535.6	131354	135478	2639932
IRR	8.27%
Optimistic(20% prob.)
Year	0	1	2	3	4	5
Asking price	-2200000
NOI		220000	226600	233398	240399.9	247611.9
Interest on loan		-96000	-92464	-88646	-84522	-80068
Sale value						2800000
Expected cash flows	-2200000	124000	134135.6	144752	155877.9	2967544
IRR	10.89%
Expected IRR for the project=
Sum of (IRR for each scenario*its respective probability)
ie.(30%*0.54%)+(50%*8.27%)+(20%*10.89%)=
6.48%
(b) Variance and standard deviation of the IRRs.
Variance=Sum ((individual IRR-Project IRR)^2*Prob.)
ie. ((0.54%-6.48%)^2*30%)+((8.27%-6.48%)^2*50%)+((10.89%-6.48%)^2*20%)=
0.1608%
Std. devn= Sq.rt. Of variance
(0.1608%)^(1/2)=
4.01%

Summary
Returns:	without loan	with loan
Pessimistic	4.98%	0.54%
Most likely	12.14%	8.27%
Optimistic	14.61%	10.89%
Expected	10.49%	6.48%
Variance	0.1386%	0.1608%
Std. devn	3.72%	4.01%